Cobb_et_al

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This manuscript (PNAS MS#2017-01090R) has been accepted with minor revisions by
Proceedings of the National Academy of Sciences of the United States of America.
How temporal patterns in rainfall determine the
geomorphology and carbon fluxes of tropical
peatlands
Alexander R. Cobb a,1 , Alison Hoyt b , Laure Gandois c , Jangarun Eri d , René Dommain e,f , Kamariah Abu Salim g , Fuu Ming
Kai a,2 , Nur Salihah Haji Su’ut h , and Charles F. Harvey a,b
a Center for Environmental Sensing and Modeling, Singapore-MIT Alliance for Research and Technology, 138602 Singapore; b Department of Civil and Environmental
Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139 USA; c EcoLab (Laboratoire Écologie fonctionnelle et Environnement), Université de
Toulouse, CNRS, INPT, UPS, Avenue de l’Agrobiopôle, F-31326 Castanet-Tolosan, France; d Forestry Department, Ministry of Industry and Primary Resources, Jalan Menteri
Besar, Bandar Seri Begawan BB3910, Brunei Darussalam; e Department of Anthropology, Smithsonian Institution, National Museum of Natural History, Washington, District of
Columbia, 20560 USA; f Institute of Earth and Environmental Science, University of Potsdam, 14476 Potsdam, Germany; g Biology Programme, Universiti Brunei Darussalam,
Bandar Seri Begawan BE1410, Brunei Darussalam; h Brunei Darussalam Heart of Borneo Centre, Ministry of Industry and Primary Resources, Jalan Menteri Besar, Bandar
Seri Begawan BB3910, Brunei Darussalam; 2 Present address: National Metrology Centre, Agency for Science, Technology and Research, 118221 Singapore.
This manuscript was compiled on April 13, 2017
Tropical peatlands now emit hundreds of megatons of carbon dioxide
per year because of human disruption of the feedbacks that link
peat accumulation and groundwater hydrology. However, no quantitative
theory has existed for how patterns of carbon storage and
release accompanying growth and subsidence of tropical peatlands
are affected by climate and disturbance. Using comprehensive data
from a pristine peatland in Brunei Darussalam, we show how rainfall
and groundwater flow determine a shape parameter (the Laplacian
of the peat surface elevation) that specifies, under a given rainfall
regime, the ultimate, stable morphology, and hence carbon storage,
of a tropical peatland within a network of rivers or canals. We find
that peatlands reach their ultimate shape first at the edges of peat
domes where they are bounded by rivers, so that the rate of carbon
uptake accompanying their growth is proportional to the area of the
still-growing dome interior. We use this model to study how tropical
peatland carbon storage and fluxes are controlled by changes in
climate, sea level, and drainage networks. We find that fluctuations
in net precipitation on time scales from hours to years can reduce
long-term peat accumulation. Our mathematical and numerical models
can be used to predict long-term effects of changes in temporal
rainfall patterns and drainage networks on tropical peatland geomorphology
and carbon storage.
tropical peatlands | peatland geomorphology | peatland hydrology |
peatland carbon storage
Tropical peatlands store gigatons of carbon in peat domes,
gently mounded land forms kilometers across and ten or
more meters high (1). The carbon stored as peat in these
domes has been sequestered by photosynthesis of peat swamp
trees (2) and preserved for thousands of years by waterlogging,
which suppresses decomposition. Human disturbance
of tropical peatlands by fire and drainage for agriculture is
now causing re-emission of that carbon at rates of hundreds
of megatons per year (2–5): emissions from Southeast Asian
peatlands alone are equivalent to about 2% of global fossil fuel
emissions or 20% of global land use and land cover change
emissions (6, 7). Because peat is mostly organic carbon, a
description of the growth and subsidence of tropical peatlands
also quantifies fluxes of carbon dioxide (1, 4, 8). Evidence
from a range of studies establishes that accumulation and loss
of tropical peat are controlled by water table dynamics (4, 9).
When the water table is low, aerobic decomposition occurs,
releasing carbon dioxide; when the water table is high, aerobic
decomposition is inhibited by lack of oxygen, production of
peat exceeds its decay, and peat accumulates. In this way,
the rate of peat accumulation is determined by the fraction of
time that peat is exposed by a low water table (Fig. 1).
The water table rises and falls in a peatland according to the
balance between rainfall, evapotranspiration, and groundwater
flow. Water flows downslope towards the edge of each peat
dome, where it is bounded by rivers. This flow occurs at a
rate limited by the hydraulic transmissivity of the peat—the
e ciency with which it conducts lateral flow—and follows the
gradient in the water table. The gradient in the water table
is slightly steeper near dome boundaries where the flow of
water is faster. A steeper gradient near boundaries implies
a domed shape in the water table, or groundwater mound,
corresponding to the domed shape of the peat surface. The
doming of the peat surface is very subtle: gradients are about
one meter per kilometer (1). Nonetheless, it is the dome’s
gentle curvature that accounts for the carbon storage within
the drainage boundary.
Once the peatland surface is su ciently domed, water is
DRAFT
Significance Statement
A dataset from one of the last protected tropical peat swamps
in Southeast Asia reveals how fluctuations in rainfall on yearly
and shorter timescales affect the growth and subsidence of
tropical peatlands over thousands of years. The pattern of
rainfall and the permeability of the peat together determine a
particular curvature of the peat surface that defines the amount
of naturally sequestered carbon stored in the peatland over
time. This principle can be used to calculate the long-term
carbon dioxide emissions driven by changes in climate and
tropical peatland drainage. The results suggest that greater
seasonality projected by climate models could lead to carbon
dioxide emissions, instead of sequestration, from otherwise
undisturbed peat swamps.
A.R.C. and J.E. established the site and installed the sensors; A.R.C., L.G., J.E., R.D., K.A.S.,
K.F.M., N.S.H.S., and C.F.H. collected peat cores; A.H., L.G., and J.E. completed the peat surface
elevation survey; A.R.C., A.H., and C.F.H. analyzed data; A.R.C. wrote the simulation code;
A.R.C. and C.F.H. designed the study and wrote the paper. All authors discussed the results and
commented on the manuscript.
The authors declare no conflict of interest.
1 To whom correspondence should be addressed. E-mail: alex.cobb@smart.mit.edu
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PNAS | April 13, 2017 | vol. XXX | no. XX | 1–10

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rain and
evapotranspiration
groundwater
flow
govern
and
topography and
transmissivity
control
creates
water
table
determines
peat
accumulation
Fig. 1. Ecosystem feedback leading to peat accumulation. Peat accumulation
occurs because of waterlogging of plant remains, and is therefore determined by the
proportion of time that peat is protected from aerobic decomposition by a high water
table. Over time, peat builds up into gently mounded land forms, or domes, bounded
by rivers. The slopes in a peat dome, though very small, govern groundwater flow
towards bounding rivers at rates limited by the transmissivity of the peat.
shed so rapidly that no more organic matter can be waterlogged
within the confines of the drainage network, and peat
accumulation stops (10). This maximally domed shape sets
a limit on how much carbon a peat dome can sequester and
preserve under a given rainfall regime (11). If the peat dome
is flatter than its stable shape for the current climate, it will
sequester carbon and grow; if it is more domed than its stable
shape, it will release carbon and subside as peat decomposes.
(In the tropical peat literature, “subsidence” is used for a
decline in the peat surface elevation, regardless of mechanism
(5).) The volume of this stable shape times the average carbon
density of the peat defines a capacity for storage of carbon as
peat within the drainage boundary.
If we can predict the stable shapes of peat domes and how
they evolve over time in a given climate, we can determine
how peatland carbon storage capacity and carbon fluxes are
a ected by changes in rainfall regime, drainage network, and
sea level. However, when predicting the stable shapes of peat
domes and their evolution towards these shapes, there are two
complicating factors: (1) the boundaries imposed by drainage
networks have complex shapes, and (2) rainfall is intermittent
and variable. The water table rises during rainstorms, and
falls during dry periods, even when the peat surface is stable.
These fluctuations in the water table seem to be important
because it is widely believed that seasonality of rainfall a ects
tropical peat accumulation (12, 13). But how should we take
these fluctuations into account to predict the slow development
and stable shapes of peat domes? Understanding the global
impact of changes in rainfall amount and variability, drainage
networks, and sea level on tropical peatland carbon storage and
fluxes requires a theory that can accommodate the complicated
drainage networks and intermittent rainfall of the real world.
Ingram (10) made the first prediction of the limiting shape
of a temperate peat dome imposed by the balance between
rainfall and groundwater flow. Assuming constant rainfall,
he computed the steady-state shape of a peat dome with
uniform permeability between parallel rivers. Clymo (14) later
developed a simple dynamic model for accumulation of peat at
a single point in the landscape. Clymo’s model assumed that
the thickness of peat above the water table would not change,
and focused on anaerobic decomposition in deeper waterlogged
peat. Hilbert et al. (15) later built on Clymo’s model to
allow a varying thickness of peat above the water table via a
simple water balance whereby drainage increases linearly with
peat surface elevation. Hilbert’s model inspired a series of
a
5° N
0°
5° S
500 km
4° N
100° E 110° E 120° E
1 km
Sumatra
DRAFT
c
d
Borneo
b
b
5° N
N
d
500 m
e
30 m
30 km
c
Brunei
c
114° E 115° E
e
Sarawak
Fig. 2. Site of field data collection in Brunei Darussalam. a, Distribution of
peatlands in Borneo, Sumatra and Peninsular Malaysia. b, Field site in Brunei
Darussalam, on Borneo island. c, Contour map of study area from airborne LiDAR
data, showing radiocarbon-dated peat cores (points) at primary site (Mendaram,
south) and degraded site (Damit, north), and the boundaries of the flow tube used
for hydrologic simulations (blue). d, Piezometers (triangles) at the Mendaram site. e,
Survey points in microtopography transect (see Fig. 3b).
increasingly sophisticated models for vegetation dynamics and
peat accumulation at a point. The most recent of these point
models computes water table depth from monthly rainfall
using a site-specific model (16). Meanwhile, numerical models
have been used to simulate peat accumulation under constant
rainfall (17, 18). Although these subsequent works simulate the
dynamics of peat production and decomposition in increasing
detail, a strength of Ingram’s model was that it provided
quantitative intuition for how peat dome morphology depends
on peat hydrologic properties and average rainfall. Could a
principle like Ingram’s exist that describes peatland dynamics
as well as statics, and remains applicable with realistic drainage
networks and rainfall regimes?
We established a field site in one of the last pristine peat
swamp forests in Southeast Asia, then used measurements
from this site to develop a new mathematical model for the
geomorphic evolution of tropical peatlands that is simpler, yet
more general than Ingram’s model for high-latitude peatlands.
Our model makes it possible to predict e ects of changes in
rainfall regime and drainage networks on carbon storage and
fluxes in tropical peatlands. The model predicted, perhaps
surprisingly, that surface peat would be older near dome margins.
We tested these predictions by radiocarbon-dating core
samples and comparing the age of each sample to the simulated
age at its location and depth. Finally, we explored the future
of tropical peatlands under climate projections by simulating
the geomorphic evolution of an idealized peat dome under
projected changes in rainfall patterns and drainage.
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Cobb et al.

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a
Elevation, m a.s.l.
8 decay
6
4
2
litter
peat
b 5.50
Elevation, m a.s.l.
5.25
5.00
pool
0 50 100 150
Position along transect, m
DRAFT
Height above
land surface, cm
20 c
0
-20
2012-09 2012-10
Fig. 3. Microtopography and water table dynamics in a tropical peatland. a, Cartoon of tropical peat cross-section showing variables: ˜p, the peat surface; p, the “land
surface,” a smooth surface fit through local minima in ˜p; H, water table elevation; ’, water table height relative to the land surface, ’ = H ≠ p. The peat surface ˜p is irregular
on a spatial scale of meters, with higher areas (hummocks) separating local depressions (hollows) that are not connected into channels. b, Total station survey of peat elevation
˜p (black circles) along a transect, and the land surface p (dashed black line). The minimum, median, and maximum water table elevation H from each of 12 piezometers along
the transect are also shown (dashed blue lines). The absolute elevation of the survey points comes from matching local minima among survey points within 20 m x 20 m squares
(white diamonds) with local minima in LiDAR last return data within the same squares (red diamonds). The land surface p is represented by the dashed horizontal black line. c,
Water table dynamics along survey transect (d) in late 2012, relative to the land surface p. What appears to be a single blue line is superimposed data from the 12 piezometers
shown in (d). Also shown are the average minimum, median, and maximum water table elevation above the land surface during the same time period for all 12 piezometers.
