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Application of Multiscale Modeling
for Characterizing Variability in
Hydrologic Processes
Praveen Kumarl
Abstract .-We describe a multiscale modeling framework applicable
for a wide range of hydrologic processes. The foundation for this work
has been laid by Basseville et al. [I9921 and Chou et al. [1994]. Their
development is based on treating the scale parameter akin to time,
such that description at a particular scale captures the features of
the process up to that scale that are relevant for the prediction of
finer scale features, as in the notion of state-space approaches. We
show the applicability of this model for modeling and estimation of a
broad range of multiscale phenomena in hydrology. The methodology
is illustrated through an example using a soil-moisture field obtained
using remote sensing observations.
INTRODUCTION
In varied guises throughout hydrologic science we encounter questions concerning the quantitative relationship between processes occurring at disparate
spatial and temporal scales. These questions are mostly triggered by our need
to understand, model and predict hydrologic processes over a broad range
of spatial and temporal scales. Questions about multiscale manifestation of
processes are at the heart of contemporary hydrologic research and answers
to these questions will significantly advance our efforts in understanding the
dynamics of the large scale hydrologic cycle.
Over the last several years there has been significant effort devoted to organizing field experiments in combination with remote sensing measurements,
such as FIFE (Sellers et al., 1992) and SIR-C [Stofan et al., 19951, to improve
our understanding of surface hydrologic and ecological processes and their impact on the atmosphere. In order to improve our understanding of a process
at a particular scale we need t o make measurements at that scale. Often it
is difficult to decide the scale of significance and the strategy adopted is to
make measurements at several scales with fine scale measurements of limited
coverage embedded within coarse scale measurements of larger coverage (see
Figure 1). Further analysis is performed to iptegrate the information across
scales, to understand what area-averaged fluxes represent in conjunction with
'Department of Civil Engineering, University of Illinois, Urbana, Illinois 61801
99
point fluxes, and what physical controls on area-averaged fluxes need to be
considered by a prognostic model [Smith et al., 19921.
Coarse Scale Grid
Figure 1: Schematic showing an idealized embedding of fine scale measurement
grid of limited coverage within a coarse scale grid of larger coverage for a two
level scheme.
Different techniques exist for analyzing observations obtained at different
resolutions and are well explained in Daley [1991]. These range from simple function fitting and stochastic interpolation (univariate and multivariate)
schemes to four dimensional data assimilation (FDDA). As indicated by Dale y
[1991], the essence of a good scheme is to account for the following factors
simultaneously :
1. Observations have errors that can be temporally and spatially correlated
with each other.
2. The process state variables are related to each other by the governing
laws and are not independent.
3. There can be large data voids where no direct observations exist.
An ideal algorithm should filter observational noise, variance associated with
scales too small to be properly resolved by the sampling scheme, and fluctuations that do not satisfy the governing laws.
Although multivariate statistical analysis or FDDA address these issues,
they do not include the crucial element of scale as an explicit parameter in the
formulation. This causes severe limitations in our ability to interpret the behavior of the phenomenon at different scales and the influence of the variability
of one process at a small scale on the variability of a related process at a larger
scale, except as a gross aggregate. For examples, Several studies (Mahrt [1987],
Wetzel and Chang [1988], Avissar and Piehe [1989]) have indicated that grid
scale flux averages and sub-grid flux variance are both critical in characterizingthe response of the atmosphere to surfaces fluxes. In addition no multiscale
modeling scheme is immediately available from these formulations. In order
to address these issues and more effectively utilize the observations obtained
from the hierarchical sampling scheme (see Figure 1) to formulate conjectures
and develop models we need an analysis-modeling-assimilation technique that
is inherently hierarchical i n scale.
Wavelet transforms have been significantly exploited in geophysics for the
analysis of processes to infer their multiscale variability. They, however, are
not easily amenable as a modeling tool. Motivated by the scale to scale hierarchical decomposition of wavelet transforms, significant advances in multiscale
modeling have been made by Basseville et al. [I9921 and Chou et al. [1994].
The key to their development is the treatment of the scale parameter akin to
time, such that description at a particular scale captures the features of the
process up to that scale that are relevant for the prediction of finer scale features, as in the notion of state-space approaches. These developments provide
a natural setting not only for analysis and modeling of multiscale phenomena
but also for multiscale data assimilation.
This paper attempts to provide more visibility to this multiscale modeling
tool in the geophysics community through illustration using a simple example.
In the next section, we briefly review the modeling framework and then provide
an application to modeling soil moisture fields and estimating missing values.
MULTISCALE STATE-SPACE MODEL
The following description briefly summarizes the basics of the ideas developed
by Basseville et al. [I9921 and Chou et al. (19921. Consider the problem of
disaggregating a random field (in two dimensions) from coarse to fine resolution. At the coarsest resolution the field will be represented by a single value
(see Figure 2). At the next resolution there will be four values and in general,
at the mth resolution we obtain 4" values. The values of the random field can
be described on the index set (m, i, j ) where m represents the resolution and
(i, j ) the location index. The scale to scale decomposition can be schematically
depicted as a tree structure (quadtree for two-dimensional processes) as shown
in Figure 3. To describe the model let us use an abstract index X t o specify
the nodes on the tree and let y X specify the parent node of X (see Figure 3).
