Assimilation of Satellite Remote Sensing Data into Land Surface Modeling Systems

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1 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 at Dept. of Environmental Sciences, Rutgers University April 5, 2006 Princeton University

2 Land Surface Hydrologic Systems Variable Infiltration Capacity (VIC) Macro-scale Land Surface Model 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 2 Princeton University

3 Water Budget in the Land Surface (and Atmosphere) Knowledge of the water budget components and their relation to climate changes are of critical importance: Drought/flood monitoring/prediction Water resources management Climate studies (McCabe et al, 2003) Atmospheric Water Budget: ds dt a = C q ( P ET ) Terrestrial (Land) Water Budget: ds l dt = ( P ET ) Q 3 Princeton University

4 Remote Sensing Soil Moisture TMI (TRMM Microwave Imager) Polar orbit, GHz, 25km resolution (only horizontally polarized component being used) LSMEM (Land Surface Microwave Emission Model) (Drusch, 2003; Gao et al., 2004) brightness temperature Radiometer Calculate Tb Retrieve SM Atmospheric Emission Volumetric soil moisture VIC TMI Mesonet surface moisture, temperature, vegetation characteristics Soil Emission Surface Reflection Vegetation Emission 4 Princeton University

5 Remote Sensing Evapotranspiration (ET) SEBS (Surface Energy Balance System) (Su, 2002) Input Output Sensible Heat Latent Heat (ET) MODIS sensors on AQUA/TERRA Bulk Aerodynamics LAI, land cover, albedo, emissivity, surface temperature, shortwave, etc. 5 Princeton University

6 Remote Sensing Rainfall TRMM (Tropical Rainfall Measurement Mission) TRMM NLDAS Gauge/Radar 6 Princeton University

7 Assimilation combining in situ, models and remote sensing In-situ Measurement Modeling Remote Sensing Variables P, ET, S, Q P, ET, S, Q P, ET, S, Q Scale/Coverage Small Any scale (::forcing) Large Remark Accurate, continuous, scattered Consistency (closure), biases Errors, intermittences, low cost Benefits Inability to close the water budget ds l /dt = P ET Q by observational approaches. Data Assimilation 7 Princeton University

8 Mathematic Statement of the Assimilation Problem: Filtering Forcing Input Dynamic System: x k = Fk ( x k 1, v k 1) xˆ k k 1 = Fk (ˆ xk 1 k 1) (land model) ˆ x k k 1 ) y k H ( x 1) = k k k (emission model) Observation: The goal is to estimate: z k = H k ( xk, u k ) ) x k k = Estimate[ x k z1, z 2Kz k ] k = k + 1 z k ˆ Filter x k k 1 Optimality Criteria: Minimum Variance (Least Squared Errors), Bayesian (conditional) mean, Maximum Posterior Probability, etc. updated ˆx k k Examples: Kalman filter, extended Kalman filter, ensemble Kalman filter, particle filter, other Bayesian Monte- Carlo methods. Truth Unfiltered Filtered 8 Princeton University

9 A Typical Monte-Carlo-based Data Assimilation System Meteorological Forcing Fields TRMM rainfall Randomizer (Ensemble/Particle Generator) VIC + LSMEM forcings VIC LSMEM Remote Sensing Observations TMI Tb, MODIS-SEBS ET T b, ET Ensemble/Particle Filter Water (Energy) Balance Constrainer Outputs states/fluexes updated states/fluexes constrained states/fluexes Reinitialize t = t + 1 = statistical model = physical model Legend forcings = {P (i), T a (i), R s (i), V wind (i) } states/fluxes = {SM (i), T s (i), T b (i), ET (i), Q (i), } 9 Princeton University

10 Data Assimilation Techniques Ensemble Kalman Filter (EnKF) and Particle Filter (PF) EnKF: PF: modify ensemble members (s.t. they re closer to obs) preferentially weight/sample particles (closer to 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. 10 Princeton University

11 Filtering in Monte Carlo Fashion Weighted Monte Carlo Sampling (PF) Unweighted (Equally-weighted) Monte Carlo Sampling (EnKF) 11 Princeton University

12 Particle Filter and Ensemble Kalman Filter PF wki wki 1 p ( z k x ik ) 12 EnKF x ik k = x ik k 1 + K kν ki Princeton University

