zmap a tool for analyses of seismicity patterns typical applications ...

ZMAP
A TOOL FOR ANALYSES OF SEISMICITY
PATTERNS
TYPICAL APPLICATIONS AND USES:
A COOKBOOK
MAX WYSS, STEFAN WIEMER & RAMÓN ZÚÑIGA
12/4/2001

Table of Content
INTRODUCTION.............................................................................................................. 3
CHAPTER I .......................................................................................................................4
What’s going on with this earthquake catalog? Which parts are useful? What
scientific problems can be tackled?............................................................................. 4
CHAPTER II.................................................................................................................... 14
Are there serious problems with heterogeneous reporting in a catalog? What is
the starting time of the high-quality data? ............................................................... 14
CHAPTER III .................................................................................................................. 22
Measuring Changes of Seismicity Rate..................................................................... 22
Comparing two periods for rate changes .................................................................. 28
Articles in which tools discussed in this chapter were used:.................................... 33
CHAPTER IV................................................................................................................... 34
Measuring Variations in b-value ............................................................................... 34
CHAPTER V .................................................................................................................... 41
Stress Tensor Inversions............................................................................................. 41
Introduction............................................................................................................... 41
Data Format .............................................................................................................. 41
Plotting focal mechanism data on a map .................................................................. 42
Inverting for the best fitting stress tensor. ................................................................ 43
Inverting on a grid..................................................................................................... 44
Plotting stress results on top of topography.............................................................. 46
Using Gephart’s code................................................................................................ 47
The cumulative misfit method .................................................................................. 49
References................................................................................................................. 52
CHAPTER VI................................................................................................................... 54
Importing data into ZMAP........................................................................................ 54
ASCII columns – the most simple way..................................................................... 54
Other options to import your data: Writing your own import filters ........................ 55
CHAPTER VII ................................................................................................................. 57
Tips an tricks for making nice figures ...................................................................... 57
Editing ZMAP graphs............................................................................................... 57
Exporting figures from ZMAP.................................................................................. 60
Working with interpolated color maps ..................................................................... 63
2

INTRODUCTION
This cookbook is aimed at entry level and experienced users of ZMAP. It described
typical applications of ZMAP to seismicity analysis. By giving enough detail on both the
scientific objective and the mechanics of using ZMAP as a tool for seismicity analysis,
we hope to provide users will a helpful document beyond the customary manual and
online help. Note that the cookbook assumes some familiarity with basic ZMAP
functions (starting ZMAP, loading catalogs, etc.), which are described in the First Steps
manual. (./help/firststeps.htm).
This cookbook is provided as PDF file for easy viewing and printing, and as a HTML
document for online browsing. Since ZMAP is continually developing, not all windows
may look in your ZMAP version just like show here. We tried to make this manual as
useful and error-free as possible while keeping the time required to produce within
tolerable bound (we do like to do science more than writing documentation). As always
in ZMAP: No guaranties, feedback is welcome!
Max Wyss and Stefan Wiemer 08/2001
3

CHAPTER I
What’s going on with this earthquake catalog? Which parts are
useful? What scientific problems can be tackled?
Step 1: Read the catalog of interest into ZMAP. The Alaska catalog used in this analysis
can be downloaded in *.mat format from the ZMAP resources page. First click on load
*.mat and go in the menu window, and select the mat-file containing your catalog.
Review the limits of the basic catalog parameters in the general parameters window
(Figure 1.1) that opens after you click on the mat-file containing the catalog.
Figure 1.1: General Parameters window
Notice at a glance: (1) This catalog contains 78028 events, (2) covers the period 1898.4
to 1999.5, (3) contains a strange flag in the field of magnitudes for some events (-999),
(4) the largest shock has M=8.7, and (5) the depth ranges from –3 to 600 km.
First decision: Decide that you are not interested in earthquakes whose M is not known.
Therefore, replace the value for Minimum Magnitude with 0.1 by typing this into the
yellow window spot. Then click on Go.
The epicenter map appears (Figure 2), sometimes displaying scales that look strange
because they are not taking into account that X and Y are coordinates on a globe. For a
nicer map with more appropriate scaling, try the Tools -> Plot map using m-map option
from the ZTools menu of the seismicity map (Figure 2). The large events (within 0.2
units of the largest one) are labeled, because you left the default option in the button
labeled Plot Big Events with M>.
4

Figure 2: Seismicity map of the entire catalog. (top): Normal ZMAP display (bottom), plotted in Lambert
projection using M-Map, bottom, plus topography.
Rough selection of the area of interest: Based on your experience, you know that the
coverage in the Aleutians is much inferior to that of mainland Alaska, you decide to
concentrate on the seismicity in central and southern Alaska. Click on the button Select
in the Seismicity Window and choose your method of selection. Select EQ inside
polygon may be the most convenient. Cross hairs appear. Click on the corners of the
polygon you like with the left mouse button and with the right one for the last point. The
Cumulative Number window will open (Figure 1.3) and display the selected events as a
function of time.
5

Rough selection of the period of interest: It is evident from Figure 1.3a that only very
large events are reported before the mid 1960s. Because this is not the subject of your
quest and you want to concentrate on the small magnitude events, you delete all data
before the reporting increase in the 1980s by selecting Cuts in Time Cursor in the
ZTools button of the Cumulative Number window and clicking with the appearing
crosshairs at the beginning and end of the period you want. This re-plots the cumulative
Figure 1.3: Cumulative number of the selected earthquakes as a function of time.
number plot (Figure 1.3b) for a period during which the rate of earthquakes reported was
approximately constant with time. This suggests that the reporting may have been
homogeneous from 1989 on. Because this is the type of catalog you want, you now click
on Keep as newcat, which re-plots the epicenter map as seen in Figure 1.4.
Figure 1.4: Epicenters after rough selection of area and period.
6

Save new mat-file: At this point it is time to save the culled catalog in a mat-file by
clicking on the Save selected Catalog (mat) button in the Catalog menu of the
seismicity map. It is a good idea, but not necessary, to now reload this new catalog.
Inspecting the catalog: From the ZTools menu select Histograms, then select Depth.
The resulting Figure 1.5 shows that there must be an erroneously deep event in the data
and
Figure 1.5: Histogram as a function of depth.
that there exists a minimum at 35 km depth, which might be the separation between the
crustal and the intraslab activity. Next, select Hour of Day from the Histogram button in
Figure 1.6: The absence of smokestacks at certain day hours suggests there are few or no explosions
the ZTools menu. The resulting Figure 1.6 shows that the data are not contaminated by
explosions (or at least not much) because the reporting is uniform through day and night.
Finally, check out the distribution as a function of magnitude by plotting the appropriate
7

Figure 1.7: Frequency as a function of magnitude.
histogram. From Figure 1.7 one learns that magnitudes near zero are sometimes reported,
but that the maximum number is near M2, suggesting that the magnitude of completeness
is generally larger than M2, but that it may be near M2 in some locations. An alternative
Figure 1.8: Frequency magnitude distribution of the over-all catalog. Plotted is both the cumulative
(squares) and non-cumulative form (triangles).
presentation in the cumulative form can be obtained by first plotting the cumulative
number as a function of time (the button to do this is found in ZTools of the seismicity
map), and then selecting the button Mc and b-value estimate (with the proper subchoice).
An automatic estimate yields Mc=2.0 for the overall catalog (Figure 1.8).
8

Narrowing the target of investigation: At this point one might decide to study just the
shallow seismicity. Because of the minimum of the numbers at 35 km depth (Figure 5),
this offers itself as the natural cut. The bulk Mc for the shallow events can be estimated
by the same steps as outlined above. It turns out to be 1.9. Therefore, you may want to
map the Mc for the shallow seismicity with a catalog without the events below M1.5,
because we know that not enough parts of the catalog can be complete at that level. The
catalog for the period and area with depths shallower than 35 km and M>1.4 contains
15078 events. The map of Mc (Figure 9) is obtained by selecting Calculating Mc and b-
Figure 9: Map of magnitude of complete reporting.
value Map from the Mapping b-values menu in the ZTools menu of the seismicity map.
The cross hairs that appear are used to click by left mouse button the polygon apexes
desired, and terminating the process by clicking the right mouse button. Once the
computation is completed, you can save the resulting grid (which also contains the
catalog used) for reloading later on. Pressing Cancel will just move on without saving.
It might be fun to interpret the b-value map that is presented at first after the calculation,
but first we should examine the Mc map. We call for it by selecting mag of
completeness map in the menu of Maps in the seismicity window (Figure 9). Here the
symbols for the epicenters are selected as none, such that they do not interfere with the
information on Mc.). In it, we see that the offshore catalog is inferior since Mc>3.5.
Before we accept the Mc map, it is a good idea to sample a number of locations to see if
we agree with the algorithm’s choice of Mc by visual inspection of the FMD plots. For
9

this quality control, we open the select menu in the seismicity map, click on select in
circle and place the cross hairs into the red zone offshore, where we click, to learn if
really the resolution is a bad as the algorithm shows. Then we repeat the selection
process, only this time we Select Earthquakes in Circle Overlay existing plot, such that
we can click on a deep blue area in the interior of Alaska and compare its FMD with the
one we already have. The two FMS are indeed vastly different and we see that the
algorithm has defined Mc correctly in both cases (Figure 10).
Figure 10: Comparison of frequency magnitude distributions for offshore (squares) and central (dots)
Alaska.
After accepting the Mc-map, we limit the study are further to the part of the catalog that
is of high quality, let’s use Mc=2.2 for the boundary. Selecting the area by the same
method as before we create a new and final catalog for study. The polygon we just
clicked to select the final area can be saved by typing into the Matlab command window
“save filename xy –ascii”.
10

