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Princeton University
Assimilation of Satellite Remote Sensing Data into Land Surface Modeling Systems
Ming Pan
Dept. of Civil and Environmental Engineering, Princeton University
Presented at the Graduate Seminar atDept. of Environmental Sciences, Rutgers University
April 5, 2006
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Land Surface Hydrologic Systems
Some characteristics of the land surface dynamic system:
Nonlinear
Non-Gaussian (e.g. rainfall)
Possibly discontinuous (e.g. snow)
Large variability in space
Complicated scaling behaviors
Variable Infiltration Capacity (VIC) Macro-scale Land Surface Model
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Water Budget in the Land Surface (and Atmosphere)
)( ETPCdt
dSq
a −−=
Knowledge of the water budget Knowledge of the water budget Knowledge of the water budget Knowledge of the water budget components and their relation to components and their relation to components and their relation to components and their relation to climate changes are of critical climate changes are of critical climate changes are of critical climate changes are of critical importance:importance:importance:importance:
• Drought/flood monitoring/prediction
• Water resources management
• Climate studies
(McCabe et al, 2003)
Atmospheric Water Budget:
Terrestrial (Land) Water Budget: QETPdt
dSl −−= )(
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Remote Sensing – Soil Moisture
VegetationEmission
Surface Reflection
Atmospheric Emission
Soil Emission
Radiometer
TMI
Mesonet
VIC
Vo l
umet
ric
soil
moi
s tur
e
LSMEM (Land Surface Microwave Emission Model)(Drusch, 2003; Gao et al., 2004)
TMI (TRMM Microwave Imager)
Polar orbit, 10.65 GHz, 25km resolution(only horizontally polarized component being used)
surface moisture , temperature, vegetation characteristics
brightness temperature
Retrieve S
M
Cal
cula
te T
b
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Remote Sensing – Evapotranspiration (ET)
SEBS (Surface Energy Balance System) (Su, 2002)
Input Output
Sensible HeatLatent Heat (ET)
MODIS sensors on AQUA/TERRA
LAI, land cover, albedo, emissivity,surface temperature,shortwave, etc.
Bulk Aerodynamics
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Remote Sensing – Rainfall
TRMM
NLDASGauge/Radar
TRMM (Tropical Rainfall Measurement Mission)
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Assimilation – combining in situ, models and remote sensing
Errors, intermittences, low costLargeP, ET, S, QRemote Sensing
Consistency (closure), biasesAny scale (::forcing)P, ET, S, QModeling
Accurate, continuous, scatteredSmallP, ET, S, QIn-situ Measurement
RemarkScale/CoverageVariables
Inability to close the water budget dSl/dt = P – ET – Q by observational approaches.
Data Assimilation
Benef
its
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Mathematic Statement of the Assimilation Problem: Filtering
(land model)
)ˆ(ˆ 1|11| −−− = kkkkk F xx
1|ˆ −kkx
Forcing Input
(emission model)
1|ˆ −kkx
Filterkz
)( 1| −= kkkk H xy )
updated kk|x̂
k=
k+
1
Truth
Unfiltered
Filtered
Dynamic System:
Observation:
The goal is to estimate:
Optimality Criteria: Minimum Variance (Least Squared Errors), Bayesian (conditional) mean, Maximum Posterior Probability, etc.
Examples: Kalman filter, extended Kalman filter, ensemble Kalman filter, particle filter, other Bayesian Monte-Carlo methods.
