Sensors 2012, 12 16312

py x

y x

∑

y x

where is the number of particles and | x is the likelihood of the observations given the modeled

states for a particle:

y x 1

exp 1 y

2 R 2R H x (22)

For the definition of R and H , see Section 2.1 on EnKF. In these expressions, it was assumed that

all observations have the same measurement error variance. For the multivariate case, the expression

for the likelihood of the observation modifies to:

y x 1

exp 1

2 |R | 2 y H x T R

y H x (23)

We see in this expression that the measurement error variances are in matrix notation, which

acknowledges that different measurement types will be associated with different uncertainties.

Measurement errors for different observations can also be correlated in space, as could be the case for

remote sensing data. The uncertainty of the different (types of) observations affects the weighting of

the particles.

5.2. Applications

Although multivariate DA seems like a relatively straightforward extension of univariate DA, most

studies in terrestrial systems assimilate only one data type. The complication of MVSS DA is not so

much of an algorithmic nature, but is related to the specification of the measurement uncertainty for all

data types involved. If different data types are assimilated, the correct weighting of the different pieces

of information becomes very important for the efficiency of the procedure. The following discussion of

the papers that deal with MVSS DA is organized according to the application area, focusing on

developments during the last decade and on EnKF, PF and VAR.

5.2.1. Groundwater

In groundwater hydrology, sequential DA focused from the outset on jointly updating states and

parameters by assimilating piezometric head data using an augmented state vector approach. The work

of Chen and Zhang was among the first in this area [74]. Initial work on multivariate DA considered

the joint assimilation of time series of piezometric heads and conditioning to hydraulic conductivity

data. In such publications, information on hydraulic conductivity was assimilated in the first step and

later preserved in the DA [151–153]. The joint assimilation of measurements for more than one state

variable is less frequently reported in the literature. Liu et al. [154] and Nowak [151] assimilated both

piezometric head and concentration data, but they provided few details about the assimilation of the

concentration data and the (expected) non-Gaussian concentration distribution. Li et al. [155] provided

a more detailed analysis of the value of additional concentration data, and found that the joint

assimilation of head and concentration data gave much better results than the assimilation of head data

only. They used the classic EnKF and a very large ensemble size (1,000 realizations). In addition to

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