Figure 11: Resolution map with the scale in km set from 5 to 100 such that radii larger than 100 km (they
reach 231 km) are lumped together by setting the scale limits in the Display menu in the b-value map,
because they are of no interest.
Parameters for Analysis: Now that we have a Mc-map, we might as well check the
resolution map (Figure 11) by selecting it from the map menu. From this map we can
learn what the range of radii is with the selected N=100 events. Of course, this is still
with Mmin=1.5, which means that in many sample there are events, which are not used in
the estimate of b, but it provides an approximate assessment of the radius we may choose
if we decide to calculate a b-value map with constant radius, from which a local
recurrence time map or, equivalently, a local probability map can be constructed. One
can see that to cover the core of Alaska with a probability map one would have to select
R=40 km.
For each map that we select there is a histogram option available (from the Maps menu).
For the radii mapped in Figure 11, the distribution is shown in Figure 12. One sees that
11

Figure 12: Histogram of radii in Figure 11.
35 km is the most common radius.
A further of quality control is offered by the standard error map for the b-value estimates
(Figure 13). This map allows the investigator to select samples from areas where
problems may exist with straight line fits of the magnitude frequency distribution.
Figure 13: Map of the standard error of the b-value estimate.
Often, these errors are due to the presence of a single large event that does not fit the rest
of the distribution, as in the case of the red pot near 63.3°/-145.8° (Figure 13). But
sometimes they flag volumes with families of events with approximately constant size.
Figure 14: Frequency magnitude distribution from 63.3°/-145.8°, where a poor fit is flagged in the error
map (Figure 12).
12

Articles in which tools discussed in this chapter were used:
Zuniga, R., and M. Wyss, Inadvertent changes in magnitude reported in earthquake
catalogs: Influence on b-value estimates, Bull. Seismol. Soc. Am., 85, 1858-1866,
1995.
Zuniga, F.R., and S. Wiemer, Seismicity patterns: are they always related to natural
causes?, Pageoph, 155, 713-726, 1999.
Wiemer, S., and M. Baer, Mapping and removing quarry blast events from seismicity
catalogs, Bulletin of the Seismological Society of America, 90, 525-530, 2000.
Wiemer, S., and M. Wyss, Minimum magnitude of complete reporting in earthquake
catalogs: examples from Alaska, the Western United States, and Japan, Bulletin of
the Seismological Society of America, 90, 859-869, 2000.
13

CHAPTER II
Are there serious problems with heterogeneous reporting in a
catalog? What is the starting time of the high-quality data?
Work done already: We assume that you have acquainted yourself with the general
properties of the catalog. You deleted the hypocenters outside the periphery of the
network and those of erroneously large depth, as well as the M0, if they are meaningless,
and the explosions. For this cases study, we use the seismicity on the San Andreas fault
near Parkfield.
Preliminary Declustering: If you want to evaluate whether or not the catalog contains
rate changes that are best interpreted as artificial, it may be that aftershocks and swarms
get in the way. If you feel that is the case, please decluster leaving all earthquakes with
meaningful magnitudes in the data. The earthquakes smaller than Mc contain important
information on operational changes in the network.
Running GENAS: Once you have loaded the catalog of interest, select RunGenas from
the ZTools in the seismicity map window. Enter the desired values into the Genas
Control Panel (Figure 2.1).
Figure 2.1: Genas Control Panel. Select the minimum and maximum magnitudes such that you calculate
rate changes for magnitude bins that have enough earthquakes in them to warrant an analysis. Base your
judgment on the distribution you saw in the histogram of magnitudes. It is not worthwhile skimping on the
increment.
tart the calculation by activating the button Genas. Habermann’s algorithm now searches
for significant breaks in slope, starting from the end of the data, and for all magnitude
bins for MMi. The purpose of separately investigating magnitude bins is to
isolate the magnitude band in which individual reporting changes occur.
14

Figure 2.2: Genas1 window. The cumulative numbers of earthquakes with M>Mi and with M

fit) button is found in the cumulative number window in the menu offered by the Ytools
button. Clicking on it opens the Time selection window shown in Figure 2.4. In this
figure we type in the limits of the periods we wish to compare. In this case the limits of
the clean periods before and after the change in 1995.
Figure 2.4: Time selection window. Limits of periods in which to compare rate changes can be selected
by typing in the times, or by cursor on the cumulative number window.
The result of the comparison is presented in two windows. The compare two rates
window (Figure 2.5A, 2.5B and 2.5C) compares the earlier and later data in a cumulative
and logarithmic-scale plot, a non-cumulative plot and a magnitude signature (Habermann,
1988), each as a function of magnitude. The frequency-magnitude distribution
window (Figure 2.5D) shows the same change in the usual FMD format, and not
normalized.
16

Figure 2.5: Comparison of the rates as a function of magnitude for two periods, which are printed at the
top. The rate change took place in 1995.5 along the Parkfield segment of the San Andreas fault (35.3° to
36.4°). The numbers are normalized by the duration of the periods. (A) Frequency-magnitude curve. (B)
Non-cumulative numbers of events as a function of magnitude. (C) Magnitude signature. (D) The rate
comparison before and after 1995.5 in the usual FMD format. The three lines below the graph give the
results of fits by two methods to the FMD of the first period (black) and the result by the WLS method for
the second period. Inside the figure, at the top, appears the summary of the data, numbers of events used,
and b-values found. Also, the probability, p, that the two sets are drawn from an indistinguishable common
set is given (Utsu, 1992).
Although the magnitude signature is an informative plot for the experienced analyst, the
non-cumulative FMD (Figure 2.5B) is probably the most helpful to understand the rate
change. It shows that the two periods experienced approximately the same rate of events
in their respective top-reporting magnitude bands, only, these bands were shifted. Before
1995, the maximum number of events was reported at Mmax(pre)=1.0, after
Mmax(post)=1.2. Many more events were reported for 0.7≤M≤0.9 before 1995.5 than
afterward. In contrast, the rate in the magnitude band 1.2≤M≤1.6 was substantially
lower, before compared to afterward. This type of opposite behavior for the smaller and
the larger events is demonstrated by positive and negative peaks on opposite sides of the
magnitude signature plot (Figure 2.5C).
That nature would produce fewer larger events, but balance this by more smaller events
after a certain date without a major tectonic event is not likely. Thus, we propose that the
rate change found by GENAS in Figure 2.4, and analyzed in Figure 2.5, should be
interpreted as a magnitude shift (e.g. Wyss, 1991). If we look at it in the presentation of
Figure 2.5D, we see that it is also a mild magnitude stretch (e.g. Zuniga and Wiemer,
1999). This result is not good news, since it happened in the Parkfield catalog at a recent
date, introducing an obstacle for rate analyses at Parkfield. The amount of shift in 1995
is approximately +0.2 units. Even though the amount of shift appears to be close to +0.2
units, it would be necessary to determine the optimal value by a quantitative method. This
can be accomplished by selecting the "Compare two rates (fit)" from the "ZTools"
pulldown menu of the cumulative number window. You will start a comparison of the
seismicity rates in the two time periods which this time includes the fitting of possible
magnitude shifts, or stretches of magnitude scale, by means of synthetic b-value plots.
Also provided are estimates for the b-value, minimum magnitude of completeness, mean
rate for each period, and z-test values comparing the rates of the two periods. For a more
detailed description on magnitude stretches, see Zuniga and Wyss (1995).
After selecting "Compare-fit" in the cumulative number plot window, you will be
prompted for the limits of the time periods under investigation, just as for the “Compare
two rates (no-fit)” option. A frequency-magnitude relation (normalized to a year) for each
interval is plotted with different symbols. The first interval is labeled as "background",
while the subsequent interval is the "foreground". You should select two magnitude endpoints
for each curve to obtain an estimated b-value for each interval; these have to be
17