Methods Summary
Field measurements. We established a field site in pristine
peat forest in Brunei Darussalam (Borneo) to study a peat
dome where current processes a ecting peat accumulation
are essentially similar to those during its long-term development
(Fig. 2). At the site, we installed 5 piezometers along
a 2.5 km trail, 12 piezometers along a 180 m transect, and 3
throughfall gauges. We completed a total station survey of
peat surface elevation along the transect to characterize peat
surface microtopography. To characterize large-scale peatland
morphology, we also obtained LiDAR data for the entire study
area. To study peat dome development, we collected 9 peat
cores from which we obtained 37 radiocarbon dates. To test
whether our undisturbed site behaved similarly to sites studied
by other groups, we installed 4 soil respiration chambers and
a piezometer at a nearby logged but undrained site.
Morphology vs. microtopography. Superimposed on the gross
morphology of a peat dome is a fine microtopography of meterscale
depressions, or hollows, separated by higher areas, or
hummocks (19, 20). The hummocks consist of partly decomposed
logs, branches, and leaves lodged among living
buttresses, stilt roots, pneumatophores, and giant rhizomes.
Whereas the microtopography in high-latitude peat bogs may
have regular and oriented patterns (21), surveys by Lampela
et al. (20) in a tropical peat swamp in Central Kalimantan
showed no orientation or regularity. Similarly, our microtopography
survey and other observations revealed no regular
patterns or channels in peat dome microtopography.
In describing the evolution of peat dome morphology, we
would like to capture the e ects of the hummock-and-hollow
microtopography without explicitly simulating its details. Measurements
from the 12 piezometers along our microtopography
transect showed that the water table is relatively smooth, even
though the peat surface is highly irregular on a spatial scale
of centimeters to meters (Fig. 3). We therefore represent the
peat surface by a reference surface p, smooth like the water
table, that underlies the actual peat surface ˜p. We refer to
this reference surface p as the “land surface.” The peat surface
˜p is a “texture” that sits on the smooth land surface p. The
bottoms of hollows provide the most readily identifiable local
reference elevation (20), so we define the land surface p as a
smooth surface fit through the bottoms of hollows (local minima
in the peat surface ˜p). On the basis of this definition, we
determined the current land surface at our site by smoothing a
raster map obtained from local minima in LiDAR last-return
points. We also used the transect survey and piezometer data
to find the land surface p along the microtopography survey
transect (SI Text).
Groundwater flow. We model the dynamics of the water table
H subject to net precipitation q n (rainfall intensity R minus
evapotranspiration ET) using Boussinesq’s equation for
essentially horizontal groundwater flow
ˆH
S y = qn + Ò · (T ÒH) [1]
ˆt
where the specific yield S y is the amount of water required
for a di erential increment in water table elevation, and transmissivity
T is the volumetric flow per perimeter driven by
a particular head gradient ÒH. Boussinesq’s equation is a
standard groundwater modeling equation for flow domains like
peatlands that are much wider than they are thick.
At high water tables, hollows become flooded from saturation
of the peat below, forming small pools. These pools
are not connected into channels (20), and therefore do not
allow open-channel flow on a large scale in the peatland. Instead,
flow through the peatland is limited by flow through
the porous matrix of the hummocks between these isolated
pools. We apply Boussinesq’s equation at scales much larger
than hummocks and hollows (tens of meters) and refer to
the flow of water through the peatland as “groundwater flow”
even though some flow occurs above the local peat surface, in
hollows, during wet periods. Boussinesq’s equation requires
only that lateral flow is proportional to the head gradient,
which is the case if the overall flow is limited by laminar flow
through hummocks. We never observed ephemeral channels
connecting hollows within the peatland in our six years at the
site. In addition, if flow were non-laminar, we would expect
di erent local flow behavior at the same water table height in
areas with di erent water table gradients, but instead water
table behavior is uniform (Results and Discussion).
Local carbon balance. A broad range of studies demonstrates
that the thickness of peat exposed above the water table determines
the rate of peat accumulation or loss (4, 22). Like
others (4, 22), we modeled the dynamics of peat accumulation
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et al.
PNAS | April 13, 2017 | vol. XXX | no. XX | 3

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Water table height , cm
a 20
0
−20
−40
−60
−80
Land surface p
(bottoms of hollows)
−6 −4 −2 0
Peat accumulation (+)
or loss (−) , cm / y
b
0 2 4
Soil surface CO 2 flux,
µmol m -2 s -1
Fig. 4. Peat accumulation and CO 2 flux vs. water table height in tropical peatlands.
Peat accumulation (a) represents the balance between peat production and
decomposition. Aerobic decomposition is one of the two main sources of peat surface
CO 2 flux (b); the other is root respiration. a, Peat accumulation or loss vs. water
table height from model calibration (solid line) and from literature subsidence data
(circles: (4); squares: (22)). The straight line was not fit to these data, but rather, arose
naturally from calibration to match the modern surface of the Mendaram peat dome
(Figure 7). b, Soil surface CO 2 flux vs. water table height at our site in Brunei Darussalam
(white circles) was very similar to fluxes in other tropical peatlands (squares
(23), diamonds (19), triangles (24); pentagons (9); hexagons (25)).
or loss ˆp/ˆt as the di erence between the rate of peat production
f p when the water table is at the land surface, and
the rate of peat loss by decomposition (p ≠ H)–, which is the
thickness p ≠ H of peat exposed above the water table times
a decomposition rate constant –
ˆp
= fp ≠ (p ≠ H)– [2]
ˆt
(Fig. 4). The peat surface is stable, neither growing nor
subsiding Ȉp/ˆtÍ =0wherever the water table fluctuates in
such a way that peat production is balanced by decomposition
over time
f p = Èp ≠ HÍ –, [3]
where angle brackets È·Í indicate a time average.
Several other studies have shown a leveling-o of soil CO 2
e ux at very low water tables (25, 26), and it is also likely
that very high water tables ultimately limit net carbon uptake
by trees (primary production) (16). However, including these
e ects did not a ect simulations because these extreme water
table heights and depths were neither observed at our site
nor predicted by simulations of our site. We also did not
include anaerobic decomposition below the water table because
analysis of peat cores from tropical sites in Asia (2), including
our site (27), do not show detectable loss of waterlogged peat
from anaerobic decomposition.
Numerical simulations. We built a numerical model of waterlogging
and peat accumulation based on Eqn. 1 and Eqn. 2
to simulate peat dome geomorphogenesis and carbon fluxes.
These two equations are coupled by the water table elevation
H and the peat surface elevation p, both of which vary in
time and space. The equations require four parameters: (1)
a specific yield function S y, (2) a transmissivity function T ,
(3) a rate of peat production f p, and (4) a decomposition rate
constant –. The model employs a finite volume scheme with
special features designed to handle the severe nonlinearity of
the transmissivity function T (SI Text).
We determined the specific yield and transmissivity functions
S y,T from the response of the water table to heavy
rain and dry spells (Results and Discussion). We then fit
the parameters for peat accumulation f p, – by simulating the
2700-year evolution of a peat dome at our field site in Brunei,
and matching the simulated modern peat surface to the peat
surface measured by LiDAR. We tested our model against radiocarbon
dates from peat cores extracted from the peatland,
then used the model to answer general questions about carbon
fluxes from tropical peatlands after perturbation by climate
change and drainage.
Limitations of modeling approach. Our goal was to build the
simplest model that can make reasonable quantitative predictions
of tropical peat dome dynamics. In most Southeast Asian
peatland complexes, every area between rivers is occupied by a
peat dome, so it is not apparent how any peat dome could now
expand to fill a larger area. However, domes tend to be larger
in older peatlands, suggesting a long-term process of dome
coalescence. We did not attempt to model these long-term
changes in river networks. We also did not consider changes in
hydraulic conductivity near the surface caused by compaction
or changes in microtopography under agriculture.
Results and Discussion
Carbon storage capacity of tropical peatlands.
Local water balance is dominated by flows near the surface. Eighteen
months of data on water table height in five piezometers
along a 2.5 km transect (Fig. 5) show two distinctive features
of water table behavior in tropical peatlands. First, when the
water table is high, it falls very rapidly; and second, the water
table height relative to the land surface remains approximately
uniform in all piezometers as the water table rises and falls,
as observed elsewhere by Hooijer (28). In what follows, we
use “water table height” ’ = H ≠ p to refer to the water
table height relative to the land surface, as distinct from the
water table elevation H above mean sea level. Because the
water table height ’ is approximately uniform, the water table
behavior can be summarized by a pair of curves describing
the uniform rise of the water table during heavy rain, and
the uniform decline of the water table during dry intervals
between rains (Fig. 5e,f). During heavy rain, the e ects of
evapotranspiration and outward flow are negligible, and the
rainfall intensity vs. rate of increase in water table height
gives the specific yield. Between rain events, the water table
DRAFT
declines because of evapotranspiration and the divergence of
groundwater flow.
Transmissivity T is a function of water table height ’ and
controls the divergence of groundwater flow Ò · (T ÒH). We
determined the e ect of water table height on transmissivity
using our water table data. The water table declines during
dry intervals because of a combination of evapotranspiration
and the divergence of groundwater flow; however, the two are
easily distinguished at low water tables because evapotranspiration
ceases at night (Fig. 5d). Therefore, we can obtain
the divergence of groundwater flow from the declining water
table during dry intervals after accounting for evapotranspiration
((28, 29); further details in SI Text). We find that
transmissivity increases exponentially at high water tables,
when water rises into hollows and flows through hummocks,
but decreases dramatically at low water tables when water
flows through fine pores in the peat matrix (Fig. 5c). Very
high permeability near the peat surface is consistent with our
observations of more void space higher in the peat profile, and
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Water table height , cm
20
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−20
20
0
−20
a
(Maximum local peat surface height in transect)
2012-02 2012-04 2012-06 2012-08 2012-10 2012-12
d
Heavy rain
Time
No rain
e
Heavy rain
0 10
Rainfall depth P, cm
f Bog plain
piezometer
Rainfall intensity R, cm / h
DRAFT
10
5
0
Water table height , cm
20
−20
b
0 0.25 0.5 10 2 10 4
Specific yield S y ,
cm / cm
Transmissivity T,
m 2 d -1
No rain
0 10 20 30
Time since rain stopped, d
Fig. 5. Site hydrology and calibration. a, Superimposed water table height (jagged blue lines) from five piezometers spanning 2.5 km and rainfall intensity (vertical lines)
from three automated rain gauges over a 10-month interval. The piezometer farthest from the river (red) lies in a region with a different surface Laplacian (the “bog plain”),
corresponding to an area of current peat accumulation (Fig. 6). Also shown are the minimum, median, and maximum local peat surface elevation (dashed horizontal lines) from
a 180 m microtopography survey transect (Fig. 3). b, c, Hillslope-scale specific yield and transmissivity curves for field site, determined from recharge and recession curves (e,
f). d, Short interval of water table data from a single piezometer selected from (a). Inset shows declining water tables during day (unshaded) and steady water tables at night
(shaded) driven by diurnal cycles of evapotranspiration. e, f, Master recharge curve (e) and recession curve (f) assembled from intervals of heavy rain and no rain, respectively,
by alignment of sequences with overlapping water table depth. During heavy rain, net precipitation intensity q n = R ≠ ET is dominated by rainfall intensity R (e); with no rain,
net precipitation consists only of evapotranspiration ET (f). Dashed black lines in (e) and (f) show water table response computed from specific yield and transmissivity (b,c),
blue translucent lines are assembled from field data in (a). As in (a), the red curve is from the piezometer in the flatter bog plain region (Fig. 6).
also with recent data from other tropical peatlands (30). The
water table curves (Fig. 5e,f) indicate that the near-surface
permeability is so great that the total thickness of deeper peat
is unimportant for groundwater flow. Therefore, transmissivity
is approximately independent of peat depth, and depends only
on the water table height ’, which is uniform in space (though
highly variable in time).