),( 11 −−= kkkk F vxx
),( kkkk H uxz =
],|[ 21| kkkk Estimate zzzxx K) =
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A Typical Monte-Carlo-based Data Assimilation System
Randomizer(Ensemble/Particle Generator)
VIC + LSMEM
Ensemble/Particle Filter
Water (Energy) BalanceConstrainer
MeteorologicalForcing Fields
Remote SensingObservations
Rei
nitia
lize
t= t
+ 1
forcings = {P(i), Ta(i), Rs
(i), Vwind(i)}
states/fluxes = {SM(i), Ts(i), Tb
(i), ET(i), Q(i), …}
updated states/fluexes
constrained states/fluexes
states/fluexes
forcings
Outputs
= statistical model
= physical model
Legend
Tb, ET
VIC LSMEM
TMI Tb, MODIS-SEBS ET
TRMM rainfall
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Data Assimilation Techniques
Ensemble Kalman Filter (EnKF) and Particle Filter (PF)
EnKF: modify ensemble members (s.t. they’re closer t o obs)
PF: preferentially weight/sample particles (closer t o obs, more weight)
Application of Equality Constraints (Water/Energy Balance)
Redistribute imbalance terms (“residues”) to various balance terms according to their uncertainty levels (error covariance).
An independent and separate step (like a “post-processor”): to work on top of any other filtering procedures
Copula Model for Observation Errors
A category of parametric probabilistic models for two/more random variables, which is more flexible than traditional Gaussian error models for allowing arbitrary marginal distributions and a large variety of parametric dependency structures.
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Filtering in Monte Carlo Fashion
Weighted Monte Carlo Sampling (PF)Unweighted (Equally-weighted)
Monte Carlo Sampling (EnKF)
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Particle Filter and Ensemble Kalman Filter
PF EnKFPF EnKFPF EnKFPF EnKF
ikk
ikk
ikk νKxx += −1||
)|(1ikk
ik
ik pww xz−∝
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Particle Filter and Ensemble Kalman Filter
Poor (recharge soil)Good (find a larger antecedent rainfall)Correcting “Dry Errors”
Nudging/PushingReweighting/ResamplingStrategy
Good (remove water directly)Very Poor (shutdown rain + raise ET)Correcting “Wet Errors”
LessMoreSample Points Needed
HighRelatively LowFiltering Efficiency
YesYesNon-additive Error
Sub-optimalYesNonlinearity
Sub-optimalYesNon-Gaussianity
EnKFParticle Filter
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Water Balance Constraining
1. Procedure (math) – perform an extra EnKF filtering step where the closure is a perfect observation
2. What it actually does – redistribute imbalance terms (“residues”) to various balance terms according to their uncertainty levels (error covariance).
3. Convenient to use – an independent and separate step (like a “post-processor”): to work on top of any other filtering procedures
Unconstrained Constrained
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TMI and VIC soil moisture
0 ~ 50 / 12.020 ~ 40 / 30.5Range / Mean (%)
2~3 overpasses/dayEvery hourAvailability
1~5cm10cmDepth
38km footprint0.125ºSpatial Resolution
TMIVIC
CDF Quantile Matching Joint Distribution
TMI versus VIC soil moisture (34.000, -98.000)
TM
I soi
l moi
stur
e
VIC soil moisture
TM
I soi
l moi
stur
e
VIC soil moisture
Regression
Soil moisture
Qua
ntile
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The Copula Approach for Bivariate distributions
FXY(x, y) = C( FX(x), FY(y) ) = C(u, v)
FXY, FX, and FY are the joint and marginal CDF’s, where u = FX(x) and v = FY(y)
C(u, v) is called the “copula” function.
• Two separated and independent components: (1) marginal distributions FX, and FY; (2) the copula function C(u, v)
• FX, and FY describe the behaviors of individual variables, and they can be fitted separately with difference probability models.
• Copula function C(u, v) characterizes the dependency structure between the two variables (a large pool of copula functions available)
• Generalizable to n-D
n: total number of samples, ncord: number of sample pairs varying in the same direction, ndiscord: number of sample data pairs varying in the opposite direction.
• Independent of marginal distributions FX, and FY
• Insensitive to outliers in data samples (robust!)