chosen on the basis of the linearity of the observed curves. Once this selection has been
performed, the routine attempts to fit the background to the foreground by assuming two
possibilities:
(1) The background is first adjusted to fit the foreground by assuming a simple magnitude
shift. The shift is estimated from the separation between the two curves and by using the
minimum magnitude at which the curve departs from a linear fit by more than one
standard deviation.
(2) The background, Mback, is matched to the foreground, Mfore, by assuming a linear
magnitude transformation (stretch or compression of magnitude scale) of the type
(Zuniga and Wyss, 1995):
Mfore = c * Mback + dM
where c and dM are constants.
Numerical results are given in a window which allow the possibility of interactively
changing any of the shift, stretch or rate factor parameters (Figure 2.6).
Results are also graphically displayed in a separate window which shows:
a) The frequency-magnitude distribution of the foreground and the frequency-magnitude
distribution of the corrected background, using the values for c and dM from the latest
run.
b) Non-cumulative histograms for both foreground and corrected background
c) Magnitude signatures (if needed)
Figure 2.6: Results of compare-fit. The values given include the best fit for two separate possibilities: a
simple magnitude shift (ideally one would like to work with this value); and a magnitude stretch. In
18

this case, the routine found that a simple shift of +0.1 units applied to the background,
would best fit the foreground, while if a stretch is chosen, one needs to apply a shift of
+0.3 and a multiplicative factor of 0.82. Notice that in the selection panels, a simple
magnitude shift of +0.1 is input, which resulted in the plot shown in Figure 2.7A.
(B)
(A)
Figure 2.7. Comparison of rates as a function of magnitude for two time periods under the Compare-Fit
option. The three panels correspond to the Frequency-Magnitude curves, normalized, the non-cumulative
histogram as a function of magnitude, and the magnitude signatures for the original two periods (circles)
and for the synthetic foreground as compared to the original background (crosses). (A) Results of applying
a simple magnitude shift of +0.1 units, without any rate change. (B) Same as in (A) but including a rate
change of 0.78 (equivalent to the -%22 change in percent given in Figure 2.5).
The magnitude signature plot is useful for asserting the goodness of the fit from the latest
run (bottom panel of Figures 2.7A and B). It shows the original magnitude signature,
which results after comparing the two time periods, and a modeled signature obtained
from comparing the synthetic foreground (i.e. the corrected background) to the original
background. The best match between both signatures indicates that we have been able to
model the observed behavior by applying the given corrections to the background. In the
example, we can see that the shape of the signature is correctly modeled by applying a
simple magnitude shift (Figure 2.7A) while a rate decrease is still necessary to model the
position of the signature (Figure 2.7B).
The Compare-fit option is also useful in case one needs to determine the relation between
two different magnitude estimations for the same period and area. For this case, you
would need to first concatenate the two data sets (“Combine two catalogs” option in the
“Catalogs” pulldown menu from the Seismicity Map window) and treat them as separate
time periods.
19

.
Another date of interest is 1980, because at this time approximately, improvements in
analysis techniques took place in all of California. The period before it is not clean
(Figure 2.3), thus we compare the rate in the periods 1972-1978 to those in 1980-1985
(the end of the clean period following the 1980 change, Figure 2.3). The rate comparison
of these periods shows an even more dramatic magnitude shift (Figure 2.7) of at least
DM=–0.5 units. In this case, the shift was accompanied by an increase in reporting of
small earthquakes. These two phenomena, magnitude shifts and increased reporting of
small events, are often seen at the same time, because a single change in the operating
procedure generated both. The conclusion is that the earthquake catalog for Parkfield can
hardly be used for seismicity rate studies. The fact that the two FMD curves show the
same slope before and after the disastrous changes in 1980 (Figure 2.7D) suggests that
the catalog can still be used for b-value studies.
Figure 2.7: Comparison of the rates as a function of magnitude for two periods, which are printed at the
top. The rate change took place in 1978/80 along the Parkfield segment of the San Andreas fault. Details
same as in Figure 2.5. The magnitude shift was at least dM=-0.5 units.
Another method to evaluate the homogeneity of reporting as a function of time, is to
inspect cumulative number curves. In a network where no magnitude shifts have taken
place but more small earthquakes are reported in recent years because of improvements
in the operations, the key to selecting the widest magnitude band which has been reported
homogeneously, is to define the smallest magnitude for which constant numbers have
been reported. This assumes that in a rather large area the production of events is
stationary, on average. Such a case is shown in the comparison for the Parkfield network
(Figure 2.8). The improvement of reporting is seen to be restricted to M

cumulative number curve for M>1. events is approximately straight, whereas the curve
for all magnitudes shows a kink upward at the time of the improvement (Figure 2.8).
Figure 2.8: Comparison of cumulative number of events for earthquake M >= 1.0 (blue) and M , 1.0 (red).
The legend was added manually.
It could be a mistake to rely solely on cumulative number curves for evaluating
homogeneity, because in the Parkfield catalog the selection of an intermediate magnitude
for cutoff (M=1.2, in this case) results in a cumulative curve with relatively constant
slope, whereas the plots for M>1.5 and for M

CHAPTER III
Measuring Changes of Seismicity Rate
Precondition: You have already selected the part of an earthquake catalog that is
reasonably homogeneous in space, time and magnitude band. All inadequate parts of the
catalog and explosions have been removed.
Measuring a Local Rate Change: Suppose you have selected earthquakes from some
volume, and, displaying it in a cumulative number curve, you notice a change in slope
(Figure 3.1a), which you want to measure.
Figure 3.1: (a) Cumulative number of earthquakes as a function of time, obtained by setting N=200 in the
window that appears if one chooses select earthquakes in circle (menu) in the pull down menu of the
select button in the seismicity map window. (b) Cumulative number of earthquakes with the AS(t)
function for which the Z-scale is on the right. The maximum of this function defines the time of maximum
contrast between the rate before and after it.
First: One might want to define the time of greatest change quantitatively (especially in
a case of change less obvious than the one in Figure 3). Open the ZTools pull-downmenu
in the cumulative number window, and point to the option Rate changes (zvalues).
Of the three options offered, choose AS(t)function. This will calculate the red
curve in Figure 3.1b, which represents the standard deviate Z, comparing the rate in the
two parts of the period before and after the point of division, which moves from (t0+tW) to
(te-tW). T0 is the beginning, te the end and tW, the window at the ends, can be adjusted by
typing the desired value into the yellow button that appears in the figure. The maximal
Z-value, and the time at which it is attained, is written in the top left corner of Figure
3.1b. (Alternatively, one could estimate the time of greatest change by eye, using the
curser. For this, one opens the ZTools menu, selects get coordinates with cursor,
moves the cursor to the point of change, and, after clicking the mouse button, the
coordinates appear in the MATLAB control window.)
22

Second: One may want to know the amount of change. For this, choose the option
Compare two rates (no fit) from the list of the Ztools menu in the cumulative number
window. A window will open, offering the opportunity for input of the end points of the
periods you wish to compare (Figure 3.2). After you activate the comparison, two
Figure 3.2: Window for selection of two periods for the comparison of seismicity rates. Instead of typing
in the end points, one may use the cursor to click at four points in the cumulative number plot. The two
periods need not be contiguous.
windows appear (Figure 3.3). One of them displays the normalized (per year) frequencymagnitude
distributions (FMD) of the two periods in cumulated and non-cumulated form
(Figure 3.3a). The other shows the FMDs in absolute values (Figure 3.3b). The periods
selected, and the symbols representing them, are given at the top of Figure 3.3a, together
with the rate change, which is –78% in the example shown.
Figure 3.3: Frequency-magnitude distribution for two periods for which we seek a comparison of the
seismicity rates. (a) normalized (per year), (b) not normalized. From these FMD plots, one cam judge in
what magnitude bands the rate change takes place.
23

In the example shown in Figure 3.3, the rate change evenly affects all magnitude bands.
This favors of the interpretation that the rate change is real. In addition, this change took
place at the time of the Landers M7.2 earthquake, at a distance of about 50 km to the east
of it. Thus, we accept the change as real and due to this main shock. The button
Magnitude Signature? was not activated in this case, because there was no reason to
attempt to interpret the change as artificial.
The regular FMD plot (Figure 3.3b) allows a comparison of the b-values during the two
periods. In this example, no change took place. The number of events used (n1 and n2),
as well as the two b-values (b1 and b2) are written into the plot. The probability,
estimated according to Utsu (1992), that the two samples come from the same,
indistinguishable population of magnitudes is p=29%, as shown in the top right corner of
Figure 3.3b.
Third: We may want to map the change of seismicity rate at the time of the Landers
earthquake, for which Figures 3.1 and 3.2 show a local example. This is done by opening
the ZTools menu in the window entitled seismicity map, and pointing to mapping zvalues.
From the several choices offered here, we select Calculate a z-value Map. This
command opens a window designed to define the parameters of the grid (Figure 3.4).
Figure 3.4: Window for the definition of the grid parameters to calculate a z-map.
Pressing the button ZmapGrid, places the cross hairs at our disposition. We now click
with the left mouse button on a sequence of points on the map, thus defining the apexes
of a polygon within which the z-values for rate changes will be calculated. The last point
is identified by clicking the right mouse button. Depending on the number of points in
the grid and the power of your computer, this calculation may take a while. At the end of
this calculation, a window opens (not shown here) in which you must enter a file name to
save this calculation of a z-map and which allows you to browse to the subdirectory
where you want to store your result. As soon as you enter the file name, a window
showing the z-menu opens (Figure 3.5).
For the example at hand, we press the button LTA under the heading Timecuts. As a
consequence, the next window opens (Input Parameters, Figure 3.6) that requires the
input of the beginning time and the duration of the time window, the rate within which
we wish to compare with the background rate, using the LTA definition. The window
may
24