Morphology of peat surface explains uniform water table behavior.
According to Boussinesq’s equation, uniform transmissivity
is not, by itself, enough to explain the uniform fluctuation of
the water table. Even in hydrologic systems where hydraulic
properties are uniform, the water table can behave di erently
at di erent locations because of topography. For example, in
most hydrologic systems a rainstorm drives a di erent water
table response at a topographic divide than it does near where
groundwater discharges to a river.
To understand the uniform water table behavior in peatlands,
we refer back to Boussinesq’s equation (Eqn. 1). If both
the specific yield S y and the transmissivity T depend only on
the local water table height relative to the surface and not on
position within the peatland, uniform water table movement
occurs if the divergence of the peat surface gradient, or the
peat surface Laplacian Ò 2 p, is uniform (Fig. S4c–e). (The
“Laplacian of the peat surface” Ò 2 p, or just “Laplacian,” is
the scalar result of applying the Laplacian operator Ò 2 to the
land surface elevation p.) To see why a uniform land surface
Laplacian explains uniform water table behavior, we re-write
Boussinesq’s equation (Eqn. 1) in terms of the water table
height relative to the land surface (’ = H ≠ p), instead of the
water table elevation H:
S y
ˆ(p + ’)
ˆt
= q n + Ò · [T Ò(p + ’)] . [4]
0
We observe that water table height is uniform (Ò’ = 0). If
transmissivity T is also spatially uniform, the groundwater
divergence term simplifies to the transmissivity times the
peat surface Laplacian (Ò · [T Ò(p + ’)] = T Ò 2 p). The time
derivative ˆp/ˆt of the land surface elevation is negligible
because peat accumulation or loss is much slower than rise
or fall of the water table, so the term p can be dropped from
the time derivative. We observe that the fluctuations in water
table height ˆ’/ˆt are uniform, as is net precipitation q n,so
the groundwater divergence term T Ò 2 p must also be spatially
uniform. Thus, Boussinesq’s equation simplifies to an ordinary
di erential equation (ODE) describing the uniform fluctuation
of the water table relative to the peat surface
S y
d’
dt = qn + T Ò2 p [5]
where the peat surface Laplacian Ò 2 p is uniform.
The peat surface Laplacian describes the curvature of the
peat surface: it is equal to the sum of the second derivatives of
the surface elevation in two perpendicular horizontal directions
(Ò 2 p = ˆ2p/ˆx 2 + ˆ2p/ˆy 2 ). Thus, analysis of water table
dynamics predicts uniform curvature of the peat surface where
water table fluctuations are uniform. This uniformity of surface
elevation curvature can be tested against elevation maps.
Maps of the peat surface Laplacian are highly sensitive to
microtopographic noise in the surface elevation map because
the Laplacian uses the second derivative of the surface elevation.
However, by the Divergence Theorem, the average
Laplacian within any closed contour is equal to the integral of
the normal gradient along the contour divided by the enclosed
area. Therefore, we can examine the uniformity of the surface
Laplacian by studying the slope of a regression between the
c
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PNAS | April 13, 2017 | vol. XXX | no. XX | 5

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b
Land surface
elevation p, m
c
Integrated
normal gradient, m
a
6
4
2
4
Bog plain
piezometer
Bog plain
Bog plain
Bog plain piezometer
Bog plain
2 0
Distance from river, km
Enclosed area, km 2
Stable
Stable
(uniform surface
Laplacian)
Stable
River flooding
influence
River
flooding
influence
River flooding
influence
Fig. 6. Estimation of peat surface Laplacian. a, Regions of different morphology
and water table behavior within the flow tube used for field site simulations, and
locations of piezometers (triangles). Furthest from the river, the land surface is
relatively flat (“bog plain”), next there is a region in which the Laplacian of the land
surface elevation is uniform (“stable”) and finally a narrow region near the river where
hydrologic processes and peat accumulation are affected by the rise and fall of the
bounding river (“river flooding influence”). b, Profile of LiDAR land surface elevation
from (a), showing piezometer locations (vertical dashed lines). c, Normal gradient
driving efflux, integrated along contours, vs. enclosed area. The slope in the “stable”
region gives the average land surface Laplacian of the land surface there and was
used for calibration of hydrologic parameters.
integrated normal gradient and the enclosed area (Fig. 6).
Indeed, we find a linear relationship between the integrated
normal gradient along each contour and the area enclosed by
the contour in our LiDAR-derived peat surface elevation map,
indicating a uniform surface Laplacian in the region of uniform
water table behavior (Fig. 6). In contrast, outside the region
of uniform Laplacian, the water table behaves di erently (‘bog
plain piezometer’ in Figs. 5,6).
Uniform surface Laplacian determines stable tropical peatland morphology.
The uniform peat surface Laplacian provides a remarkably
simple way to compute a stable morphology for a tropical
peat dome. By “stable morphology,” we mean a morphology
in which the peat surface and water table continue to fluctuate
with the vagaries of climate, but there is no long-term average
change in the peat surface or water table elevation (they are
stationary; Ȉp/ˆtÍ =0, ȈH/ˆtÍ =0). Uniform water table
height is the simplest behavior that could make an entire
peatland stable, because if the water table height is spatially
uniform, the local rate of peat accumulation is also uniform.
In a stable peatland, there is no long-term change in the water
table height, so any water added by net precipitation must
eventually be removed by groundwater flow
e f
d’
0= S y = Èq nÍ + ÈT ÍÒ 2 p Œ. [6]
dt
Thus the Laplacian Ò 2 p Œ of the stable peatland surface p Œ
is minus the average net precipitation divided by the average
transmissivity
Ò 2 p Œ = ≠ ÈqnÍ
ÈT Í . [7]
We can compute the stable topography of any tropical
peatland by solving Poisson’s equation (Eqn. 7) for the stable
peat surface morphology p Πusing the appropriate Laplacian
value for that climate. The average transmissivity ÈT Í is a
complicated function of the temporal pattern of rainfall and
the hydrologic-biological system. However, for any rainfall
regime, one can find the stable surface Laplacian Ò 2 p Œ by
repeatedly simulating water table fluctuations (Eqn. 5) with a
trial Laplacian Ò 2 p, and adjusting the Laplacian value until
peat production balances decomposition (Eqn. 3) everywhere
in the peatland (SI Text). In this way, one finds a shape
parameter (Ò 2 p Œ) that describes stable peatland morphology
under a given rainfall regime in any drainage network.
Climate and drainage network determine tropical peatland carbon
storage capacity. By specifying the stable peatland topography,
the uniform Laplacian principle gives the peat carbon storage
capacity inside any drainage boundary and in any given
climate. The volume under the surface satisfying Poisson’s
equation times the mean carbon density of the peat gives the
carbon storage capacity of the peatland. For example, the
peat dome at our primary site currently has a mean peat
depth of 3.88 m (max 4.92), and stores about 1535 t C ha ≠1 ;
however, if the climate remains similar to the climate during
its 2300-year development, we predict that in about 2500 y it
will reach a stable shape with a mean peat depth of 4.54 m
(max 7.10 m) and store 1800 t C ha ≠1 (Fig. S3; simulations of
dynamics are described in the next section).
The uniformity of the stable peat surface Laplacian is an
approximation that requires that (1) peat accumulation rate
ˆp/ˆt is a non-decreasing function of water table height; (2)
flow of water is proportional to water table gradient (Boussinesq’s
equation); and (3) transmissivity is independent of location
because flow through deep peat is negligible compared to
DRAFT
near-surface flow. In reality, groundwater flow through deeper
peat will result in a deviation of the stable peat dome surface
from the uniform Laplacian shape in very large peat domes.
Specifically, groundwater flow through deep, low-permeability
peat will tend to flatten the dome center, because of slow
infiltration of water into the deep peat, and steepen the dome
margin, because of exfiltration of water back into the high permeability
near-surface peat near the boundary. Deep groundwater
flow should be manifested as a downward (dome center)
or upward (dome margin) trend in the water table during
nights without rain when the water table is low; no such trend
is apparent in our piezometer data (Fig. 5d), suggesting that
deep groundwater flow is small. A small deep groundwater flow
term is further supported by radiocarbon dating of porewater
DOC at our site (31), which suggests a maximum downward
velocity of water of about one meter per year, or at most a
1.4 mm water table decline during a single twelve-hour night,
one-sixteenth of the 22 mm water table decline from evapotranspiration
during the day (Fig. 5). (Evapotranspirative
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Cobb et al.

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Land surface elevation p, m
a
b
Simulated peat age,
cal y BP × 1000
6
3
0 Clay
2
1
4
0
0
1 2
Peat sample age, cal y BP × 1000
Modern peat surface (LiDAR)
2 0
Distance from river, km
Age at 25–67 cm depth,
cal y BP × 1000
c
1.0
0.5
0
Simulated peat surface vs time
Primary site
Deforested site
Radiocarbon-dated
core sample
River
3 2 1 0
Distance from river, km
Fig. 7. Morphogenesis of Mendaram peat dome. a, Shape of peat dome over time,
including modeled peat surface (contours), modern peat surface from LiDAR (dashed
black line), and calibrated radiocarbon dates from peat core samples (colored points).
The deepest peat layers prior to 2250 cal y BP represent a uniformly deposited
mangrove peat on a gently sloping clay plain (27). b, Simulated age of peat vs.
calibrated radiocarbon ages from samples in the Mendaram peat dome. c, Age of
shallow peat samples (25 cm–65 cm depth) vs. distance from river at primary site
(solid markers) and a nearby deforested site (open markers). Note the old peat near
the surface close to the river as predicted by the model.
flux is about one-tenth of the rate of decline of the water table
from evapotranspiration because about one tenth of the deep
peat cross-section is available for water flow; see specific yield
curve, Fig. 5b.)
A shape parameter related to our stable peatland Laplacian
(Eqn. 7) appeared in Ingram’s model for temperate peatland
morphology (10) assuming constant precipitation, uniform
hydraulic conductivity, and simple river geometry (Ingram’s
parameter is net precipitation divided by hydraulic conductivity,
instead of average transmissivity). Our result is more
general, because it handles varying rainfall and arbitrary landscapes,
but is also mathematically simpler, because of our
finding that transmissivity in tropical peatlands is approximately
independent of peat depth.
Dynamics of tropical peatland topography and carbon fluxes.
Peat accumulation parameters regulate dome dynamics. Our analysis
shows how the rate of peat production f p and decomposition
rate constant – a ect both the stable morphology and the
dynamics of tropical peat domes. These parameters of the
peat accumulation function (Eqn. 2) have an indirect but
strong e ect on the stable peat surface Laplacian and hence
peatland carbon storage capacity via the mean transmissivity
ÈT Í (Eqn. 7) because the mean water table depth must be
equal to the ratio of the peat production rate to the decomposition
rate constant (f p/–; Eqn. 3). A higher decomposition
rate constant implies a higher mean water table in stable
peat domes, meaning a higher transmissivity, a smaller stable
surface Laplacian, and less carbon storage. If both peat
production f p and the decomposition rate constant – increase
together, carbon storage capacity does not change, but peat
dome dynamics are faster.
Fit parameters match literature data. Peat accumulation parameters
fit to the topography of a peat dome at our Brunei field
site agree with published data from other sites, and also with
our other field data (see next section). We obtained peat
accumulation parameters f p, – by simulating the evolution of
the dome (Fig. 7) and minimizing the least-squared di erence
between the simulated peat surface and the modern peat surface
measured by LiDAR. We then compared our calibrated
peat accumulation function to literature data on subsidence
in drained, vegetated peat swamps (4, 22). Our linear peat
accumulation function was not calibrated to these subsidence
data from the literature—only to the modern peat surface—
but nonetheless matched the subsidence data almost exactly
(Fig. 4a; f p =1.46 mm y ≠1 , – =1.80 d ≠1 ). Our soil CO 2
chamber measurements were also very similar to those from
other sites, suggesting that the e ect of water table on fluxes
is similar at our site and in other tropical peatlands (Fig. 4b).
