Assimilation of Satellite Remote Sensing Data into …hydrology.princeton.edu/.../Rutgers_2006_DA.pdfPrinceton University Assimilation of Satellite Remote Sensing Data into Land Surface

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

2 Princeton University

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


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 −−= )(


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


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


Remote Sensing – Rainfall

TRMM

NLDASGauge/Radar

TRMM (Tropical Rainfall Measurement Mission)


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


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) =


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


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.


Filtering in Monte Carlo Fashion

Weighted Monte Carlo Sampling (PF)Unweighted (Equally-weighted)

Monte Carlo Sampling (EnKF)


Particle Filter and Ensemble Kalman Filter

PF EnKFPF EnKFPF EnKFPF EnKF

ikk

ikk

ikk νKxx += −1||

)|(1ikk

ik

ik pww xz−∝


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


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


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


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)


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


Copula-based Joint Distribution


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


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


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

(%)


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


Large Scale Applications – Results

Compensate the missing rainfall in TRMM satellite data by monitoring the soil moisture through TMI Tb measurement


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


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.


The End

Thank you.


Stop!!

back-ups from here on


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)


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.

Documents

Assimilation of Satellite Remote Sensing Data into …hydrology.princeton.edu/.../Rutgers_2006_DA.pdfPrinceton University Assimilation of Satellite Remote Sensing Data into Land Surface