Figure 3. 5: Z-menu window. Choosing LTA opens a window that asks for the definition of the time
window for which a comparison with the background rate is to be mapped.
be positioned anywhere within the observation period, and it may have any length that
fits. The background rate in LTA is defined by the sum of the rate before and after the
window selected for comparison. In our example, we defined the beginning time and the
duration of the window such that we compare the rate before with the rate after the
Landers main shock.
Figure 3.6: Input parameter window for calculating a Z-map.
The resulting zmap (Figure 3.7a) appears often in a distorted plot, because MATLAB
does not know that the axes should be geographical coordinates. For a final map of the
rate changes (Figure 3.7b), one can select the button Plot map in Lambert projection
using m_map in the ZTools menu of the Z-value Map window.
25

Figure 3.7: Z-map of the rate change at the time of the Landers earthquake. (a) Automatic scales, (b)
Lambert projection. Stars mark the epicenters of the Landers and Big Bear main shocks of June 1992.
Various tools are available to modify what is plotted and how it is plotted in the Z-map.
For example, the epicenters, which are plotted automatically have been suppressed in
Figure 3.7a, by selecting none from the choices of Symbol Type that appear, if one
selects the Symbol menu in the Z-Value-Map window. Also, the radius for volumes for
which the calculated Z-value is plotted, was limited by typing the number 25 into the
yellow button labeled MinRad(in km) in the Z-Value-Map window and pressing Go
afterward. This was done, because in areas where the seismicity is too low for a local
estimate of the rate change, it makes no sense to plot a value for Z that would be derived
from what occurred in relatively distant volumes.
The number of events, ni, used for calculating the Z-values, appears in a gray button in
the upper right corner, below the button Go. When one uses the select button in this
window, the number of events selected equals the number visible next to the label ni. If
one wishes to select a different number of events, one may replace the value in the ni
button and then press the set ni button.
Fourth: Finding the strongest rate change anywhere in time and space can be done by
calculating an alarm cube. From the ZTools menu in the seismicity map, select
zmapmenu. The window shown in Figure 3.5 opens. Clicking on Alarm opens the
window shown in Figure 3.8, in which one can define the window length of interest (7
years in our example) and the step width in units of bin length (14 days in our example,
which was defined in the calculation of the zmap).
Figure 3.8: Selection of alarm cube parameters.
Pressing the button LTA in the window shown in Figure 3.8 starts the calculation of the
alarm cube. The code slides a time window (of 7 years in the example) along the data at
26

each node and calculates for every position the Z-value comparing the rate in the window
to that outside of it. The resulting array of z-values is then sorted, and the location in
time and space of the largest z-values are displayed in the alarm cube (Figure 3.9a). The
location of these alarms are also shown in the seismicity map as three red dots (Figure
3.9b).
Figure 3.9: (a) In the alarm cube the x- and y-axes are the longitude and latitude, the z-axes is time.
Features like fault lines and epicenters of main shocks at the top and bottom are guides to find one’s
position. Red circles with blue lines following show the position in time and space of all occurrences of Zvalues
larger or equal to the value given in the yellow button labeled Alarm Threshold. (b) The locations
of the alarms selected in the alarm cube are shown as red dots.
The parameters that can be set in the alarm cube window include the maximum radius
allowed for samples to be displayed (MinRad(in km) at upper right; set at 25 km in the
example). More importantly, in the button labeled Alarm Threshold, one can type any
value for Z, above which one wishes to see all occurrences (called alarms).
Before setting a different alarm level than the one selected automatically, one may want
to inform oneself about the distribution of alarms. The distribution can be plotted by
selecting Determin # Alarmgroups (zalarm) from the ZTools menu in the alarm cube
window. Making this selection opens a small window into which one has to type the
minimum alarm level to be plotted and the step (not shown, selected as 6 and 0.1 in the
example). The resulting pot (Figure 3.10a) shows that in our example one alarm with
Z=9.1 towers in significance above the others. The next two alarm groups appear at a
value of 7.3 and a third appears at 7.1. In order to find the position of the three additional
27

Figure 3.10: (a) Number of alarm groups as a function of alarm level. Alarm groups are defined as a
group of contiguous nodes at which an alarm starts at the same time. (b) Fraction of the alarm volume as a
function of alarm level.
alarm groups, one could select 6.5 as the Alarm Threshold in the alarm cube window
and repeat the calculation. In that case, the locations of the additional nodes with alarms
above that level would appear in the seismicity map.
Alternatively, one may be interested in estimating the fraction of the study volume
occupied by alarms at a given level (Figure 3.10b). This may be accomplished by
selecting the option Determin Valarm/Vtotal(Zalarm) in the ZTools menu of the
alarm display window.
This alarm cube routine with its various options is especially useful for determining the
uniqueness of a seismic quiescence that one wishes to propose as a precursor. Many
authors proposing quiescence or other precursors do not show how often the proposed
phenomenon occurs at a similar significance at other times and in other locations than the
one possibly associated with a main shock. If the proposed precursor occupies the
number one position in the alarms, the phenomenon can be accepted as unusual. If,
however, the supposedly interesting anomaly occupies number 45, for example, in level
of significance, one has to accept that this phenomenon occurs often and most likely
appears associated with a main shock by chance.
Comparing two periods for rate changes
In addition to the old (i.e. ZMAP 3 – 5) tools for mapping seismicity rate changes,
ZMAP6 offers a new, simplified analysis procedure for evaluating seismicity rate
changes. This tool is most suited if you want to compare two specific periods. the
example analyzed here is again drawn form the Landers region. The dataset analyzed can
be downloaded form the dataset ftp site. (landerscat.mat).
Lets assume we are interested din the rate changes before and after the 1992.48 Landers
earthquake. We load the declustered Landers data set, and cut it at M1.6, in order to have
a fairly homogeneous dataset. From the map window, we now choose the option ZTools -
> Map seismicity rates - > Compare two periods. You now need to define four times, T1
– T4. Compared in the map will be period T1 – T2 with period T3-T4. Note that T2 and
T3 do not have to be identical. You also need to define the grid parameters.
28

Again the grid is chosen by using the left mouse button to define the perimeter, trying to
exclude low seismicity areas, and clicking the right mouse button for the final point. The
result of the computation will be displayed in a map.
Several different comparisons are computed at the same time and can be selected from
the Map menu.
1) z-values. Note that positive values by definition indicate seismicity rate decreases.
2) Change in percent.
3) beta values, using the definition [Reasenberg, 1992].
4) significance based on beta or z.
5) Resolution map: Radius of the selected circles.
The probability based on beta and z map will (once it is fully implemented) show a map
of the significance of a rate change, as compared to a random simulation. If you don’t
like the colormap, click on the plotedit option (the arrow next to the printer symbol), then
click within the plot, left mouse click, then properties, and select the color tab.
30

The map can be limited in range, plotted in lambert projection, or on top of topography
(if you have the mapping toolbox). To plot on top of topography, use the option Display
-> Plot on topo map. You will need to define several input parameters:
The data-aspect refers to the steepness of the topography. You may need to experiment a
little, since it depends on the specific region. You can again limit the range of the map.
Values above the selected range will be set to the maximum value. It is often sensible to
use a range symmetric around zero. If you have not yet imported a topography and
plotted it, you will be reminded to do so. This is done from the mapping window Ztools -
> plot topographic map -> and selecting the appropriate resolution. For the Landers
region, ETOPO2 results in rather poor maps, ETOPO30 looks just fine. Therefore, we
import the W140N40.HDR GTOPO30 t topography The final map is about the same
shown in the article by [Wyss and Wiemer, 2000]. The view can be rotated if desired –
but the labels and overlay may not be quite in the right place any longer. You may also
have to edit the light position, or add ambient light to get the right effect.
31