The uniform Laplacian principle predicts a central bog plain, and old
peat near the surface at bog margins. We find that a tropical peat
dome reaches its stable shape first at its boundaries, because
the stable dome surface is lowest there (Fig. 7,8,S4). Meanwhile,
the interior of the peat dome continues growing at an
approximately uniform rate, forming a relatively flat (smaller
magnitude Laplacian) central “bog plain.” The vegetation of
tropical bog plains may not be distinct (1), unlike the unforested
bog plains of high-latitude peatlands (21); instead,
we define the bog plain of a tropical peat dome as the central
region that has not yet reached its stable Laplacian. While
the dome center continues to accumulate peat and sequester
carbon, the margin has reached its stable shape and stopped
growing, so peat near the surface is older there.
Older peat near dome margins had not been predicted
before, so we collected 21 additional radiocarbon dates from
basal and near-surface peat samples to test this prediction.
These radiocarbon dates confirmed that near-surface peat was
older near dome margins than at the same depths towards
the interior of the same domes (Fig. 7c). We also compared
radiocarbon dates in deeper peat to simulated ages at the
same locations and depths, excluding basal samples from the
mangrove peat prior to the establishment of the peat swamp
forest (Fig. 7, SI Text; (1, 27)). Radiocarbon dates and
simulated ages at the same locations and depths matched
well (Fig. 7b). We did not expect radiocarbon dates from
cores to match simulated peat ages exactly because (1) the
drainage network may have shifted during the 2300 years of
dome growth; (2) tree root growth may inject young carbon
into peat below the surface; and (3) tree falls in peat swamp
forests remove older peat to form tip-up pools which then fill
with younger peat. In an earlier study, we estimated that
replacement of older peat by younger peat in tip-up pools
would bias radiocarbon dates of deep peat to about 500 years
later than when material was first deposited in that stratum
(Fig. 11 in (27)), consistent with the o set between measured
radiocarbon dates and ages simulated by our model (Fig. 7b).
DRAFT
Carbon sequestration rate is proportional to bog plain area. The centripetal
pattern of dome development makes the rate of carbon
sequestration roughly proportional to the area of the central
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et al.
PNAS | April 13, 2017 | vol. XXX | no. XX | 7

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approx.
5 m
Effect of peat depth
on transmissivity
is negligible
Prediction of ultimate
CO 2 sequestration
from boundary and climate
growing
rain - ET
stable
approx. 10 km
Centripetal
convergence
to stable shape
CO 2 sequestration rate
proportional to area
of central bog plain
Fig. 8. Model of tropical peat dome development. The surface p of a tropical peat
dome evolves towards a shape completely described by a uniform surface Laplacian
Ò 2 p Œ given by the ratio of average net precipitation Èq nÍ to average hydraulic
transmissivity ÈT Í. The surface Laplacian Ò 2 p Œ defines the stable shape and
carbon storage capacity of a peat dome inside any drainage boundary. When the
dome surface has a uniform Laplacian, the water table height fluctuates uniformly,
and peat production is balanced by decomposition everywhere in the dome. When
a peat dome is growing, it sequesters carbon at a rate proportional to the area of a
flatter (smaller-magnitude surface Laplacian) area in the middle, the central bog plain.
Gray boxes, established results; black boxes, findings presented here.
bog plain (Fig. 8). Under a given climate, the rate of sequestration
decreases as the dome approaches its stable shape and the
central region of peat accumulation—the bog plain—shrinks in
area. For example, our simulations imply that the current rate
of CO 2 sequestration at our site (0.80 t ha ≠1 y ≠1 ,100-year
average) is less than a quarter of its initial rate about 2300
years ago (3.81 t ha ≠1 y ≠1 ), and CO 2 sequestration is more
than five times faster at the dome interior (1.89 t ha ≠1 y ≠1 ,
6.37 km from river) than at its edge (0.36 t ha ≠1 y ≠1 ,1km
from river; Fig. S3). The mechanism of tropical peat dome
development that we describe therefore creates landscape-scale
patterns in local carbon fluxes and radiocarbon date profiles.
Local measurements of carbon fluxes or radiocarbon dates cannot
be upscaled to regional fluxes without considering dome
morphology because the flatter interior of each peat dome
sequesters carbon while the margins do not (Fig. 8). Old peat
near the peatland surface (2), although in some cases caused
by local climate change or disturbance, also can be expected
at the margin of any peat dome.
Future effects of changes in drainage networks and climate.
Our analysis provides a simple way of predicting long-term
change in peat dome morphology and carbon storage in response
to changes in drainage network, climate or sea level
because the stable peat surface Laplacian completely specifies
the stable peat topography with given drainage boundary
conditions. If the drainage network changes, we can solve
Poisson’s equation in the new drainage boundary to compute
the gain or loss of peat, and the net carbon emissions, as the
peat surface approaches its new stable topography. If the
climate changes, we can compute a new stable Laplacian value
for the new climatic conditions, and determine how much a
currently stable peatland will grow or subside.
Subdivision of a peatland by drainage canals reduces carbon storage.
The average surface elevation of a stable peat dome is
proportional to the area of the dome because of the uniform
Laplacian principle. If we scale the area of a peat dome by
some factor k by multiplying both x and y coordinates by Ô k,
the surface elevation p must increase by the same factor k
to keep the same Laplacian. Therefore, the carbon storage
capacity of a peat dome scales with its area. For example, a
peat dome that is cut into halves of approximately the same
shape as the original dome will have one-half the carbon storage
capacity (half the mean stable peat depth) of the original
dome. This provides a straightforward way to estimate the
long-term impacts of artificial drainage networks that are now
a ecting over 50% of the peatlands of Southeast Asia (32)
and from which a robust quantification of carbon emissions is
urgently needed (6).
The dynamic response of a peat dome to changes in rainfall
and sea level also depends on its area because of the centripetal
pattern of dome development (Fig. 8). Because of their higher
stable mean depth, larger-area domes reach their stable shape
more slowly than smaller-area domes.
Relative effects of climate change on carbon storage capacity are independent
of drainage network. Although peatland drainage networks
play a central role in determining absolute carbon storage
and dynamics, we can calculate the proportional e ect of
climate change on long-term carbon storage of a tropical peatland
independent of the drainage network. Poisson’s equation
(Eqn. 7) must be solved in each drainage boundary to obtain
the topography of the stable peat surface. However, we can
then predict the e ects of changes in climate independent of
the drainage network because of the linearity of the Laplacian
operator. By the definition of linearity for a mathematical
operator, a peat surface Laplacian that is larger by some factor
k corresponds to a peat surface that is vertically stretched
by the same factor (kÒ 2 p = Ò 2 kp), and which therefore has
a mean peat depth that is larger by the same factor. Thus,
carbon storage capacity per area ¯p Œ is proportional to the
stable peat surface Laplacian ¯p Œ ÃÒ 2 p Œ.
DRAFT
Dynamic simulations converge to new stable morphologies after
changes in conditions. Our simulations of peat dome dynamics
demonstrate the convergence of initially stable domes to
new, stable, uniform-Laplacian morphologies after perturbations
(Fig. 9). The simulations show the e ect of increased
total rainfall (Fig. 9a,e), which is a recognized climate feedback
for tropical peatlands (12), and also show that artificial
drainage for agriculture (Fig. 9d) can dominate all natural
feedbacks if not curtailed (4, 16). In addition, our simulations
demonstrate a third feedback: the increase in rainfall variability
from warming climates (33) can cause peat loss if not
compensated by an increase in total rainfall (Fig. 9c,f). For
these simulations, we generated new rainfall time series as similar
to current rainfall as possible but with larger annual and
El Niño–Southern Oscillation (ENSO) fluctuations (Fig S1a,b;
SI Text). Either greater seasonality or a stronger ENSO decreased
peatland carbon storage capacity, but an increase in
seasonality had a larger maximum e ect, partly because the
magnitude of the ENSO fluctuation is smaller. In contrast, sea
level rise could drive peat accumulation in the long term by
elevating the tidal rivers draining most peat domes (Fig. 9b,e).
In general, losses can be much more rapid than accumulation
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a
c
e
Average peat
depth , m
f
0
CO 2 flux,
t ha -1 y -1
1.0
0.5
0.0
2
0
-2
Response of peat topography to perturbations
sea level rise
more rain
b
initial stable surface
increased seasonality
drainage
300
sea level rise
drainage
increased seasonality
sea level rise
Time, y
more rain
increased seasonality
0 200 400
Time, y
d
drainage
600 900
Fig. 9. Dynamic effects of climate change on carbon storage in tropical peatlands.
a–d, Simulated peat surface elevation vs. time of an initially stable peat dome
after different perturbations. The dashed line indicates the stable morphology for the
peat dome between two parallel rivers, colored lines give the peat dome morphology
at subsequent time steps. a, Annual rainfall increase from 2237 to 2430 mm / y
causes peat accumulation until the peat dome reaches a new stable morphology.
b, Sea level rise of 0.5 m leads to an upward shift in peat surface elevation as tidal
rivers bounding the peat dome rise. c, Increase in seasonal fluctuation in rainfall
from 902 to 1095 mm / y causes loss of peat. d, Sustained drainage to a depth
of 50 cm drives rapid peat loss from aerobic decomposition. e, Spatially-averaged
peat depth vs. time for simulations with more rain (a, long dashes), sea level rise (b,
dot-dash), increased seasonality (c, dot-dot-dash) drainage (d, short dashes), or with
no change in conditions (solid) or increased ENSO signal (long dot-dash). f, Average
CO 2 emission (negative) or sequestration (positive) vs. time for simulations as in (e).
Because peat is mostly organic carbon, peat accumulation or loss causes uptake or
release of carbon, respectively. The initial CO 2 emission for the drainage scenario is
off the chart at ≠24 t ha ≠1 y ≠1 .
(Fig. 9e), because subsidence of drained peatlands can be far
faster than typical accumulation rates (4). For example, the
estimated area-averaged current CO 2 sequestration rate at our
site is 0.80 t ha ≠1 y ≠1 , whereas Hooijer et al. estimated CO 2
emissions of at least 73 t ha ≠1 y ≠1 from tropical peatlands
under plantation agriculture (5).
Intermittency of rainfall reduces tropical peatland carbon storage.
We find that fluctuations in net precipitation on time scales
from hours to years can reduce long-term peat accumulation.
We further explored the e ects of variability in rainfall seen
in our dynamic simulations (Fig. 9) by computing the e ect
of interstorm arrival time, annual and ENSO fluctuations on
peatland carbon storage capacity (Fig. 10). The simulations
demonstrate that long-term peat accumulation is controlled
by variation in rainfall, not only by mean rainfall, because
fluctuations in the water table cause exponential changes in
groundwater flow. The high outward flow during peak water
tables is not compensated by low flow rates after the water
table declines. For example, a steady drizzle at the same
average intensity as the intermittent rainfall actually observed
at our site would sustain more than 10 times more long-term
carbon storage (19.5 kt ha ≠1 vs. 1.80 kt ha ≠1 ; Fig. S1d,e).
The intermittency of tropical convective storms significantly
a ects long-term carbon storage: carbon storage capacity can
a
320
Mean annual rainfall, cm
240
160
Normalized carbon
storage capacity
b 4
Annual amplitude, mm d -1
0
0.4 0.8 1.2 0.0 0.5 1.0
Interstorm arrival time, d ENSO amplitude, mm d -1
DRAFT
2
Normalized carbon
storage capacity
increased
seasonality
Fig. 10. Effects of climate change on carbon storage capacity of tropical peatlands.
a, Simulated carbon storage capacity (contours) versus time-averaged rainfall
and interval between storms in a simple rainfall model (Poisson process for storm
incidents, exponentially distributed rain depth per storm). The balance between
rainfall and groundwater flow sets a limit on the curvature of the peat surface, and
therefore limits the amount of carbon that can be stored as peat in a peatland. This
carbon storage capacity is proportional to the Laplacian of the stable peat surface
elevation (Results and Discussion), so the relative effect of changes in rainfall patterns
on carbon storage capacity can be calculated independent of the drainage network.
