Parameter Estimation and Data Assimilation Techniques for Land Surface Modeling Qingyun Duan Lawrence Livermore National Laboratory Livermore, California August 5, 2006

A Schematic of A Land Surface Modeling System Based on S.V. Kumar, C. Peters-Lidard et al., Land Surface Model

Land Surface Model (LSM) From A Control System’s Point of View M(U t,X t,  )

System Equation General discrete-time nonlinear dynamic land surface modeling system: Y t = M(X t, U t, θ) + V t where Y t = Measured system response X t = System state variable U t = System input forcing variable θ = System parameter V t = System measurement noise

Uncertainties Exist in All Phases of the Modeling System Y(t) Forcing (Input Variables) System invariants (Parameters) Output (Diagnostic Variables) p(U t ) p(X t ) p(Θ ) p(Y t ) U(t) X(t) p(Mk)p(Mk)

Goals in Land Surface Modeling Obtain the best prediction of land state and ouptut variables Quantify the effects of various uncertainties on the prediction

Two Approaches Parameter estimation –Adjust the values of the model parameters to reduce the uncertainties associated with input forcing, parameter specification, model structural error, output measurement error, etc. Data assimilation –Use observed data for system state and output variables to update the land state variables

Part I – Parameter Estimation

Model identification Problem U – Universal model set B – Basin M i (θ) – Selected model structure

Parameter Estimation As An Optimization Problem true input true response observed input simulated response measurement output time f parameters prior info observed response optimize parameters Minimize some measure of length of the residuals

Why Parameter Estimation? Model performance is highly sensitive to the specification of model parameters Model parameter are highly variable in space, possibly in time Existing parameter estimation schemes in all land surface models are problematic: –Based on tabular results from point samples –Many parameters are only indirectly related to land surface characteristics such soil texture and vegetation class –They have not been validated comprehensively using observed retrospective data

Liston et al., 1994, JGR

Steps in Parameter Estimation or Model Calibration Select a model structure: –BATS, Sib, CLM, NOAH, … Obtain calibration data sets Define a measure of closeness Select an optimization scheme Optimize selected model parameters Validate optimization results

The Measure of Closeness …

Objectives in Hydrologic Modeling Daily Root Mean Square Error (DRMS) Monthly Volume Mean Square Error (DRMS) Nash-Sutcliffe Efficiency Correlation Coefficient Maximum Likelihood Estimators: –Uncorrelated (Gaussian) –Correlated –Heteroscedastic (Error is proportional to data values) Bias …

Difficulties in Optimization

Single Objective Single Solution Optimization Schemes: Local methods: –Direct Search methods –Gradient Search methods Global methods: –Genetic Algorithm –Simulated Annealing –Shuffled complex evolution (SCE-UA)

Local Methods

Optimization Scheme – Local Search Method

Optimization Scheme – Global Search Method The Shuffled Complex Evolution (SCE- UA) method, Q. Duan et al., 1992, WRR

The SCE-UA Method – How it works

SCE-UA – Initial Sampling

SCE-UA – Complex Evolution

SCE-UA – Complex Shuffling

SCE-UA – Final Convergence

Local vs Global Optimization

Methods such as SCE-UA give only a single solution …

Single Objective Probabilistic Solution Optimization There is no unique solution to an optimization problem because of model, data, parameter specification errors Single objective probabilistic solution optimization treats model parameters as probabilistic quantities Monte Carlo Markov Chain methods: –Metroplis-Hasting Algorithm –Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm

The Shuffled Complex Evolution Metropolis (SCE-UA) Algorithm

SCE-UA Optimization Results

SCE-UA Optimization Results - Prediction Uncertainty Bound

Multi-Objective Multi-Solution Optimization Methods Minimize F(θ)={F 1 (θ), F 2 (θ), …, F m (θ)} Multi-objective Complex Evolution Method (MOCOM-UA)

MOCOM Algorithm Simultaneously find several Pareto optimal solutions in a single optimization run Use population evolution strategy similar to SCE-UA 500 solutions require about 20,000 function evaluations

MOCOM-UA Optimization Results

Model calibration for parameter estimation – Pros & Cons Parameters are time-invariant properties (i.e., constants) of the physical system … Traditional Model Calibration methods ü long-term systematic errors properly corrected ü Parameter uncertainties considered û State uncertainties ignored û Estimated parameters could be biased if substantial state and observational errors

A challenging parameter estimation problem … How do we tune the parameters of atmospheric models to enhance the predictive skills for precipitation, air temperature and other interested variables? –Many parameters –Large domain –Large uncertainty in calibration data

Part II – Data Assimilation Data assimilation offers the framework to correct errors in state variables by: – optimally combining model predictions and observations – accounting for the limitations of both

Background Bias represents model error which can be corrected through tuning of model parameters, while state errors can be corrected through data assimilation techniques Observation Assimilation with Bias Correction Assimilation No Assimilation From Paul Houser, et al., 2005

Data Assimilation Techniques Direct Update forced to equal measurements where available, insertioninterpolated from measurements elsewhere Nudging:K = empirically selected constant OptimalK derived from assumed (static) covariance Interpolation: ExtendedK derived from covariances propagated with a linearized Kalman model, input fluctuations and measurement errors must be filter:additive. EnsembleK derived from a ensemble of random replicates propagated Kalman with a nonlinear model, form of input fluctuations and filter:measurement errors is unrestricted.

Illustration of direct insertion

The Ensemble Kalman Filter time x t x t+1 X = measurement = model = updated

The Ensemble Kalman Filter

Application of EnsKF in Land Surface Modeling State estimation: –Soil moisture estimation –Snow data assimilation –Satellite precipitation analysis –Others …

EnsKF Snow Data Assimilation Results – Slater & Clark, 2005

SODA – Simultaneous Optimization and Data Assimilation (Vrugt et al, 2005) SODA combines a parameter optimization scheme - SCEM (Shuffled Complex Complex Metropolis) with Ensemble Kalman Filter

A Schematic of SODA

How SODA Works

SODA Results – Flood forecasting application, Vrugt & Gupta, 2005

Summary Parameter estimation –A way to remove long-term system bias. –A technique popular in other fields (i.e., surface hydrology) –Local vs global methods –Single solution vs probabilistic solution –Single-objective vs multi-objective approach –Potential in land surface modeling and atmospheric modeling

Summary – conti. Data assimilation is a technique that should be used to correct the errors in state variables –A way to take advantage of massive amount of remote sensing and in-situ data –Many methods available –EnsKF is gaining popularity in land surface modeling –SODA is a method that should be exploited more, as it seeks to remove long term bias and correct short term errors in state variables simultaneously