Then the multiscale stochastic process can be represented as
Resolution 1
,
,
I
I
,
I
,
I
Figure 2: The structure of a multiscale random field is shown. The values
at various grid locations (i, j ) are given as x,(i, j ) where m is the resolution
index. At the coarsest resolution (m = O), the field is represented by a single
value or state vector, and generally at the mth resolution there are 4" state
vectors (scale and resolution are inversely related).
A(X) and B(X) are matrices of appropriate sizes. We can assume that Xo 5
X(O) and W are normally distributed with known covariance matrices, i.e.,
where I is the identity matrix. Interpreting the states at a given level of the
tree as a representation of the process at one scale, we see that equation (1)
describes the evolution of the process from coarse to fine scale. The term
A(X)X(-IX) represents the interpolation or prediction down to the next finer
level and B(X)W(X) represents new information added as the process evolves
from one scale to the next. The covariance PAG E [ x ( X ) X ~ ( X ) ]
of the state
at node X evolves as
Notice that the form of equations (1) and (4) do not change whether we deal
with one-dimensional or multi-dimensional processes.
Figure 3: Multiscale decomposition. The abstract index X refers to a node in
the tree and yX refers to the parent node.
A particular form of model (1) is already popular as disaggregation model
in hydrologic time series analysis [see Valencia and Schaake, 19731. Equation
( I ) ,however, provides a more general modeling framework. For instance, it is
possible to choose A(X) and B(X) such that ( 1 ) leads to a fractal model. For
example, the model
X ( A ) = X(yX)
+ B ~ - O lo"
W(X)
where B is a constant
(5)
leads to fractal processes with spectrum l / f P [Chou et al., 19921 (P = 2
corresponds to Brownian motion).
Multiscale stochastic models can be used to make estimates of processes
from noisy measurements and for assimilation of data obtained at different
resolutions. If we are given noisy measurements Y(X)of the process X ( X ) we
can develop an estimation problem using the combination of equation (1) and
the following equation
-
where V ( X ) N ( 0 , R(X)). The Matrix C(X) can specify, in a very general
way, measurements taken at different spatial locations and at different scales.
R(X) specifies the covariance of the measurement errors V(X). Equations (1)
and (6) can be solved jointly to obtain estimates of X ( X ) . Notice that this is
a very attractive technique as it enables us to combine estimation and filtering
while exploiting hierarchical measurements at different scales. Also, note that
coarse scale observations provide the "context" or the background field for the
next finer scale, i.e., scale plays a role akin to time in existing data assimilation
schemes.
APPLICATION TO A SOIL MOISTURE RANDOM FIELD
We describe the applicability of the multiscale model for modeling a soil moisture field and estimating missing values using a fractal prior model. The data,
collected during the Washita' 92 Experiment [Jackson et al. 19931, were obtained using ESTAR - Electronically Steered Thinned Array Radiometer images and converted to volumetric soil moisture content. These data sets,
which correspond to several days of daily measurements of soil moisture in the
top 5 cm layer, show scaling or fractal characteristics [Rodriguez-Iturbe et a[.,
19951. We study the field obtained on the second day of measurement (June
11, 1992) by extracting a sub-field with grid dimension 64 x 64 (Figure 4), from
the original of 228 x 93. As is evident from the Figure there are several missing
values, indicated by dark patches, either due to the presence of township or
roads, or measurement errors. For this study we do not distinguish between
these and treat all as missing values due to measurement errors.
We use the results of analysis presented by [Rodriguez-Iturbe et al., 19951
to estimate the parameter /? = 0.79 in equation 5. The multiscale model we
use is give by the equations:
Y(X) =
{
:(A)
+ V(h)
if 2' = 64
otherwise,
N(0, o2 = 25) gives the noise distribution at the finest
where V(X = 6)
scale. The algorithm first estimates the coarser scale values from the given
finer scale values, i.e., going upward in the tree in Figure 3, and after reaching
the root node proceeds downward to predict the estimates at all grid points
and all scales, thereby providing estimates of the missing values. The results
of the multiscale estimation are given in Figure 4. As is evident, the algorithm
does an extremely good job of estimating the missing values. The estimation
error are within the range of the data values.
The above is a simple example of the applicability of the multiscale model,
although the model is vary general. In particular the state at any node can
be described as a vector. For example, one may represent the state at any
node by a vector whose elements are the components of the surface energy
balance equation. The model can then be used to estimate the energy balance
at various scales by combining point and remote sensing estimates. These
issues are currently under investigation.
Original Field
Estimated FieM
Estimated Error
Estimated Obsewed Field
-
Figure 4: This figure shows the original volumetric soil moisture field (top-left)
and the estimated field using a fractal prior model. The lower row shows the
estimation error and the difference field.
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BIOGRAPHICAL SKETCH
Praveen Kumar is an Assistant Professor in the Department of Civil Engineering at the University of Illinois in Urbana-Champaign, Illinois. He obtained his PhD from the University of Minnesota in 1993. His research interests
include analysis and modeling of multiscale variability in surface hydrologic
processes, large scale hydrologic processes, and hydrometeorology and hydroclimatology.