Higher rainfall increases carbon storage capacity, whereas increased time between
storms reduces it. b, Carbon storage capacity (contours) as in (a), but driven by
rainfall at our site (diamond), or with a weakened or strengthened annual or El Niño-
Southern Oscillation fluctuation in rainfall. The vertical shift to lower carbon storage
with increased annual variation in rainfall (up arrow) corresponds to the simulated
effect of increased seasonality in Fig. 9.
decrease by a third depending on whether convective storms
arrive every fourteen hours on average, as at our site, or every
twenty four hours, with the same mean rainfall (Fig. 10a).
Our simulations with smoothed rainfall intensity and evapotranspiration
show that models must consider the e ects
of subdiurnal fluctuations in rainfall to correctly predict the
long-term evolution and carbon storage of tropical peatlands.
The exact details of the fluctuations in rainfall are not important,
in the sense that many distinct rainfall time series
can give the same stable surface Laplacian, and the same
carbon storage capacity. However, carbon storage capacity
can be severely overestimated by simulations that entirely
ignore the e ects of fluctuations in rainfall. We explored the
e ects of neglecting fluctuations in rainfall by computing the
stable surface Laplacian after averaging net precipitation on
hourly and longer intervals. Treating rainfall intensity and
evapotranspiration as constant each hour, instead of every 20
minutes, increased the simulated stable surface Laplacian by
a few percent, but averaging over a day led to an overestimate
by 20%, a week 100%, a month 400%, and a year more than
1000% (Fig. S1d,e).
Conclusions
The mathematical and numerical models presented here predict
the long-term e ects of changes in rainfall regimes and
drainage networks on the morphology of tropical peat domes.
Because tropical peat domes are mostly organic carbon, these
predictions of peat dome morphogenesis also quantify peat
dome carbon storage capacity and carbon fluxes. Our approach
shows that tropical peatlands approach a limiting shape in
which the Laplacian of the land surface is uniform. This stable
peatland surface Laplacian can be computed from any rainfall
time series, and completely summarizes the e ects of the rain-
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fall pattern on the stable morphology and storage capacity of
carbon within the peatland drainage boundary.
The uniform Laplacian principle is supported by a range of
observations: (1) the peat surface Laplacian is approximately
uniform in a region near the dome edge (Fig. 6c); (2) water
table behavior is uniform where the surface Laplacian is uniform,
and is di erent in the dome interior (Fig. 5a); (3) water
table behavior is the same in areas with di ering gradients
within the uniform-Laplacian region (Fig. 5a); (4) transmissivity
increases exponentially at high water tables, so that local
water balance is dominated by flow near the surface (Fig. 5c);
and (5) peat accumulation parameters match literature data,
even though those data were not used for calibration (Fig. 4a).
Our analysis underscores the importance of considering
geomorphology when measuring and modeling carbon fluxes
in tropical peatlands. On a growing peat dome, the perimeter
of the dome reaches a steady elevation first while central
areas continue to accumulate carbon (Fig. 8). This pattern
of dome morphogenesis implies that the locations of groundtruth
carbon flux measurements within tropical peat domes
are important considerations for earth system models (34).
For example, measurements of carbon flux in the center of a
growing dome overestimate the average flux for the whole dome,
because peat accumulation is fastest in the center (Figs. 8,S3).
The distribution of peat dome areas within a peatland complex
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is also important, because smaller domes reach their stable
shapes faster after a change in conditions. Improved earth
system models could use the uniform Laplacian principle to
e ciently account for the e ects of changing rainfall, sea level,
and drainage on tropical peat carbon storage given a realistic
distribution of peat dome sizes. The approach outlined here
also provides a framework for including the e ects of other
long-term processes that remain understudied, such as shifts
in river networks, changes in tree community composition and
salt water intrusion from rising sea levels.
ACKNOWLEDGMENTS. We thank Mahmud Yussof of Brunei
Darussalam Heart of Borneo Centre and the Brunei Darussalam
Ministry of Industry and Primary Resources for their support of
this project; Hajah Jamilah Jalil and Jo re Ali Ahmad of the
Brunei Darussalam Forestry Department for facilitation of field
work and release of sta ;AmyChuaforlogisticalsupport;and
Bernard Jun Long Ng, Rahayu Sukmaria binti Haji Sukri, Watu bin
Awok, Azlan Pandai, Rosaidi Mureh, Muhammad Wafiuddin Zainal
Ari nandSylvainFerrantforfieldassistance.WealsothankPaul
Glaser and two anonymous reviewers for their detailed comments
on the manuscript. This research was supported by the National
Research Foundation Singapore through the Singapore-MIT Alliance
for Research and Technology’s Center for Environmental Sensing
and Modeling interdisciplinary research program and by the USA
National Science Foundation under Grant Nos. 1114155 and 1114161
to C.F.H.
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of Sarawak, Malaysia. Tellus 57B:1–11.
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on peatland in Sumatra, Indonesia. Biogeosciences 9:617–630.
26. Hirano T, et al. (2012) Effects of disturbances on the carbon balance of tropical peat swamp
forests. Global Change Biology 18:3410–3422.
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(Cambridge University Press, Cambridge, UK), pp. 447–461.
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in tropical peatlands. Geochimica et Cosmochimica Acta 137:35–47.
32. Miettinen J, Shi C, Liew SC (2016) Land cover distribution in the peatlands of Peninsular
Malaysia, Sumatra and Borneo in 2015 with changes since 1990. Global Ecology and Conservation
6:67–78.
33. Huang P, Xie SP, Hu K, Huang G, Huang R (2013) Patterns of the seasonal response of
tropical rainfall to global warming. Nature Geoscience 6:357–361.
34. Quéré CL, et al. (2009) Trends in the sources and sinks of carbon dioxide. Nature Geoscience
2:831–836.
35. Moore ID, Grayson RB (1991) Terrain-based catchment partitioning and runoff prediction
using vector elevation data. Water Resources Research 27(6):1177–1191.
36. Stuiver M, Reimer PJ (1993) Extended 14 C data base and revised CALIB 3.0 14 C age
calibration program. Radiocarbon 35(1):215–230.
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years cal BP. Radiocarbon 55(4):1869–1887.
38. Ferziger JH, Perić M (1999) Computational Methods for Fluid Dynamics. (Springer-Verlag,
Berlin, Germany), 2nd edition.
39. Patankar SV (1980) Numerical Heat Transfer and Fluid Flow. (Taylor & Francis, Boca Raton,
FL, USA).
40. Moerman JW, et al. (2013) Diurnal to interannual rainfall ” 18 O variations in northern Borneo
driven by regional hydrology. Earth and Planetary Science Letters 369–370:108–119.
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Pacific warm pool hydrology since the Last Glacial Maximum. Nature 449:25–30.
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SI Text
S1. Extended Materials and Methods
Field data collection. Field data were collected in the Ulu
Mendaram Conservation Area of the Belait District in Brunei
Darussalam (4.359863 N, 114.352252 E; Fig. 2). The site
is described in more detail in Dommain et al (2015) (27).
Average rainfall at the nearby Seria and Kuala Belait weather
stations is 2880 mm / y (1947 through 2004). The area has
never been logged in recorded history, although commonly
domesticated tree species planted along the river suggest that
there were settlements there some decades ago. The peat dome
shows a typical north-west Borneo peat swamp catena (1),
with mixed swamp forest at the river’s edge, followed by alan
batu community (scattered, tall Shorea albida in upper canopy,
large gaps mostly dominated by Pandanus andersonii), and
then alan bunga community furthest from the river (closed
upper canopy of Shorea albida 50 m tall). The forest floor is
a dense tangle of buttresses, Pandanus rhizomes and woody
debris, with no clearly defined channels for the flow of water
(Fig. 3). Data are archived at the Dryad Digital Repository,
http://datadryad.org/.
Soil respiration. Soil respiration measurements were made with
an automated dynamic chamber system (LI-8100 with LI-
8150 multiplexer, LI-COR Biosciences, Lincoln, NE, USA),
in which carbon dioxide concentrations were measured after
periodic closure of four chambers connected to an infrared gas
analyzer via a gas multiplexer. Each chambers was closed for
6 minutes and a flux computed every hour using instrument
software (LI-8100 software version 2.0.0). Chambers were
supported 30 cm above the peat by four two-inch PVC pipes,
1 m long, to avoid submergence of the chambers. Although
temperature a ected individual flux measurements around
the diurnal cycle, the e ect of temperature on daily mean
fluxes was negligible. Water table height was simultaneously
monitored using a logging pressure transducer as described
below (Piezometers).
Microtopography transect. We used a total station (TC-GTS-
105N (60536) 5" Totalstation, B.L. Makepeace, Inc.) to survey
peat microtopography and 12 piezometers along a 180 m transect
at the field site in August 2012 (Fig. 2d,e). We refer
to the piezometers in this transect as “transect piezometers”
to distinguish them from the 5 “trail piezometers” along the
2.5 km trail. We located the transect survey points in geographic
space by rotating the points to match the compass
orientation between the first two piezometers, then shifting
the rotated points so that the position of the first piezometer
matched its GPS coordinates (Fig. 2e).
To obtain the elevation of the survey points above mean
sea level, we matched the survey data to the LiDAR data.
The LiDAR DEM was defined by finding local minima in
last-return elevations on a 20 m x 20 m grid. Therefore, we
found local minima in surveyed peat surface points on the
same grid, and then aligned the (arbitrary) vertical coordinate
of the minimal survey points and the DEM elevation in the
corresponding grid cell (Fig. 3b).
We then defined a reference land surface p in the transect
as follows. First, we fit a line to survey points by orthogonal
distance regression, then projected all points on to that line.
To find the water table elevation in each piezometer, we subtracted
the water table depth (Solinst Model 101 Water Level
Meter, Solinst Canada, Georgetown, Ontario, Canada) from
the elevation of the piezometer casing. We then found the
calibrated water table elevation vs. time in the piezometer by
matching the measured water table elevation to the piezometer
output at the time of measurement (Fig. 3b,c). We defined
the land surface p in the microtopography transect as a linear
interpolant of the average water table in each piezometer,
shifted up to align with the local minima in the surveyed peat
surface points (Fig. 3b). By this definition, the land surface p
is smooth because its shape comes from the water table, and
it passes through the bottom of hollows because it is shifted
to the local minima in the peat surface.
We next used the transect hydrologic and microtopographic
data to define the land surface at each trail piezometer. Our
definition of the land surface p as a smooth surface fit through
the bottom of hollows makes it easy to estimate the local land
surface elevation by looking at a nearby hollow. However,
because this reference land surface is smoothed, we cannot
easily find the exact height of trail piezometers relative to
the land surface. The trail piezometer nearest the transect
piezometers (80 m to 130 m away) had almost identical water
table behavior to the transect piezometers during the time
interval of the transect piezometer data (Fig. 3c,5a). Therefore
to find the land surface elevation for each trail piezometer,
we first found the land surface for this nearest piezometer by
matching the data on the overlapping time interval. Because
the behavior of the water table was so similar in all trail
piezometers except the bog plain piezometer, for plotting in
Fig. 5 we simply matched the water table height ’ = H ≠ p
for these piezometers to that of the piezometer nearest the
transect by subtracting the di erence in mean. Although this
may introduce some error for the piezometer outside the area
of uniform water table behavior (the “bog plain piezometer”),
we did not use that piezometer to determine the specific yield
and transmissivity functions for simulations.
LiDAR data and topography. LiDAR data were obtained from the
Brunei Survey Department from a 2009 aerial mission. We
constructed a digital terrain model (DTM) from the LiDAR
data by taking the minimum last-return elevation within 20 m
◊ 20 m grid cells, and then smoothing the resulting surface,
corresponding to our definition of a reference land surface p as
a smooth surface fit through hollows (local minima in ˜p). We
then created a contour vector map from the DTM (GRASS
GIS 6.4.3, http://grass.osgeo.org/) and outlined a flow tube
containing our piezometer and cores by constructing boundary
flow lines from contours using the TAPES-C algorithm (35),
tracing back to an estimated location of the groundwater
divide on the Malaysian side of the border (Fig. 2c). Although
the groundwater divide is on the Malaysian side of the border
where LiDAR data are unavailable, the tapered shape of the
flowtube ensures that the simulations are insensitive to the
precise location of the divide (Fig. 6).