Articles in which tools discussed in this chapter were used:
Wiemer, S., and M. Wyss, Seismic quiescence before the Landers (M=7.5) and Big Bear
(M=6.5) 1992 earthquakes, Bull. Seismol. Soc. Am., 84, 900-916, 1994.
Wyss, M., and A.H. Martyrosian, Seismic quiescence before the M7, 1988, Spitak
earthquake, Armenia, Geophys. J. Int., 124, 329-340, 1998.
Wyss, M., K. Shimazaki, and T. Urabe, Quantitative mapping of a precursory quiescence
to the Izu-Oshima 1990 (M6.5) earthquake, Japan, Geophys. J. Int., 127, 735-743,
1996.
Wyss, M., A. Hasegawa, S. Wiemer, and N. Umino, Quantitative mapping of precursory
seismic quiescence before the 1989, M7.1, off-Sanriku earthquake, Japan, Annali di
Geophysica, 42, 851-869, 1999.
Wyss, M., and S. Wiemer, How can one test the seismic gap hypothesis? The Case of
repeated ruptures in the Aleutians., Pageoph, 155, 259-278, 1999.
33

Measuring Variations in b-value
CHAPTER IV
Precondition: You have already selected the part of an earthquake catalog that is
reasonably homogeneous in space, time and magnitude band. All inadequate parts of the
catalog and explosions have been removed. Also, you have culled the events with
magnitudes significantly below the Mc, such that the algorithm that finds Mc for the local
samples cannot mistakenly fit a straight line to a wide magnitude band below Mc.
Assumption: The b-value is relatively stable as a function of time. The first order
variations are expected as a function of space.
Mapping b-values: In the seismicity map window, open the ZTools menu and point to
Mapping b-values. From the sub-menu select Calculate a Mc and b-value map. The
window for Grid Input Parameters (Figure 4.1) will open. After defining the
parameters and pressing Go, the cross hairs appear. Define the apexes of the polygon
within which you want to calculate a b-value map, using the left mouse button, until the
last point, for which you use the right mouse button. Once the calculation is done, a
window opens that allows you to save the grid in a file and place it in the subdirectory of
your choice by browsing. (Bug: If an error results, type Prmap=0 in the MATLAB
window and repeat the calculation).
Figure 4.1: Grid Input Parameters for b-value maps. In addition to the total number, N, of events (or
radius) of the samples, one must enter a minimum number of events above the local value of Mc estimated
for each sample. Although one does not expect this number to drop below about 80% of N, one wants to
eliminate the possibility that it drops to an unacceptably small number.
The b-value map that appears is calculated using the weighted least squares method
(Figure 4.2b), but we use mostly the map calculated using the maximum likelihood
method (Figure 4.2a). These two figures should be approximately the same. Volumes
containing a main shock substantially larger than the rest of the events stand out with
lower b-values in the WLS map.
34

Figure 4.2: b-value maps of southern California for the period 1981-1992.42. (a) Maximum likelihood
method, (b) weighted least squares method.
The default scale for the b-values, with which the maps are presented, includes the
minimum and the maximum values that are found. However, it is usually better to select
limits that result in a map in which the blue and red are balanced. This is done by
selecting Fix color (z) scale from the Display menu in the b-value map window.
The first item of business when viewing a b-value map, is to check if the results can be
trusted. For this, one can click on any location of interest and view the FMD plot. For
example, the distribution at a location of high b-values is compared to that at a location of
low ones in Figure 4.3a. This plot was obtained by first selecting Select EQ in Circle
and then Select EQ in Circle overlay existing Plot from the Select menu in the b-value
map window. According to the Utsu test, the two distributions are different at a
significance level of 99% (p=0.1 in the top right corner).
35

Figure 4.3: Frequency-magnitude distributions for quality control, comparing distributions which were
judged as different in the maps. (a) Comparison of data sets with a high and a low b-value. (b)
Comparison of datasets with a high and a low Mc.
For all of the maps (Figures 4.2 and later) a histogram showing the distribution of the
values can be plotted by selecting Histogram in the menu of Maps of the b-value map
window. Figure 4.4, for example, shows the distribution of b-values based on the max.
L. method.
Figure 4.4: Histogram of the b-values that appear in Figure 4.2a.
Important additional options in the Maps menu of the b-value map window are the mag
of completeness (Figure 4.5a) map and the resolution map (Figure 4.5b). Using the
information already stored in the array computed for the b-value maps (Figure 4.2) one
can display the Mc. In the example (Figure 4.5a), the NE corner seems to show a higher
Mc than the rest of southern California. To check if the algorithm estimates Mc
correctly, one can open the Select menu and click first on Select EQ in Circle (placing
the cross hairs near a brown node of Figure 4.5a), and then clicking on Select EQ in
Circle (overlay existing plot), which results in the comparison of the two FMDs (Figure
4.3b). The Mc one would select by eye agrees with that estimated by the algorithm.
36

Figure 4.5: (a) Magnitude of completeness map with scale in magnitude. (b) Resolution map with scale in
kilometers.
The resolution map (Figure 4.5b) shows the radii necessary to sample the N events
specified in the grid computation (N=100 in the example). The radius is, of course,
inversely proportional to the a-value, a map of which can also be plotted using the Maps
menu (not shown). The resolution map is more informative for the analyst than the avalue
map, because it demonstrates how local the computed values are. From the
example (Figure 4.5b), one can see that volumes along some of the edges need large
radii, thus they are not worth studying. Also, near the center to the left side, one notices
relatively large radii because this area contains few earthquakes. As a result of restricting
the b-value plot to R

opens, showing the hypocenters in cross section and offering several buttons at the top as
choices for the next step (Figure 4.6b). Because we wanted to calculate b-values, we
selected the button at the left top labeled b and Mc grid. The two Figures 4.6 are ideal
for documenting the location and position of cross sections.
Figure 4.6: (a) Lambert projection of epicenters that appears when one chooses to work in a cross section
in the seismicity window. The earthquakes selected by the choice of endpoints and cross-section width are
highlighted. (b) The hypocenters in cross section selected in (a), with buttons at the top designed for
executing the next step (mapping the b-value, in our example).
By selecting the topmost button on the right in Figure 4.6b, one opens a window like
Figure 4.1 that allows the definition of the grid properties. After they have been selected,
cross hairs appear, which must be used to click in a polygon, as in the case of calculating
the b-value map, in the cross section, within which the b-values are to be calculated. The
result of this calculation is shown in Figure 4.7.
Figure 4.7: b-value cross section of a 20 km wide section of the San Jacinto fault defined in Figure 4.6.
In Figure 4.7, one has again the option of limiting the radius, and setting the number of
events in samples one may want to extract (top right corner). Also, this window has a
button labeled Maps that offers the same options as that button in the b-value map
window discussed before. Also, as in the b-value map, the Display button offers a
number of ways to modify the display.
38

Changes of b-values as a function of time may be identified by pointing to the option
Mc and b-value estimation from the ZTools menu in the cumulative number window
and selecting b with time from the possibilities offered. There are several other options
offered, such as b with depth and b with magnitude. After the selection, a window
opens (not shown) requesting input of the number of events to be used in the sliding time
window. In the example for which the result is shown in Figure 4.10, we selected a
volume around the Landers epicenter and used 400 events per b-estimate.
Figure 4.10: b-values in sliding time windows of 400 events as a function of time. WLS method above
and max L method below.
Both methods of estimating b-values show a brief decline of b after the Landers
earthquake, followed by a substantial increase. The result in Figure 4.10 does not
guarantee that the observed change of b is a change in time, because it could be that the
activity shifted from a volume of constant and low b-value to one of constant, but high
value. In order to determine which of the two possibilities was the case, ZMAP offers the
option Calculate a differential b-value Map (const R) in the sub-menu Mapping bvalues
that appears in the ZTools menu of the seismicity map. Selecting this option
brings up a window (not shown) in which the starting and ending times of the periods to
be compared need to be defined. After that, another window of the type of Figure 4.1
opens for defining the grid parameters. Once these are defined, the cross hairs appear
and the analyst has to click at the locations of the apexes, as usual.
39

