Data%20Assimilation:%20Data%20assimilation%20seeks%20to%20characterize%20the%20true%20state%20of%20an%20environmental%20system%20by%20combining%20information%20from%20measurements,%20models,%20and%20other%20sources.

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• Data Assimilation Data assimilation seeks to
  characterize the true state of an environmental
  system by combining information from
  measurements, models, and other sources.
• Typical measurements for hydrologic/earth science
  applications
• Ground-based hydrologic and geological
  measurements (stream flow, soil moisture, soil
  properties, canopy properties, etc.)
• Ground-based meteorological measurements
  (precipitation, air temperature, humidity, wind
  speed, etc.)
• Remotely-sensed measurements (usually
  electromagnetic) which are sensitive to
  hydrologically relevant variables (e.g. water
  vapor, soil moisture, etc.)
• Mathematical models used for data assimilation
• Models of the physical system of interest.
• Models of the measurement process.
• Probabilistic descriptions of uncertain model
  inputs and measurement errors.

A description based on combined information
should be better than one obtained from either
measurements or model alone.
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State estimation -- System is described in terms
of state variables, which are characterized from
available information Multiple data sources --
Estimates are often derived from different types
of measurements (ground-based, remote sensing,
etc.) measured at different times and
resolutions. State variables may fluctuate over a
wide range of time and space scales -- Different
scales may interact (e.g. small scale variability
can have large-scale consequences) Spatially
distributed dynamic systems -- Systems are often
modeled with partial differential equations,
usually nonlinear. Uncertainty -- The models used
in data assimilation applications are inevitably
imperfect approximations to reality, model inputs
may be uncertain, and measurement errors may be
important. All of these sources of uncertainty
need to be considered in the data assimilation
process. The equations used to describe the
system of interest are usually discretized over
time and space -- Since discretization must
capture a wide range of scales the resulting
number of degrees of freedom (unknowns) can be
very large.
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• State-space concepts provide a convenient way to
  formulate data assimilation problems. Key idea
  is to describe system of interest in terms of
  following variables
• Input variables -- variables which account for
  forcing from outside the system or system
  properties which do not depend on the system
  state.
• State variables -- dependent variables of
  differential equations used to describe the
  physical system of interest, also called
  prognostic variables.
• Output variables -- variables that are observed,
  depend on state and input variables, also called
  diagnostic variables.

Classification of variables depends on system
boundaries
Atmosphere
Atmosphere
Precip.
Precip.
ET
ET
Land
Land
System includes coupled land and atmosphere --
precipitation and evapo-transpiration are state
variables
System includes only land, precipitation and
evapo-transpiration are input variables
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Components of a Typical Hydrologic Data
Assimilation Problem
The data assimilation algorithm uses specified
information about input fluctuations and
measurement errors to combine model predictions
and measurements. Resulting estimates are
extensive in time and space and make best use of
available information.
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• When models are discretized over time/space there
  are two sources of output measurement error
• Instrument errors (measurement device does not
  perfectly record variable it is meant to
  measure).
• Scale-related errors (variable measured by device
  is not at the same time/space scale as
  corresponding model variable)

3.5
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Large-scale trend described by model
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3
Instrument error
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2.5

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True value
2
Scale-related error

0
1.5
Measurement
-1
1
-2
100
101
102
90
95
100
105
110
115
120
When measurement error statistics are specified
both error sources should be considered
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Zi z1, z2, , zi Set of all measurements
through time ti
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Downscaling Characterize system at scales
smaller than output measurement resolution
Upscaling Characterize system at scales larger
than output measurement resolution
States (y1 y4)
Measurement (z1 )
Measurements (z1 ...z4)
State (y1)
Downscaling and upscaling are handled
automatically if measurement equation is defined
approriately
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Characterizing Uncertain Systems
What is a good characterization of the system
states and inputs, given the vector Zi z1,
..., zi of all measurements taken through
ti? The posterior probability densities p(y Zi)
and p(u Zi) are the ideal estimates since they
contain everything we know about the state y or
input u given Zi.
In practice, we must settle for partial
information about this density

• Variational DA Derive mode of py(t) Zi by
  solving batch least-squares problem.
• Sequential DA Derive recursive approximation of
  conditional mean (and covariance?) of py(t) Zi

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The Variational/Batch Approach
The state equation is often incorporated as a
constraint, using adjoint methods.
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Sequential methods are designed to propagate and
update the conditional pdf in a series of
discrete steps
In practice various approximations must be
introduced.
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A common approximation is to assume that the
conditional PDF is multivariate Gaussian. The
update for conditional mean has the form
K weights measurements vs. model predictions
Some common approximations Direct Update
forced to equal measurements where
available, insertion interpolated from meas.
elsewhere Nudging K empirically selected
constant Optimal K derived from assumed (static)
covariance Interpolation Extended K
derived from covariances propagated with a
linearized Kalman filter model, input
fluctuations and measurement errors must be
additive. Ensemble K derived from a ensemble of
random replicates propagated Kalman filter with
a nonlinear model, form of input fluctuations and
measurement errors is unrestricted.
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Example -- Microwave Measurement of Soil Moisture
L-band (1.4 GHz) microwave emissivity is
sensitive to soil saturation in upper 5 cm.
Brightness temperature decreases for wetter
soils. Objective is to map soil moisture in real
time by combining microwave meas. and other data
with model predictions (data assimilation).
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SGP97 Experiment - Soil Moisture Campaign
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Typical precipitation events
Plan View Estimation pixels (large) Microwave
pixels (small)
ESTAR observation
0.8 km
0.8 km
4.0 km
170 6/19/97
Vertical Section Soil layers differ in
thickness Note large horizontal-to-vertical scale
disparity
For problems of continental scale we have 105
est. pixels, 105 meas, 106 states,
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NOAH soil class
NOAH vegetation class
Meteor. Stations
RTM Inputs
Clay fraction
El Reno
0
2
4
6
8
0
2
4
6
8
10
12
NOAH Inputs
0
0.05
0.1
Estimation region 50 by 200 km (12 by 50 pixels
4 km on a side)
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Brightness Temp. and Precip Time Series El Reno
Brightness temp. deg. K.
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Moisture Content and Precip. Time Series El Reno
Moisture content
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Extended Kalman filter

• Can be adapted for real time or smoothing
  problems
• Provides info. on estimation accuracy
• Computationally demanding, limited capability to
  deal with model errors -
• Linearization approximation may be poor, tends to
  be unstable -

Ensemble Kalman filter

• Well-suited for real time applications, not
  optimal for smoothing problems /-
• Provides information on estimation accuracy
• Very flexible, modular, able to accommodate wide
  range of model error descriptions
• No need for adjoint model or for linearizations
  or other approximations during propagation step
• Approach is robust and easy to use
• Update assumes states are jointly normal
• Can be computationally demanding