We sampled the magnitude of the gradient along
each contour using a spatial database (SpatiaLite 4.2.0,
http://www.gaia-gis.it/gaia-sins/). By the Divergence Theorem,
the average Laplacian of surface elevation within an
enclosed region is equal to the normal surface gradient integrated
around its boundary divided by the enclosed area. We
estimated the Laplacian in the region around the piezometer
DRAFT
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by linear regression of the integrated normal gradient against
the enclosed area in the four piezometers nearest the river,
where the slope was approximately uniform (Fig. 6).
Peat cores. Methods for the two peat cores and 15 samples taken
for radiocarbon dating with a modified Livingstone corer are
available in Dommain et al (2015) (27), and for the seven peat
cores and 22 radiocarbon samples taken with a Russian peat
sampler in Gandois et al (2014) (31). Median calendar ages
were estimated from radiocarbon ages using Calib 7.1.0 with
the IntCal13 calibration curve (36, 37). Because cores were
taken in hollows, the top of each core corresponds roughly to
the reference land surface p, which is fit through the bottoms
of hollows.
Piezometers. Water table height was recorded every 20 minutes
at each of 5 piezometers along a 2.5 km transect using logging
pressure transducers (Solinst Levelogger Edge 3001, Solinst
Canada, Georgetown, Ontario, Canada; Fig. 2c). Transducer
measurements were corrected for barometric fluctuations using
a barometer (Solinst Barologger Gold 3001). Each transducer
was suspended on a steel cable inside a 2” PVC pipe 1.5 m
in length installed to 1.4 m depth and screened at 1.3–1.45 m
below the top of the piezometer casing. Piezometer locations
were determined with a GPS (GPSmap 60csx, Garmin Ltd.,
Olathe, Kansas, USA).
An additional 12 piezometers were installed along an 180
m transect 1 km from the river (Fig. 2d,e; Microtopography
transect). These piezometers were screened from 1 m below the
peat to 5 cm above the peat and were attached to 2” PVC pipes
anchored in the underlying clay. Data from these piezometers
were recorded from 2012-08-17 through 2012-11-03.
Throughfall measurement. Three siphoning tipping-bucket rain
gauges (Texas Electronics TR-525S, Dallas, Texas, USA) were
installed along the microtopography transect to record throughfall
in the forest understory. Each gauge was mounted on an
acrylic plate supported 50 cm above the peat surface on a 2”
PVC tube pushed into the peat. Large fronds of understory
plants (mostly Pandanus andersonii) were cleared from above
gauges to reduce the spatial variability in measurements. Tip
events were recorded by a switch closure event logger (UA-
003-64, Onset Computer Corp, Bourne, MA, USA) enclosed
in a small PVC case attached to each gauge.
Calculation of stable peat dome topography. We found
the stable peat surface Laplacian for a given rainfall
regime by a one-dimensional search as follows. The
water table in a stable peatland can be simulated efficiently
using an integrating ODE solver (SUNDIALS,
http://computation.llnl.gov/projects/sundials). These simulations
are fast because the water table behaves the same
everywhere, so there is only one variable to simulate (Eqn. 5).
Initially, we do not know the Laplacian of the stable peat
surface Ò 2 p Œ for the ODE. However, by taking a guess at the
surface Laplacian and simulating the water table using the
ODE, we can determine whether we have the correct Laplacian
value. If our guess of the surface Laplacian is too large in
magnitude, the average water table height will be too low,
and decomposition will exceed peat production; if our guess
of the surface Laplacian is too small, peat production will
exceed decomposition. Thus, we find the stable peat surface
Laplacian by simulating the water table (Eqn. 5) and checking
for balance between peat production and decomposition over
time (Eqn. 3), successively refining our guess until we have the
stable surface Laplacian for the given rainfall regime (Brent’s
method).
Dynamic model of peat dome development. We solved local
peat accumulation and groundwater flow equations numerically
to simulate development of peat domes. We simulated the
dynamics of groundwater flow (Eqn. 1) and the dynamics
of peat accumulation (Eqn. 2) using a finite volume scheme
in one horizontal dimension representing a flow tube on the
peat surface (Fig. S2, S3, Expanded description of peat dome
simulation). Simulating tropical peat swamp hydrology is
di cult because both transmissivity and specific yield depend
strongly on the water table elevation relative to the surface.
We avoided the numerical issues that arise from the severe
nonlinearity of the flow problem by expressing Boussinesq’s
equation in conservation form (38) via a change of variables
from the water table elevation to the local water storage
(volume per area). The switch to conservation form made it
feasible to simulate storm responses over thousands of years
with time steps of minutes.
We first split the water table elevation H into the surface
elevation p and the water table elevation relative to the surface
’ = H ≠ p. Evolution of the peat surface is extremely slow
relative to movement of the water table, so we treated the
peat surface as quasi-steady while simulating changes in the
water table, and re-wrote Boussinesq’s equation (Eqn. 1) in
terms of the water elevation above the surface ’
ˆ’
S y = qn + Ò · (T Ò’)+Ò · (T Òp) . [8]
ˆt
A di erential change in water storage W at a point in the
landscape is related to a change in water table via the specific
yield, dW = S y dH, so we can express the water table elevation
relative to the surface in terms of a water storage above the
land surface, or “shallow storage” W s
’ =
⁄ Ws
DRAFT
W Õ =0
S ≠1
y (W Õ )dW Õ [9]
where W Õ is a dummy variable of integration. Because the
specific yield function S y(W ) is not a function of time, we can
then re-write the groundwater flow equation in conservation
form
ˆW s
= q n + Ò · (DÒW s)+Ò · (T Òp) [10]
ˆt
where the aquifer di usivity D is defined conventionally as
transmissivity divided by specific yield D = T/S y. We then
discretized time and space following standard methods and
solved the discretized form of the water conservation equation
(Eqn. 10) using a Crank-Nicolson scheme with a no-flow boundary
at the groundwater divide and specified surface and water
table elevation dynamics at the other boundary (S3, Expanded
description of peat dome simulation). For each hydrologic time
step, multiple internal steps might be taken for stability and
accuracy based on a standard stability heuristic (39). After
each hydrologic time step, the surface was incremented by
straightforward use of equation 1, and the change in water
storage below the surface was calculated and deducted from
the shallow storage W s for conservation of water volume.
Model calibration.
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Master recharge and recession curves. The master recharge and
recession curves were computed based on the uniform water
table behavior among the four piezometers nearest the river.
A related approach, without the assembly of master curves
but using the division of water table time series into periods
of heavy rain, days without precipitation, and nights without
precipitation, has been used previously in other studies
of wetland hydrology (28, 29). Under very heavy rain the
groundwater divergence term Ò · (T ÒH) in the groundwater
flow equation (Eqn. 1) becomes negligible
S y
ˆH
ˆt
¥ qr, [11]
and because specific yield S y is a function of water table
relative to the surface ’ only, an integral form of the simplified
di erential equation
⁄
P =
S y d’ + constant [12]
corresponds to the master recharge curve, giving the cumulative
rainfall depth P driving a change in water table height.
During dry intervals, “net precipitation” consists entirely of
evapotranspiration, and taking evapotranspiration as constant
and equal to its average through the diurnal cycle q et (nonnegative)
yields a simplified di erential equation without rain
S y
d’
dt = ≠qet + T Ò2 p [13]
which, in its integral form,
⁄
S y(’)
t =
d’ + constant [14]
≠q et + T Ò 2 p
is equivalent to the master recession curve (Fig. 5f) turned on
its side. The Laplacian Ò 2 p was obtained from topographic
data as described above (“LiDAR data and topography”).
We assembled the master recharge and recession curves
by using throughfall data (above) to identify intervals of the
time series either with no rain, or with heavy rain (rainfall
intensity greater than 4 mm / h). Within each interval, we resampled
a spline of the head versus time data to yield time on
a uniform grid of water table height. We then found an o set
in time or rain depth for each interval to minimize the leastsquares
di erence between overlapping intervals, creating the
master recharge and recession curves (S4, Master recession and
recharge curve assembly). At low water tables, the water table
was nearly constant at night and decreased linearly during the
day (Fig. 5f), so we estimated evapotranspiration as 2.246 mm
/ d using the slope of the daytime decrease in water table and
specific yield obtained from the master recharge curve.
We estimated parametric functions for the specific yield
(cubic spline) and transmissivity (piecewise-linear logarithm of
hydraulic conductivity) by simultaneously fitting the recharge
and recession curves to model results (Levenberg-Marquardt,
http://www.pesthomepage.org). We simulated recharge (water
table height vs. cumulative rainfall depth ’(P )) or recession
(water table height vs. time since rain ’(t)) bysolving
the relevant di erential equation (Eqn. 12 for recharge,
Eqn. 14 for recession) using an integrating solver (SUNDI-
ALS, http://computation.llnl.gov/projects/sundials) with the
current parameter estimates. Although the data provide no
information on how conductivity is distributed in the peat
below the lowest excursion of the water table, the dynamics
of the water table are always dominated by other factors,
either the conductivity of peat higher in the soil profile or
evapotranspiration.
Calibration for peat surface evolution parameters. After estimation
of hydrologic parameters T,S y, we estimated the peat accumulation
parameters f p, – by fitting the simulated shape
of our peat dome to the LiDAR surface using a nonlinear
least-squares optimization routine (Levenberg-Marquardt,
http://www.pesthomepage.org). The coastal peat swamp
forests of Brunei Darussalam are underlain by a broad, very
gently sloped mangrove clay (1, 27), so as the initial condition
we used a flat, sloped surface corresponding to a gradient in
the elevation of basal peat in our peat cores (Fig. 7a).
We drove our simulation of the peat dome using a precipitation
time series derived from our own throughfall data,
meteorological data, and nearby speleothem ” 18 O records using
an approach similar to that of Kurnianto et al. (16). Our
goal was not to reconstruct the actual rainfall at our site
over the last 3000 years, but to reasonably approximate the
fluctuation in rainfall at sub-annual, decadal, and millennial
time scales. For sub-annual rainfall, we cycled through field
throughfall intensity data from 2012 on a 20-minute grid from
our site. We then scaled throughfall intensities by a factor
specific to each simulation year and month as follows. First,
we followed the approach of Moerman et al. (40) and calculated
a regression of two-month rainfall means at a nearby
meteorological station (Brunei International Airport (BWN),
90 km north-east of our site) against ” 18 O at Gunung Mulu airport,
61 km south-east of our site, from August 2006 through
April 2011 (P =2.53006 mm / d ≠ ” 18 O ◊ 1.00825 mm / d,
R 2 =0.208513). We then used this regression to compute
mean rainfall intensities for each decadal to centennial interval
captured by the speleothems at Mulu, which are believed to
characterize regional patterns of rainfall (40, 41). We then
repeated 44 years of monthly precipitation totals from the
meteorological station (1966–2009, mean annual rainfall 2.90
m) for 3000 years, adjusting monthly means to match the linearly
interpolated precipitation averages from the speleothem
data regression. Finally, we scaled 20-minute intensities in
each month of the 3000 year time series to match the month
precipitation total determined by the meteorological data
DRAFT
and speleothem record, scaled by a constant for all months
and years so that precipitation in the measured year (2012)
matched throughfall at our site. Evapotranspiration was modeled
as zero at night and constant by day, equal to e ective
evapotranspiration estimated from piezometer data as described
above. Because the details of changes in river stage
over the last 3000 years are unknown, we used dates in the
core closest to the river to drive the gradually shifting surface
elevation and assumed that average river stage remained in
the same position relative to the surface at the boundary.
Sensitivity to environmental and anthropogenic change. We
explored the e ects of fluctuations in rainfall by using a very
simple model for rainfall (Poisson process, exponential depth
distribution with two parameters: average storm depth and
average inter-storm arrival time. Because rain storms are
instantaneous in this simple model, we computed the increase
in head from each storm by numerical integration of S y(’),
then found the recession of head using an ODE solver as when
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calibrating to the master recession curves (Model calibration,
above). We then found the stable Laplacian for each net
precipitation time series by root-finding as described above
(Calculation of stable peat topography). As a point of reference
for intermittency of rainfall, we calculated the average
inter-storm arrival time at our site as the average duration of
contiguous sequences of 20-minute intervals without rain.