Figure 4.11: b-value changes at the time of the Landers 1992 earthquake in its vicinity (R=15 km).
The map of the b-value changes at the time of the Landers earthquake reveals that
changes as a function of time have indeed taken place, but that they are positive as well
as negative (Figure 4.11). This demonstrates how important it is to map temporal
changes and not to rely on figures like Figure 4.10, showing b as a function of time only.
Articles in which tools discussed in this chapter were used:
Wiemer, S., and J. Benoit, Mapping the b-value anomaly at 100 km depth in the Alaska
and New Zealand subduction zones, Geophys. Res. Lett., 23, 1557-1560, 1996.
Wiemer, S., and S. McNutt, Variations in frequency-magnitude distribution with depth in
two volcanic areas: Mount St. Helens, Washington, and Mt. Spurr, Alaska,
Geophys. Res. Lett., 24, 189-192, 1997.
Wiemer, S., S.R. McNutt, and M. Wyss, Temporal and three-dimensional spatial analysis
of the frequency-magnitude distribution near Long Valley caldera, California,
Geophys. J. Int., 134, 409 - 421, 1998.
Wiemer, S., and M. Wyss, Mapping the frequency-magnitude distribution in asperities:
An improved technique to calculate recurrence times?, J. Geophys. Res., 102,
15115-15128, 1997.
Wyss, M., K. Nagamine, F.W. Klein, and S. Wiemer, Evidence for magma at
intermediate crustal depth below Kilauea's East Rift, Hawaii, based on anomalously
high b-values, J. Volcanol. Geotherm. Res., in press, 2001.
Wyss, M., D. Schorlemmer, and S. Wiemer, Mapping asperities by minima of local
recurrence time: The San Jacinto-Elsinore fault zones, J. Geophys. Res., 105, 7829-
7844, 2000.
Wyss, M., K. Shimazaki, and S. Wiemer, Mapping active magma chambers by b-values
beneath the off-Ito volcano, Japan, J. Geophys. Res., 102, 20413-20422, 1997.
Wyss, M., and S. Wiemer, Change in the probability for earthquakes in Southern
California due to the Landers magnitude 7.3 earthquake, Science, 290, 1334-1338,
2000.
40

CHAPTER V
Stress Tensor Inversions
Introduction
There have been significant changes in the way ZMAP performs stress tensor inversions
from ZMAP 5 to ZMAP6! To do the inversions, ZMAP now uses software by Andy
Michael USGS Menlo Park. The advantage is that that (1) inversions can now be
performed on a PC as well as on UNIX, precompiled Windows executables are included
in the ZMAP distribution; (2) The linearized inversion by Michael is much faster, taking
only seconds rather than minutes to complete. Results between the two methods have
been show to be equivalent for the most part [Hardebeck and Hauksson, 2001]. A first
application of the ZMAP tools to map stress can be found in [Wiemer et al., 2001].
When you use these codes included in ZMAP, please make sure to give credit to the
author of the code, Andy Michael. [Michael, 1984; Michael, 1987a; Michael, 1987b;
Michael, 1991; Michael et al., 1990].
On a PC, the inversion should work without the need to compile the software. On other
platforms, you will need to run the makefiles found in the ./external directory:
makeslick
makeslfast
makebtslw
This should compile the necessary executables.
Data Format
The input data format for stress tensor inversions remains identical to the ZMAP5
versions, and is compliant with the USGS hypoinverse output The data imported into
ZMAP needs to contain three additional columns:
column 10: Dip-direction
column 11: Dip
column 12: Rake
column 13: Misfit - fault plane uncertainty assigned by hypoinverse (optional)
Shown below is an example of the input data.
41

Table 1: Fault plane solution input data format
dip-direction dip rake misfit
230.0000 75.0000 137.3870 0.03
325.0000 90.0000 55.0000 0.04
145.0000 80.0000 -55.0000 0.12
140.0000 75.0000 50.0000 0.01
50.0000 50.0000 140.0000 0.10
45.0000 50.0000 -135.0000 0.03
Data import is only supported through the ASCII option. Select the EQ Datafile (+focal)
option when importing your data into ZMAP. Several precompiled datasets are available
through the online dataset web page (use the ‘online data’ button in the ZMAP menu).
Plotting focal mechanism data on a map
Using the Overlay -> Legend by … -> Legend by faulting type option from the
seismicity map, a map differentiating the various faulting styles of the individual
mechanisms by color can be plotted (Figure 5.1)
Figure 5.1 Map of the Landers regions. Hypocenters are color coded by faulting style.
42

Inverting for the best fitting stress tensor.
Stress tensor inversions can either be performed for individual samples, or on a grid. The
inversion for individual samples is initiated form the cumulative number window. Select
a subset from the seismicity window (generally 10 < N < 300). Select the ZTOOLS ->
Stress Tensor Inversion -> Invert using Michael’s Method option. The inversion is started
and will take several seconds, depending on the sample size and speed of your machine.
The inversion is performed by first saving the necessary data into a file, then calling
Michael’s inversion program unix(' slfast data2 ') to find the best solution. To estimate
the confidence regions of the solution, a bootstrap approach is used by Michael (unix(['
bootslickw data2 2000 0.5' ]); ). In the defaults setup, fault planes and auxiliary planes
are assumed equally likely to be the rupture plane (expressed by the 0.5 in the bootslickw
call). Results are displayed in a stereographic projection (Wulff net, Figure 2).
Figure 5.2: Output of the stress tensor inversion
The faulting type is determined based on Zoback’s (1992) classification scheme; the info
button will link to a web page describing the faulting styles. Seer Michael’s papers for
details on variance and Phi.
In addition, the stress tensor can be investigated as a function of time and depths.
Inversions will be performed for overlapping windows with a constant, user defined
number of events, and plotted against time or depth.
43

Figure 5.4: Stress tensor inversion results plotted as a function of time
Inverting on a grid
From the seismicity window, the ZTOOLS -> Map stress tensor option will open an input
window for the inversion on a grid.
The input structure is similar to the b- and z-value mapping. Grids node spacing is
degrees. You can either choose a constant number near each node, or a constant radius in
kilometer. When choosing the latter. it might be a good idea to set the minimum number
of events to a value above its default of zero (e.g., 10). A grid is defined interactively,
excluding areas of low seismicity (left mouse: new node; right mouse button: last node).
Results of the inversion are displayed in two windows: A map that sows the orientation
44

of S1 as a bar, and color codes the faulting style, and a map of the variance of the
inversion at each node, under laying again the orientation of S1 indicated by bars.
Figure 5.5: Stress tensor inversion results for the Landers region/ Th etop frame shows the orientatio of S1
(bars), differentiating various faulting regimes. The Bottom plot shows in addition the variance of the stress
tensor at each node.
Red areas are regions where only a poor fit to a homogeneous stress tensor could be
obtained. The Select -> Select EQ in circle option will plot the cumulative number at this
node, then perform an inversion and plot the results in a wulff net.
45

Figure 5.6: Typical inversion results for a ‘red’ region, i.e. high variance and a poor fit to a homogeneous
stress filed, and a blue region.
Plotting stress results on top of topography
A nice looking map of the variance and orientation of S1, plotted on top of topography,
can be obtained using the Maps -> Plot map on top of topography option from the
variance map. However, you must have access to the Matlab Mapping toolbox to use this
option, and you must have already loaded/plotted a topography map using the options
from the seismicity map. The script called to do the plotting in dramap_stress.m. It may
be necessary to change the script in order to adjust the labeling spacing, color map etc.
Note that the map cannot be viewed from a perspective different from straight above,
since the bars are all at one height of 10 km.
Figure 5.7: Variance map plotted on top of topography.
46

Using Gephart’s code
An alternative code to compute stress tensor was given by Gephart [Gephart, 1990a;
Gephart, 1990b; Gephart and Forsyth, 1984]. His code performs a complete grid search
of the parameter space. The ZMAP6 version of the code is essentially unchanged from
the ZMAP5 version, with the exception that now a precompiled PC version is also
available. The data input format is identical to the one for Michael’s inversion. Note that
significant differences between the two methods have been observed in special cases.
For UNIX or LINUX version, you need to precompiled a few files, that are located in the
external/src_unix directory. Check the INFO file in this directory for information on
compiling.
To initiate a stress tensor inversion, select the "Invert for stress tensor" option from the
Tools pull down menu of the Cumulative Number window. The dataset currently selected
in this window will be used for the inversion. The actual inversion is performed using a
Fortran code based on Gephart and Forsyth [1984] algorithm, and modified by Zhong
Lu.
The actual program is described and discussed by Gephart and Forsyth [1984], Gephart
(1990), [Lu and Wyss, 1996; Lu et al., 1997] and [Gillard et al., 1995]. Two main
assumptions need are made: 1) the stress tensor is uniform in the crustal volume
investigated; 2) on each fault plane slip occurs in the direction of the resolved shear
stress. In order to invert the focal mechanism data successfully for the direction of
principal stresses, one must have a crustal volume with faults representing zones of
weakness with different orientations in a homogeneous stress field. If only one type of
focal mechanism is observed, then the direction of the principal stresses would be poorly
constraint (modified from Gillard and Wyss, 1995)
47