13 Particle Filter and Ensemble Kalman Filter Strategy Non-Gaussianity Nonlinearity Non-additive Error Filtering Efficiency Sample Points Needed Correcting Dry Errors Correcting Wet Errors Particle Filter Reweighting/Resampling Yes Yes Yes Relatively Low More Good (find a larger antecedent rainfall) Very Poor (shutdown rain + raise ET) EnKF Nudging/Pushing Sub-optimal Sub-optimal Yes High Less Poor (recharge soil) Good (remove water directly) 13 Princeton University

14 Water Balance Constraining Unconstrained Constrained 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 14 Princeton University

15 TMI and VIC soil moisture TMI versus VIC soil moisture (34.000, ) VIC TMI TMI soil moisture Spatial Resolution Depth Availability Range / Mean (%) 0.125º 10cm Every hour 20 ~ 40 / km footprint 1~5cm 2~3 overpasses/day 0 ~ 50 / 12.0 VIC soil moisture CDF Quantile Matching Joint Distribution TMI soil moisture Regression Quantile VIC soil moisture Soil moisture 15 Princeton University

16 The Copula Approach for Bivariate distributions F XY (x, y) = C( F X (x), F Y (y) ) = C(u, v) F XY, F X, and F Y are the joint and marginal CDF s, where u = F X (x) and v = F Y (y) C(u, v) is called the copula function. Two separated and independent components: (1) marginal distributions F X, and F Y ; (2) the copula function C(u, v) F X, and F Y 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 Measure of dependency/coherence Kendall sτ: τ = ( n cord n discord ) / 2n(n+1) n: total number of samples, n cord : number of sample pairs varying in the same direction, n discord : number of sample data pairs varying in the opposite direction. Independent of marginal distributions F X, and F Y Insensitive to outliers in data samples (robust!) 16 Princeton University

17 Fitting a Copula Model TMI versus VIC soil moisture F x (x) versus F Y (y), τ = TMI soil moisture VIC soil moisture Simulate F x (x) versus F Y (y), τ = Simulate TMI versus VIC soil moisture 17 Princeton University

18 Copula-based Joint Distribution 18 Princeton University

19 Advantages of Copula Approach Copula based models are a superset of some other methods, and thus more powerful and flexible than them, for example: More choices of marginal distributions More choices of dependency structures Joint Gaussian Gaussian marginals + Gaussian copula; CDF quantile matching Arbitrary marginals + Kendall s τ = 1 Kendall s τ < 1, with uncertainty properly addressed Copula-based joint distribution is parametric (as long as C(u, v), F X (x), and F Y (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 EnKF and PF. 19 Princeton University

20 Assimilation experiments at Selected Locations Study Location: 5 Oklahoma Mesonet Stations Testing Strategy: Benchmark + Open-loop + Assimilation Experiments Forcing Data Grid Obs Assimilated Benchmark NLDAS (ground obs) º none Open-loop TRMM rainfall 0.25 º none Assimilation TRMM rainfall 0.25º TMI T b, MODIS ET 20 Princeton University

21 Results RMS Errors in Top Layer Soil Moisture (%) in Open-loop and Assimilation Runs (Computed against the Ground Observations Driven Benchmark) RMS Error in Top Layer Soil Moisture (%) 21 Princeton University

22 Large Scale Applications The Final Exam Large Scale Applications with Real Satellite Data Study Area: Red-Arkansas River Basin (~645,000 km 2 ) 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 T b and MODIS ET VIC Model Grid Size: 0.25 deg VIC Time Step: 1 hour 22 Princeton University

23 Large Scale Applications Results Compensate the missing rainfall in TRMM satellite data by monitoring the soil moisture through TMI T b measurement 23 Princeton University

24 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 Open-loop Assimilation Impact Top soil moisture (mm) % Precipitation (mm/h) % ET (mm/h) % Total Runoff (mm/h) % 24 Princeton University

25 Summary Assimilation of satellite data into land surface hydrologic models is a promising approach to utilize the remote sensing data which is increasingly 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 the assimilation of remote sensing data, and their potential in real practice is well shown in the experiments performed over satellite data. 25 Princeton University

26 The End Thank you. 26 Princeton University

27 Stop!! back-ups from here on 27 Princeton University

28 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) 28 Princeton University

29 Flexibility of the Copula Model Equivalent measurement error variances as functions of measurement values in the copula based error models fitted from NLDAS/VIC simulations and TMI satellite measurements. 29 Princeton University