In a second set of simulations, we changed the amplitudes
of annual and El Niño Southern Oscillation (ENSO) fluctuations
in simulated rainfall time series. We first created a
reference rainfall time series by superimposing the 44-year
variation in monthly mean rainfall from meteorological data
on our throughfall rainfall intensities (as in Calibration for
peat surface evolution parameters, above, but without superimposing
the speleothem signal). We define the amplitudes
of the annual and ENSO fluctuations as the amplitudes of
sinusoids with arbitrary phases fit to daily rainfall intensities
by least-squares. For each desired combination of annual and
ENSO amplitudes, we then found the rain time series with
the same mean rainfall and desired amplitudes at annual and
ENSO periods that was least-squares closest to this reference
rainfall time series, while requiring that rainfall intensity was
positive or zero and the pattern of storms was the same, using
a quadratic programming solver (S6, Setting annual and
ENSO amplitudes of rainfall).
We simulated the e ects of environmental and anthropogenic
change in the future by imposing di erent forcing
and boundary conditions on simulations starting with an idealized
peat ridge between two parallel rivers. We generated
the ridge with the simulated precipitation time series used
for model calibration, and chose the breadth of the ridge so
that it would be approximately stable at the end of the time
series (2308 years). The base case for the projections was
the reference rainfall time series described above. Current climate
projections indicate increased average rainfall, increased
seasonality of rainfall, and probably both in the tropics (33).
Therefore, we performed simulations with a stronger seasonal
signal, and scaled the rainfall time series to explore the effects
of higher average rainfall. In the first perturbation, we
increased the seasonal fluctuations to 3.0 mm / d to match
projections for average fluctuations at Brunei’s latitude (4 N)
during 2081–2100 (33) (increase of 0.52866 mm / d). In a
second projection, we increased mean rainfall by the same
amount. The tidal rivers draining coastal tropical peatlands
will be directly a ected by sea level changes, so in a third
perturbation, we increased the boundary water level by 50 cm
over an interval of 100 y to mimic the e ect of sea level rise
(42). (Isostatic subsidence or uplift, such as the postglacial
rebound important in many northern peatlands (11), could
be accommodated by the model in the same way.) Finally, we
simulated the e ects on dome shape of drainage to a constant
50 cm below the peat surface, the recommended best practice
of the Roundtable for Sustainable Palm Oil (43), to model the
e ects of conversion to agriculture on dome shape and carbon
e ux.
S2. Hydrologic budget for our site
We can use water table and throughfall measurements to calculate
an estimated water budget for our site (28). Evapotranspiration
and divergence of groundwater flow can be distinguished
in the decline of the water table between rain events because
at low water tables, when divergence of groundwater flow is
small, evapotranspiration creates a distinct diurnal pattern of
steady water tables at night and declining water tables during
the day (Fig. 5d). Therefore, after estimating specific yield
from the ascending curve (Fig. 5e), we can estimate evapotranspiration
from the daytime decline in the water table when
the water table is low (Fig. 5f). Based on this estimate of
evapotranspiration and our throughfall gauge data, there was
307 cm of throughfall to the peat at our site in 2012, of which
82 cm was lost as evapotranspiration from the peat. Overall
precipitation falling on the peat forest, and evapotranspiration
from the forest, are higher because some rain is intercepted by
the canopy and evaporates before it reaches the peat surface.
The increase in stored water in the peatland over the year,
or recharge, was about 4 mm: the water table was 1.3 cm
higher at the end of the data interval (2013-01-20) than at the
beginning (2012-02-06), and the specific yield in that water
table height range (5.2 cm to 6.5 cm) averages 0.3 (Fig. 5).
Because the recharge term was only 4 mm, nearly all of the net
precipitation (225 cm) was lost via divergence of groundwater
flow.
S3. Expanded description of peat dome simulation
Implementation of numerical model. The water table in a tropical
peatland can rise by centimeters in a matter of hours in
response to intense convective rainstorms, and fluctuations
on these short time-scales a ect the changes in the landscape
over millennia (Fig. S1d,e). Therefore simulations required an
extremely large number of steps. Two practical consequences
of this are that simulations are too large to fit in physical memory
on a current mid-range hardware, and there is a risk of
numerical losses in computing cumulants to verify water conservation
by the model. To overcome these two issues, we stored
forcing data and simulations in Hierarchical Data Format
(http://www.hdfgroup.org/HDF5), periodically prefetching
data from disk to construct interpolants of net precipitation
intensity, and computed cumulants using exact arithmetic (Hettinger’s
lsum, http://code.activestate.com/recipes/393090/).
The overall simulation driver was implemented in Python
(version 2.7.5; http://www.python.org); rate-limiting calculations
were implemented in Cython (version 0.21.1;
http://cython.org).
DRAFT
Flow line coordinate system. Simulations of groundwater flow
and peat accumulation used a flow line coordinate system.
Flow lines, contours, and elevation on the peat dome form
an approximately orthogonal coordinate system: streamlines
and contours are always orthogonal, and elevation is nearly
orthogonal to the plane of the surface because gradients are
very small. In general, in orthogonal curvilinear coordinates
with three coordinates (r 1,r 2,r 3) and unit vectors e 1, e 2, e 3,
there is also an arc length parameter associated with each
coordinate, h 1,h 2,h 3. In Cartesian coordinates, all three arc
length parameters are identically 1, but in general they can
be functions of the coordinates. The gradient operator in
orthogonal curvilinear coordinates is
ÒV = 1 h 1
ˆV
ˆr 1
e 1 + 1 h 2
ˆV
ˆr 2
e 2 + 1 h 3
ˆV
ˆr 3
e 3 [15]
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and the divergence operator is
Ò · F =
Ë
1 ˆ
(h 2h 3F 1)
h 1h 2h 3 ˆr 1
+ ˆ (h 3h 1F 2)
ˆr 2
+ ˆ È
(h 1h 2F 3)
ˆr 3
[16]
We worked in a 2-dimensional coordinate system of streamlines
and contours, consistent with the use of the essentially
horizontal flow approximation for hydrologic calculations, and
considered elevation z approximately constant. The gradient
operator is defined with reference to a streamline arc length
parameter h s relating a di erential change in that coordinate
to geometric distance
ÒH = 1 ˆH
es. [17]
h s ˆs
The divergence operator depends on both the streamline arc
length parameter and an analogous contour arc length parameter
h „ (s) describing the distance associated with a di erential
change in a coordinate „ along a contour,
Ò · T ÒH = 1
h sh „
ˆ
ˆs
1
h„
T ˆH
h s ˆs
2
. [18]
In the special cases of Cartesian coordinates the arc lengths
parameters are both identically 1, whereas for cylindrical polar
coordinates the contour arc length parameter is equal to the
radial position h „ = r.
Along the reference streamline, the streamline arc length
parameter is 1 by definition, and we approximated the contour
arc length parameter h „ as the ratio of the arc length ¸(s)
along that contour to an adjacent streamline relative to the
distance ¸(s ú) to that neighboring streamline at the boundary
h „ (s) = ¸(s)
¸(s ú) . [19]
Then in flow tube coordinates, the groundwater flow equation,
rewritten in terms of shallow storage W s (Eqn. 9) and surface
elevation p, becomes
ˆW s
ˆt
= q n + 1¸
1
ˆ
ˆs
¸T ˆH
ˆs
2
+ 1¸
1
ˆ
¸D ˆp 2
. [20]
ˆs ˆs
Discretization of groundwater flow equations. We discretized the
groundwater flow problem in flow tube coordinates (Eqn. 20)
using a standard one-dimensional finite-volume scheme (39),
with a Neumann boundary condition at the origin (zero gradient
at groundwater divide) and a Dirichlet condition at the
drainage boundary. We use n as an index for time, written
as a superscript, and j as an index for space, written as a
subscript. By integrating through the cell j and dividing by
its area A j, we discretize the volume-conservation equation
(Eqn. 20)
ˆW s
ˆt
= q n + 1 Ë È
¸ D ˆWs s2j
+ 1 Ë
¸T ˆp È s2j
. [21]
A j ˆs s 2j≠1
A j ˆs s 2j≠1
E ectively, we have decomposed the flux in and out of the
cell into a component associated with the gradient in the local
water table elevation relative to the surface, and another component
for the gradient in the surface itself. As a mnemonic,
we call these components the shallow flux Q s and the deep
flux Q o, although this has nothing to do with where in the
soil profile the water transport occurs.
We approximate the gradients with finite di erences, and
find fluxes through the cell faces. Geometry is handled in
much the same way in both cases. The shallow fluxes are
approximated as
Q s;1,j ¥ (Ê 2,j≠1D j≠1 + Ê 1,jD j)(W s;j ≠ W s;j≠1) [22]
Q s;2,j ¥ (Ê 2,jD j + Ê 1,jD j+1)(W s;j+1 ≠ W s;j) [23]
and the deep fluxes as
Q o;1,j ¥ (Ê 2,j≠1T j≠1 + Ê 1,jT j)(p j ≠ p j≠1) [24]
Q o;2,j ¥ (Ê 2,jT j + Ê 1,jT j+1)(p j+1 ≠ p j) [25]
where Ê is a dimensionless aspect ratio, accounting for both
the ratio of cell flow line segment length to the length of the
face between cells ¸ and the position of the face along the
distance between nodes
Ê 1,j =
DRAFT
¸2,j≠1(s1,j ≠ s2,j≠1)
(s 1,j ≠ s 1,j≠1) 2 [26]
and
¸2,j(s2,j ≠ s1,j)
Ê 2,j =
(s 1,j+1 ≠ s . [27]
1,j) 2
This approach ensures that the downstream flow Q 2,j out of
acellj equals the upstream flow Q 1,j+1 into its downstream
neighbor j +1for volume conservation.
After integrating with respect to time, the discretized problem
for interior points j œ {2, 3,...,J ≠ 1} has equation
W n+1
s;j ≠ W n s;j = P n + a s;2,j (W s;j+1 ≠ W s;j)
≠ a s;1,j (W s;j ≠ W s;j≠1)
+ a o;2,j (p j+1 ≠ p j)
≠ a o;2,j (p j ≠ p j≠1)
in which coe cients a s = – s t for shallow fluxes
and a o = – o
[28]
– s;1,j = — 1,jD j≠1 + — 2,jD j
– s;2,j = — 3,jD j + — 4,jD j+1
[29]
t for deep fluxes
– o;1,j = — 1,jT j≠1 + — 2,jT j
– o;2,j = — 3,jT j + — 4,jT j+1
[30]
depend on computed values of transmissivity and specific yield
and on a geometric parameter — that summarizes all spatial
information in the problem
— 1j = Ê 2,j≠1 A ≠1
j
— 2j = Ê 1,j A ≠1
j
— 3j = Ê 2,j A ≠1
j
— 4j = Ê 1,j+1 A ≠1
j
.
[31]
The Neumann condition (zero flux) at the origin and the
Dirichlet condition (specified shallow storage and surface elevation)
at the right boundary are implemented in the standard
way (39), so that the matrices describing the shallow flux A n s
are constructed as
S
W
U
– s n 2,1 ≠– s n 2,1
≠– s n 1,2 – s n 1,2 + – s n 2,2 ≠– s n 2,2
. .. . .. . ..
X
≠– s n 1,J≠1 – s n 1,J≠1 + – s n 2,J≠1 ≠– s n V
2,J≠1
0
[32]
T
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PNAS | April 13, 2017 | vol. XXX | no. XX | 5

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and analogously the matrices describing the deep flux A n o are
S
T
– o n 2,1 ≠– o n 2,1
≠– o n 1,2 – o n 1,2 + – o n 2,2 ≠– o n 2,2
.
W
.. . .. . ..
X
U
≠– o n 1,J≠1 – o n 1,J≠1 + – o n 2,J≠1 ≠– o n V
2,J≠1
0
[33]
Avector÷ represents changes in storage with the time step
forced either by net surface-atmosphere flux q n or by boundary
conditions
;
÷ n+ P n , j œ {1, 2,...J ≠ 1}
j =
W n+1
J
≠ WJ n [34]
, j = J.
where
Then the system of equations to be solved is
B n s W n+1
s + B n o p n = C n s W n s + C n o p n + ÷ n [35]
B n s = I + f n tA n+1
s
B n o = I + f n tA n+1
o
C n s = I +(1≠ f n ) tA n+1
s
C n s = I +(1≠ f n ) tA n+1
o
[36]
and f represents a weighting for explicit and implicit steps;
a Crank-Nicolson scheme (f = 1/2) was used for all calculations.