Figure 5.8. Schematic representation of the misfit angle (Figure provided by Zhong Lu)
To determine the unknown parameters, the difference between the prediction of the
model and the observations needs to be minimized. This difference, is called the misfit,
and is defined as the minimum rotation about any arbitrary axes that brings the fault
plane geometry into coincidence with a new fault plane. A grid search over the focal
sphere is performed - at first with a 90-degree variance with 10 degree spacing
(approximate method) then with a 30-degree variance and 5 degree spacing. Each
inversion takes a significant amount of time to run, which depends mainly on the number
of earthquakes to be inverted. As a rule of thumb: 30 earthquakes take about 15 minutes
to be inverted on a SUN Sparc 20, about 3 minutes on a PC 1.7 GHz. Please wait until the
inversion is completed, do not attempt to continue using ZMAP. The inversion creates a
number of temporary files in the directory `~/ZMAP/external. The final result can be
found in the file `stress.out'
Table 2: Output of the stress tensor inversion in file `stress.out' and out95
S1
(az)
S1
(plun)
S2
(az)
S2
(plun)
S3
(az)
S3
(plun)
PHI R Misfit
13 46 5 314 76 201 -5.6 0.9 3.597
The ratio is defined as: .
For the definition of PHI, see Gephart (1990). The file out95 contains the entire gridsearch,
where each line is in the same format as shown in Table 2. To plot the best fitting
stress tensor (the one with the smallest misfit value), type `plot95' in the Matlab
command window. This will load the file plot95 and calculate the 95 percent confidence
regions using the formula (Parker and McNutt, 1980)
were n is the number of earthquakes used in the inversion and MImin the minimum
achieved misfit. All grid-points with a misfit MI

#define VARIANCE_30 30
change to:
#define VARIANCE_30 90
and re-compile (cc -o msiWindow_1 msiWindow_1.c)
The cumulative misfit method
Stress tensor inversions are time consuming, and the resulting tensor is not easily
visualized. To identify crustal volumes that satisfy one homogeneous stress tensor Lu and
Wyss (1995) and Wyss and Lu (1995) introduced the cumulative misfit method. The
misfit, f, for each individual earthquake can be summed up in a number of different ways,
for example along the strike of a fault or plate boundary. If the stress direction along
strike is uniforms within segments, but different from other segments, the cumulative
misfit will show constant, but different slope for each segment (Figure 84). We can
also study the cumulative misfit as a function of latitude, depth, time, or magnitude, and
try to identify segments with constant but different slope.
ZMAP allows taking the cumulative misfit method one step further: A grid (in map view
or cross-section) is used, and the average misfit of the n closest earthquakes in
an Euclidean sense is calculated. The distribution of this average misfit can be displayed
using a color representation. Maps of this type, calculated for a number of different
assumed homogeneous stress tensor can identify homogeneous volumes, which then can
be inverted using the stress tensor inversion method described earlier.
Figure 5.9. Schematic explanation of the cumulative misfit method. Changes in the slop of the cumulative
misfit curve (blue) indicate a change in the stress regime. Figure courtesy of Zhong Lu
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Figure 5.10. Input parameters for the misfit calculation
To initiate a cumulative misfit analysis, a reference stress model needs to be defined. The
misfit between the observed and the theoretical slip directions estimated based on the
reference stress model will then be calculated. The reference stress mode is defined by: 1)
Plunge of S1 or S3; 2) Azimuth of S1 or S3; 3) R value; and 4) Phi value. Hit `Go' to start
the analysis. Once the calculation is complete, a map will display the misfit f of each
individual event with respect to the assumed reference stress tensor. The symbol size and
gray shading represents the misfit: small and black indicate a small misfit, and large and
white symbols a large misfit.
Figure 5.11 Map of the individual misfit f to an assumed homogeneous stress field
Also displayed will be the cumulative misfit F as a function of Longitude (Figure 88).
Using the `Tools' button the catalog can be saved using the currently selected sorting, the
standard derivative z can be calculated to quantify a change in slope, and two segments
can be compared. Selecting the X-Sec button in the Misfit map will create a cross-section
view of the misfit f of each individual earthquake. Again, the size and color of the symbol
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depicts the misfit value f. Please note that in order to show this cross-section view a
cross-section must have been defined previously.
To calculate a map the grid spacing needs to be defined (in degrees) and the number of
earthquakes sampled around each grid-node. The distribution of average misfit values
will then be shown in a color image (Figure 89). A low average misfit will be indicated in
red, a high misfit in blue. A study by Gillard and Wyss (1995) showed that in many cases
average misfit values of F