Measure of dependency/coherence – Kendall’s τ : τ = ( ncord – ndiscord) / 2n(n+1)
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Fitting a Copula Model
TMI versus VIC soil moisture T
MI s
oil m
oist
ure
VIC soil moisture
Fx(x) versus F Y(y), ττττ = 0.4398
Simulate F x(x) versus F Y(y), ττττ = 0.4398
Simulate TMI versus VIC soil moisture
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Copula-based Joint Distribution
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Advantages of Copula Approach
• Copula based models are a superset of some other methods, and thus more powerful and flexible than them, for example:
– Joint Gaussian Gaussian marginals + Gaussian copula;– CDF quantile matching Arbitrary marginals + Kendall’s τ = 1
• Copula-based joint distribution is parametric (as long as C(u, v), FX(x), and FY(y) are parametric), so analytical formula can be easily derived for conditional probability and conditional simulation can be done. This makes it possible to incorporate the copula model into filters like EnKFand PF.
More choices of marginal distributions
More choices of dependency structures
Kendall’s τ < 1, with uncertainty properly addressed
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Assimilation experiments at Selected Locations
TMI Tb, MODIS ET
none
none
Obs Assimilated
0.25ºTRMM rainfallAssimilation
0.25 ºTRMM rainfallOpen-loop
0.125 ºNLDAS (ground obs)Benchmark
GridForcing Data
Study Location: 5 Oklahoma Mesonet Stations
Testing Strategy: Benchmark + Open-loop + Assimilation Experiments
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Results
RMS Errors in Top Layer Soil Moisture (%) in Open-loop and Assimilation Runs
(Computed against the Ground Observations Driven Benchmark)R
MS
Err
or in
Top
Lay
er S
oil M
oist
ure
(%)
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Large Scale Applications
The Final Exam – Large Scale Applications with Real Satellite Data
Study Area: Red-Arkansas River Basin (~645,000 km2)
Climate and Vegetations: east-west gradient of decreasing rainfall and vegetation thickness
Study Period: July ~ August, 2003
Rainfall Forcing: TRMM rainfall and NLDAS ground observed rainfall (as a benchmark)
Remote Sensing Data: TMI Tb and MODIS ET
VIC Model Grid Size: 0.25 deg
VIC Time Step: 1 hour
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Large Scale Applications – Results
Compensate the missing rainfall in TRMM satellite data by monitoring the soil moisture through TMI Tb measurement
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Large Scale Applications – Results
Impact of Data Assimilation – Difference between the RMS Errors (Top Layer Soil Moisture) in Open-loop and Assimilation Runs Computed against the Ground Observations Driven Benchmark
-4.7 %0.00042160.0004026ET (mm/h)
0.03 %
1.7 %
6.7 %
Impact
0.000067340.00006736Total Runoff (mm/h)
0.010040.01021Precipitation (mm/h)
0.030160.03233Top soil moisture (mm)
AssimilationOpen-loop
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Summary
Assimilation of satellite data into land surface hydrologic models is a promising approach to utilize the remote sensing data which isincreasingly available.
The behaviors of both the land surface dynamic system and related radiative transfer processes make the assimilation a challenging task and also motivate us to develop more sophisticated procedures.
A number of techniques (statistical tools) are identified, proposed and tested to handle a variety of problems that arise during theassimilation of remote sensing data, and their potential in realpractice is well shown in the experiments performed over satellite data.
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The End
Thank you.
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Stop!!
back-ups from here on
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Outline
Introduction
•Land surface hydrologic systems, water and energy budget (1~2)
•Modeling the land surface hydrology (1)
•Observational techniques (1)
•Remote sensing data, techniques, retrievals (2~3)
•Data assimilation problem
•Flow chart of a typical DA, and a filtering procedure
•Difficulties
Traditional and new techs in probabilistic estimations:
EnKF vs PF (2~3)
Additive Gaussian and Copula (3)
Water/energy balance constrainer
Point validations
Large scale applications (Red-Arkansas)
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Flexibility of the Copula Model
Equivalent measurement error variances as functions of measurement valuesin the copula based error models fitted from NLDAS/VIC simulations and TMI satellite measurements.
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