Solution of the system was first attempted using
Picard iteration; if that approach failed, the system
was solved using Powell’s method (GNU Scientific Library,
https://www.gnu.org/software/gsl). If Powell’s method also
failed, the step size was halved and Picard’s method tried
again. The convergence criterion for both Picard and Powell
solvers was reduction of the L 2 norm of the residual error by
5 orders of magnitude (38).
We chose the time step conservatively for stability by requiring
that the diagonals of the explicit coe cient matrix for
shallow fluxes C s and for deep fluxes C o be kept positive (39),
that is,
! #
t Æ min min (–s 1,j + – s 2,j) ≠1 ,
jœ2,3,...,J≠1
(– o 1,j + – ≠1$" [37]
o 2,j) .
Strictly speaking, this is a heuristic for the stability of explicit
steps, and can be exceeded when using the Crank-Nicolson
approach adopted here. However, because the nonlinear solvers
sometimes failed to converge with large step sizes because of
the very strong nonlinearity of the problem, in practice it was
most e cient to choose time steps exceeding this heuristic
maximum by no more than a small factor (e.g., 2).
S4. Master recession and recharge curve assembly
We assembled the master recession curve and master recharge
curve by least-squares alignment of short water table time
series representing intervals of either intense rain or no rain.
We describe first the procedure for the master recession curve,
which is slightly simpler. The head versus time sequence is
split into J inter-storm series sorted by initial head so that
the lowest series index has the lowest initial head. For each
series, the head versus time data were interpolated with a
cubic spline and resampled to give time versus head data with
head values integer multiples of a uniform step size H. In
general, a series may hit the same head at multiple times
because of measurement noise and near-constant heads low in
the peat at night. The equations are nearly the same, except
for weighting, if those times are replaced by their mean, so
that each series j has no more than one time t ij for each head
i: for each head value in each short series, all distinct times
t at a particular head were replaced with their average ¯t to
give distinct values (k, j, ¯t) where k is the head as an integer
multiple of the head step size and j is the series index.
Sets of time-versus-head series can be assembled into a
recession curve if one can navigate from the lowest to the
highest head via regions of overlap between series. In most
cases, there were some short sequences at the very highest and
very lowest heads where a subset of series did not overlap with
the rest. These small subsets of series were eliminated, leaving
a single large connected component of time-versus head series.
We also removed heads at which there was only one series
because these contribute no information when finding time
o sets.
We then solved for a time o set t j for each series by
minimizing the mean squared di erence in all times at the
same heads, as follows. Ideally, after the o set for a series t j
has been applied, the time t ij of that series equals the mean
time at that head of all J(i) series with a value at that head i:
A
t ij + t j = 1
J(i)
B
ÿ
t j + t ij [38]
J(i)
j=1
so at each discrete water level i, for each series j we have a
single equation with an unknown time o set t j
A
B
J(i)
1 ÿ
t j ≠ t j = t ij ≠ 1
J(i)
ÿ
t ij [39]
J(i)
J(i)
#
1
J(i)
j=1
1
...
J(i)
1
≠ 1 ... 1
J(i) J(i)
DRAFT
$
W
U
j=1
which can be re-written in matrix form with typical row
S
t 1
T
t 2 X
=
C
t ij ≠ 1
J(i)
.
t j
.
t J≠1
A ÿJ(i)
BD
t ij .
j=1
X
V [40]
The prior elimination of disconnected series ensures that there
is an equation at each head, and the removal of heads with only
one series ensures that there are no trivial equations. Note
that we have excluded a reference series J from the unknown
vector; its o set is fixed to 0 to make the problem non-singular.
After least-squares solution of this system of equations (LU
decomposition), the solution vector contains the time o sets
t for the corresponding series.
We constructed master recharge curves using an approach
similar to that used for master recession curves, based on
the idea that when rain is intense, discharge is negligible by
comparison
dH
S y ¥ qr [41]
dt
with q r the rainfall. We calculated the cumulative total rain
depth as a function of time for each interval of heavy rain, then
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found the rain depth o set for each series using the master
recession curve least-squares approach. The resulting master
recharge curve represents head as a function of cumulative
rain depth, and is equivalent to the integral of specific yield
s
Sy dH across all heads H.
Rain storms did not arrive simultaneously at throughfall
gauges and all piezometers, so it was necessary to match
storms as recorded at throughfall gauges to the time of the
storm’s arrival at each piezometer. We treated each contiguous
interval of rapidly increasing head as a storm, then searched
the throughfall time series for the corresponding record of
that storm, considering that it might be slightly o set in time.
Usually there was only one heavy rain that overlapped with the
increasing head interval; in rare cases, there were two. In those
cases we chose the storm with the higher total depth, assuming
that it would be more likely to cause the rapid increase in
head. If the time o set for onset of the storm di ered greatly
(by more than 1 h), the candidate storm was discarded.
Evapotranspiration estimated from the master recession
curve represents loss from the connected porewater and surface
water, not the total evapotranspiration from the forest. Total
evapotranspiration also includes water intercepted by live
foliage and water perched on the abundant leaf litter suspended
above the peat surface, and is probably considerably higher
(28).
S5. Effects of rainfall aggregation time
We explored the importance of short-term fluctuation in rainfall
on long-term simulations of peat accumulation by experimenting
with di erent time grids. In our simulations,
we represented rainfall intensity and evapotranspiration as
piecewise-constant on 20-minute intervals. Most models represent
rainfall on a coarser time grid, making rainfall constant
on time scales of months, years or centuries, but we chose
a much finer grid to represent rainfall intensity because our
results showed that short-term fluctuations in rainfall can have
a significant e ect on the simulated long-term evolution of peat
domes. We started with our 20-minute time grid, on which
rainfall is modeled as piecewise-constant on each consecutive
time interval:
R(t) =R i,tœ [t i,t i+1), iœ {1, 2,...,n≠ 1}. [42]
We then computed new representations of rainfall intensity
and evapotranspiration such that the rainfall intensity on
a piecewise-constant interval in the output is equal to the
average intensity on the same interval in the input:
Rj Õ 1
=
t Õ j+1 ≠ tÕ j
⁄ t
Õ
j+1
t=t Õ j
R(t)dt, j œ {1, 2,...n Õ ≠ 1} [43]
where the overall time span is the same (t Õ 1 = t 1,t Õ nÕ = tn) but
there are fewer total time steps (n Õ

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To ensure that the new rainfall intensity vector ˆr has the same
mean µ and the specified harmonic content a, b we therefore
require
S T
C +ˆr = U â
ˆb V [49]
ˆ“
where C + is the Moore-Penrose pseudoinverse of C. Subtracting
the equation defining the harmonic content of the original
rainfall intensity vector C + r = [a, b, “] T , the constraint is
equivalently expressed in terms of the di erence between the
new and old rainfall time series x = ˆr ≠ r
C +ˆx =
S
U â ≠ a
ˆb ≠ b
ˆ“ ≠ “
T
V . [50]
Now that we have written the constraints as linear equality
constraints and box constraints, the overall problem of finding
the new rainfall intensity vector ˆr can be written as the
quadratic program
minimize
subject to
1
2 xT x
Gx ∞ h
Ax = b
[51]
where h is an m-vector consisting of the non-zero entries of r,
G is a m ◊ n matrix defined by
;
≠1, ri ”= 0,i= j
g ij =
[52]
0, otherwise
and A is a block matrix enforcing the linear equality constraints:
that zero entries remain zero, that the mean is the
same, and that amplitudes at the chosen frequencies are as
specified in the new time series:
5 6
F
C +
and
A =
S
W
b = U
0
â ≠ a
ˆb ≠ b
ˆ“ ≠ “
T
[53]
X
V [54]
with F an (n ≠ m) ◊ n matrix that keeps the zero entries equal
to zero
;
1, ri =0,i= j
f ij =
. [55]
0, otherwise
The quadratic program (Eqn. 51) was then solved using a
sparse QP solver (CVXOPT, http://cvxopt.org).
DRAFT
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SI Figure Legends
Figure S1 | Rainfall intensity data used to drive simulations.
a, Annual mean rainfall intensity from Brunei Airport (BWN)
meteorological station after correction to match throughfall
(black) and with stronger ENSO signal added (gray). b,
Monthly mean rainfall intensity from BWN, rescaled to match
throughfall at site as used in base simulation (black) and
with stronger seasonal signal (gray), showing medians, upper
and lower quartiles, outliers (+) and means. c, Rainfall
intensity used for simulations of past peat dome evolution.
Oxygen isotope data from speleothem data at Mulu Park,
Sarawak (Partin et al 2007, Moerman et al 2013) was used to
approximate fluctuations in past rainfall (SI Text, Sensitivity
to environmental and anthropogenic change). d,e, Bias in
stable surface Laplacian as a function of rainfall averaging
time. Using a coarser grid for rainfall time series results in
overestimation of the stable Laplacian of the surface because
real fluctuations in the water table driven by intermittent rain
increase the average transmissivity, reducing long-term peat
accumulation.
Figure S2 | Flow tube discretization used for peat growth and
hydrologic calculations. a, Discretization of flow tube for finite
volume hydrologic simulation. b, Piecewise-constant and
piecewise-linear approximations of variables for hydrologic
simulation.
Figure S3 | Simulated past and future morphology and fluxes
of Mendaram peat dome. a, Simulated past and future development
of Mendaram peat dome towards its stable shape with
uniform Laplacian (dashed line). b, Current modeled CO 2
sequestration rate vs. distance from groundwater divide at
Mendaram peat dome. c, Modeled CO 2 sequestration rate
vs. position and time at Mendaram peat dome. d, Spatial
average of modeled CO 2 sequestration rate of Mendaram peat
dome vs. time.
Figure S4 | Convergence to uniform surface Laplacian. a,
Simulated development of a peat ridge growing on an initially
flat substrate under simulated precipitation (average net
precipitation 1477 mm / y, solid lines) or under constant net
precipitation that yields the same stable shape, only 190 mm
/ y (dashed lines), and their convergence to a surface with a
uniform Laplacian (dotted line). The factor-of-eight di erence
in mean rainfall required to generate the same peat dome
morphology with steady (dashed) and intermittent (solid) precipitation
illustrates the strong e ect of fluctuations in rainfall
on long-term peat accumulation. Colors from time color bar,
bottom. b, Laplacian from (a); with simulated rainfall (solid)
and with constant net precipitation (dashed), and convergence
to ultimate Laplacian (horizontal dotted line). c, Surface with
a uniform negative Laplacian (dashed) and with a positive
Laplacian (solid). d,e, Simulated water table at three points
on surfaces from (c): uniform negative Laplacian (d) and
positive Laplacian (e). The water table behaves the same at
all three locations shown in (c) with a uniform negative land
surface Laplacian (d), but behaves di erently at the three
locations if the surface Laplacian is positive (e).
DRAFT
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PNAS | April 13, 2017 | vol. XXX | no. XX | 9

a
Rainfall intensity
R, mm / d
c
Rainfall intensity
R, mm / d
Year
b
Rainfall intensity R, mm / d
d
Year
e

a
b
node
face

a
Land surface
elevation p, m
b
Current CO 2 flux,
t ha -1 y -1
12
8
4
0
1.5
1.0
0.5
0.0
Time, y
-2000 0
2000 4000 6000
6 4 2
0
Distance from river, km
Distance from
river, km
Average CO 2 flux,
t ha -1 y -1
CO 2 flux, t ha -1 y -1
0 2
4
c
6
4
2
0
d
3.0
1.5
0
0 2500 5000
Time, y

Land surface elevation p, m
a
b
Laplacian of land surface
elevation , km -1 x 1000
1.2
0.8
0.4
0
0
−5
−10
1000
750
500 250
0
Distance from river, m
Land surface elevation p, m
Water table height , cm
c
1.2
0.8
0.4
0
1000 750
500
250
0
Distance from river, m
20
0
−20
20
d
e
0
−20
0 30 60 90
Elapsed time, d
−3000
−2000
Time, y
−1000
0