Figure 5.14. Image showing the distribution of average misfit values F in map view. Red colors indicate a
low average misfit and thus good compliance with the assumed theoretical stress field. This map shows the
Parkfield segment of the san Andreas fault. The theoretical stress filed was given as (151 deg az, 2 deg
plunge, R=0.9, Phi = 1).
Figure 5.15. Image showing the distribution of average misfit values F in cross-section
view.
References
Gephart, J.W., FMSI: A FORTRAN program for inverting fault/slickenside and
earthquake focal mechanism data to obtain the original stress tensor, Comput. Geosci.,
16, 953-989, 1990a.
Gephart, J.W., Stress and the direction of slip on fault planes, Tectonics, 9, 845-858,
1990b.
Gephart, J.W., and D.W. Forsyth, An Improved Method for Determining the Regional
Stress Tensor Using Earthquake Focal Mechanism Data: Application to the San Fernando
Earthquake Sequence, Journal of Geophysical Research, 89, 9305-9320, 1984.
Gillard, D., M. Wyss, and P. Okubo, Stress and strain tensor orientations in the south
flank of Kilauea, Hawaii, estimated from fault plane solutions, Journal of Geophysical
Research, 100, 16025-16042, 1995.
Hardebeck, J.L., and E. Hauksson, Stress orientations obtained from earthquake focal
mechanisms: What are appropriate uncertainty estimates?, Bulletin of the Seismological
Society of America, 91 (2), 250-262, 2001.
Lu, Z., and M. Wyss, Segmentation of the Aleutian plate boundary derived from stress
direction estimates based on fault plane solutions, Journal of Geophysical Research, 101,
803-816, 1996.
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Lu, Z., M. Wyss, and H. Pulpan, Details of stress directions in the Alaska subduction
zone from fault plane solutions, Journal of Geophysical Research, 102, 5385-5402, 1997.
Michael, A.J., Determination of Stress From Slip Data: Faults and Folds, Journal of
Geophysical Research, 89, 11517-11526, 1984.
Michael, A.J., Stress rotation during the Coalinga aftershock sequence, Journal of
Geophysical Research, 92, 7963-7979, 1987a.
Michael, A.J., Use of Focal Mechanisms to Determine Stress: A Control Study, Journal
of Geophysical Research, 92, 357-368, 1987b.
Michael, A.J., Spatial variations of stress within the 1987 Whittier Narrows, California,
aftershock sequence: new techniques and results, Journal of Geophysical Research, 96,
6303-6319, 1991.
Michael, A.J., W.L. Ellsworth, and D. Oppenheimer, Co-seismic stress changes induced
by the 1989 Loma Prieta, California earthquake, Geophysical Research Letters, 17, 1441-
1444, 1990.
Wiemer, S., M.C. Gerstenberger, and E. Hauksson, Properties of the 1999, Mw7.1,
Hector Mine earthquake: Implications for aftershock hazard, Bulletin of the
Seismological Society of America, in press, 2001.
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Importing data into ZMAP
CHAPTER VI
ASCII columns – the most simple way
Importing data is often the first major hurdle that users have in running ZMAP. Within
ZMAP, data are stored internally in the Matrix ‘a’ in the following format:
Column 1 2 3 4 5 6 7 8 9
Longitude Latitude Year Month Day Magnitude Depth Hour Minute
The trick is to get your data into ‘a’. The most stable and easy solution is to create an
ASCII file with exactly these columns, separated by at least one blank or tab. Note that
zeros in column 4 and 5 will create errors. Western Longitudes are by convention
negative. Such a file could look like this:
-120.819 36.251 1980 1 1 3.7 6.60 2 9
-120.825 36.249 1980 1 1 3.6 7.80 2 9
-120.809 36.251 1980 1 2 1.5 6.54 0 54
-120.817 36.255 1980 1 2 2.8 6.44 10 39
-120.615 36.048 1980 1 3 0.5 4.58 12 19
-120.815 36.265 1980 1 4 1.3 4.97 19 49
-120.470 35.925 1980 1 7 0.8 4.62 7 39
-120.565 36.016 1980 1 9 0.9 6.48 1 51
-120.472 35.928 1980 1 9 2.9 10.35 17 54
-120.643 36.083 1980 1 10 0.9 3.19 22 54
-120.951 36.371 1980 1 15 1.9 8.44 2 39
-120.934 36.357 1980 1 16 2.0 5.87 10 2
You can test if it is an acceptable file by loading it into Matlab. Lets assume the file is
stored in the filename mydata.dat, then typing
load mydata.dat
should create a variable mydata in the workspace, with 9 columns and as many row as
earthquakes in you catalog. Quite frequently, you will encounter the following message:
??? Error using ==> load
Number of columns on line 6 of ASCII file C:\ZMAP6\out\park.dat
must be the same as previous lines.
You should check line 6 for inconsistencies and try again. Once this file can be loaded
into Matlab, there are three ways to load it into ZMAP:
• Using the ASCII file import option. From the ZMAP menu, select the “Create
or Modify *.mat file’ option, the “EQ Datafile” option ASCII columns.
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• First rename mydata into a: a = mydata; then start zmap and type: startwitha
• Use the ‘Import Filter’ option from the ZMAP menu, and select the ASCII
import filter.
To get your data into ASCII column format, you can use various tools. On a PC, I often
use textpad (http://www.textpad.com/). In block selection mode, you can delete columns,
and by copying and pasting a blank column, you can add blanks between dates. On a
UNIX workstation, cut and paste work well:
cut –c4-8 catalog.dat > year
cut –c9-10 catalog.dat > month
…
paste long lat year month … > mydata.dat
which will be a tab separated file that can be readily read into Matlab.
Other options to import your data: Writing your own import filters
A second option to import your data is to write your own import filter. A number of
existing filters are listed in the ./importfilters directory. When you choose “Import Data”
from the ZMAP menu, an window will list all available filters:
Select a filter and choose “Import”, then select the name of the file to be imported. The
filters generally will try to read the entire data matrix at once. If this fails, it will read line
by line and report lines that could not be imported.
To design a filter that fits you data format is relatively easy. Lets assume that you would
like to read in the SCEC DC earthquake catalog for 2001
(http://www.scecdc.scec.org/ftp/catalogs/SCEC_DC/2001.cat). The first few lines are
shown below.
2001/01/01 01:16:25.7 le 1.0 h 35.123 -118.538 5.2 A 9172298 0 0 0 0
2001/01/01 01:21:55.9 le 1.6 h 35.047 -119.083 10.9 A 9172300 0 0 0 0
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2001/01/01 01:58:37.0 le 1.3 h 34.082 -116.636 3.7 D 9172302 0 0 0 0
2001/01/01 02:19:45.3 le 2.3 l 33.300 -116.209 5.6 C 9608497 124 225 0 0
2001/01/01 03:06:16.6 le 2.8 l 34.060 -116.712 13.8 A 9608501 0 366 0 0
2001/01/01 03:10:31.3 le 2.1 l 34.057 -116.722 13.7 A 9608505 131 284 0 0
2001/01/01 03:22:45.0 le 2.9 l 35.701 -118.231 13.2 A 9608509 0 205 0 0
The critical lines of the filter scecdcimp.m are:
uOutput(k,1) = str2num(mData(i,41:48)); % Longitude
uOutput(k,2) = str2num(mData(i,34:39)); % Latitude
uOutput(k,3) = str2num(mData(i,1:4)); % Year
uOutput(k,4) = str2num(mData(i,6:7)); % Month
uOutput(k,5) = str2num(mData(i,9:10)); % Day
uOutput(k,6) = str2num(mData(i,26:28)); % Magnitude
uOutput(k,7) = str2num(mData(i,51:54)); % Depth
uOutput(k,8) = str2num(mData(i,12:13)); % Hour
uOutput(k,9) = str2num(mData(i,15:16)); % Minute
To modify this script for your data, you need to change these lines to fit your data format.
If the year, for example, would be in column 8:12, this would work:
uOutput(k,3) = str2num(mData(i,8:12)); % Year
The script will first try to convert all lines at once. If this fails, it will try again, reading
each line individually and ignoring lines with errors or inconsistencies. It will print out
the line number where the error occurred.
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CHAPTER VII
Tips an tricks for making nice figures
Most ZMAP figures are not publication or presentation quality right away. Below are
some ideas on how to 1) tweak the ZMAP figures within Matlab such that they look
nicer, 2) Export the figures out of Matlab, and 3) post-process them in various editing
software.
Editing ZMAP graphs
The edit options in Matlab have improved dramatically. While Matlab 5.3 had some
option, that were not very stable, Matlab 6 now offers a full array of editing option.
Therefore, I recommend strongly to use Matlab 6 whenever possible. I personally create
Figures mostly on a PC, because Editing tends to be more stable on a powerful PC than
on HP or SUN workstations, and because using copy & paste, progress can be made very
quickly.
Figure 7.1: Starting point of the edit
Lets start with a simple example: A cumulative number curve, comparing seismicity
above and below M1.5 in the Parkfield area. This plot was made using the ZTOOLS –
overlay another plot (hold) option, then The original ZMAP way (left) is ok for display
on the screen, but not useful for publication. The fonts are too small, there is no legend,
axes scales are not quite right, and lines should be gray. First we activate the Edit option
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y clicking on the arrow next to the small printer symbol. Now we can click on any
element and view/change its properties. Lets first change the lines. Select the line you
want to change with a left mouse click, then select the available options with a right
mouse. You can change some options right there, for more advanced options, like
MarkerType, you need to open the Properties menu.
Selecting the labels, we delete the title (park.mat) and change the size and position of the
axes to bold. We also increase the size of the star, and change it color.
Figure 7.3: First iteration
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Now lets change the axes setup. Select the main axes, and open the properties box.
We will change the axes font size, and Yaxes ticklabels. We could also change the axes
background color. By selecting the axes, and then unlocking in, we can resize the figure
aspect ratio to our liking, and relock the axes. Finally, we select the axes, and use the
‘Show legend’ option to plot a legens. Its axes can then be unlocked, moved. We also edit
the text in the axes by selecting it until a text edit cursor appears. Finally, we could
change the figure background color by using Property edit- Figure Menu (double click in
the figure, or use Edfit -> Figure properties). You might add Annotations using the “T”
option, or lines and arrows. Below is the final result:
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Figure 7.4. Final edited figures
Exporting figures from ZMAP
In Figure 3, we created a decent looking figure. What to do next depends largely on what
you need to do. The best option to make publication quality figures of simple graphs such
as figure 3 is to print the above figures into a postscript file. To do this, either use the
Print … button from the File Menu, and select the Print to file option (you need to have a
postscript printer driver installed to do this). Note that the output may not have the same
aspect ratio, unless you use the PageSetup Menu options “Use Screen size, ceneterd on
Page” or FixAspectRatio. You also want to select the right paper format (A4 or letter) to
avoid later complications.
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Alternatively, you can print from the command prompt in Matlab, using, for example:
print –dps –noui myplot.ps
The figure you want to print must be the active one. The –noui option avoid printing the
menus. See help print for details on drivers etc. The same PageSize setup applies.
In addition, you could keep a copy as a Matlab *.fig file. These Figures can be reloaded
into any Matlab session, and edited within Matlab. See hgsave and hgload for details.
Postscripts files can be printed well, converted into PDF, but they cannot always be
edited. On Unix workstations, IslandDraw does a good job opening simple postscript
figures from Matlab – but fails with interpolated color maps and too many points etc.
FrameMaker works will for printing and text editing, but postscript cannt be edited, just
resized. Postscript generally cannot easily be edited in Microsoft Word or PowerPoint,
but Designer or CoralDraw on a PC or Mac often works if the files are small.
If you Work on a PC, an alternative to postscript is the emf Format (enhanced Meta file).
You can get this either by using the File -> Export option, or by setting the Edit -> Copy
options to Windows metafile and then selecting “Copy Figure”. The clipboard content
can then be simply pasted into Word or Powerpoint. In Powerpoint (or other PC editing
programs), they can be ungrouped and edited.
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lets, for example, assume we want to give a colorful presentation using PowerPoint. The
options are almost limitless … but it does take some time.
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Working with interpolated color maps
Interpolated color maps in Matlab look nice, but the tend to be not exportable readily into
any program. For quick documentation, such as this document, the copy as bitmap and
paste option, or Alt PrintScreen options works out well. If a higher resolution is needed, I
often end up using the following approach: 1) Finalize the Figures as much as possible in
Matlab. 2) Export it to a jpeg file. The resolution can be set interactively when printing
from the command line:
print –djpeg –r300 –noui myfig
Also, If you want a dark background and Matlab defaults to white, try:
set(gcf,’Inverthardcopy’,’off’)
The resolution for Founts is OK at 300 dpi:
In PowerPoint, the imported Figure will be too large for a Page, but can be resized while
keeping the dpi resolution. You will often notice that thicker lines and bigger, bold fonts
work better. Make sure that the background in Matlab is the same you will use in the
slide, since it cannot easily be changed. To switch a Matlab figure from black to white
and vice versa, use the command line option: whitebg(gcf). In PowerPoint, it is then
readily possible to add elements on top, add captions or Figures numbers etc. These
figures are generally high enough quality for publication, if they have been save with at
least 300 dpi.
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