Hydrologic Data Assimilation with a Representer-Based Variational Algorithm Dennis McLaughlin, Parsons Lab., Civil & Environmental Engineering, MIT Dara Entekhabi, Parsons Lab., Civil & Environmental Engineering, MIT Rolf Reichle, NASA Goddard Space Flight Center Problem context - Mapping continental-scale soil moisture from satellite passive microwave measurements. Problem is spatially distributed, nonlinear, and has many degrees of freedom O(10 6 ). Available models of hydrologic system and measurement process are highly uncertain. Variational data assimilation Results from a synthetic experiment (OSSE)

Soil Moisture Soil moisture is important because it controls the partitioning of water and energy fluxes at the land surface. This effects runoff (flooding), vegetation, chemical cycles (e.g. carbon and nitrogen), and climate. Precipitation Runoff Infiltration Evapotranspiration Soil moisture Soil moisture varies greatly over time and space. Measurements are sparse and apply only over very small scales. Soil moisture Solar Radiation Ground Heat Flux Sensible and Latent Heat Fluxes

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).

Relevant Time and Space Scales Plan View Estimation pixels (small) Microwave pixels (large) Vertical Section Soil layers differ in thickness Note large horizontal-to-vertical scale disparity 5 cm 10 cm 5 km Typical precipitation events and measurement times For problems of continental scale we have ~ 10 5 est. pixels, 10 5 meas, 10 6 states,

State equations are derived from mass and energy conservation Soil moisture is governed by a 1D (vertical) nonlinear diffusion eq (PDE). Soil temperature and canopy moisture are linear ODEs. Essential Model Features Canopy moisture, soil moisture and temperature States: Canopy moisture Soil moisture Soil temperature Soil properties and land use Land Surface Model (State equations) Uncertain initial conditions Uncertain land- atmosphere boundary fluxes Radiative transfer model (Measurement equations) Microwave radiobrightness (deg. Kelvin, L-band) Random meas. errors

The Estimation (Data Assimilation) Problem Some options: Variational Approaches: Derive mode of p[y(t)| Z i ]. Good for smoothing problems (t < t i ). Requires adjoint model, limited capabilities for handling model error (process noise), does not give info. about accuracy of state ests. Extended Kalman Filtering: Uses Gaussian assumption to approximate conditional mean and covariance of p[y(t)| Z i ]. Good for filtering/forecasting problems (t  t i ). Requires computation and storage of very large covariance matrices. Tends to be unstable. Provides some info. about estimation accuracy. Suppose we are given a vector Z i = [z 1,..., z i ] of all meas. taken through t i. Ideally, we wish to derive the posterior density p[y(t)| Z i ] at any time t..... In practice, we must settle for partial information about this density Is there a more efficient and complete way to characterize p[y(t)| Z i ] ?

Operating System Simulation Experiment (OSSE) “True” microwave radiobrightness “Measured” microwave radiobrightness Canopy moisture, soil moisture and temperature Soil properties and land use Land surface model Mean initial conditions Mean land- atmosphere boundary fluxes Radiative transfer model Random model error Random initial condition error Random meas. error Data assimilation algorithm Estimated microwave radiobrightness and soil moisture Soil properties and land use, mean fluxes and initial conditions, error covariances Estimation error OSSE generates synthetic measurements which are then processed by the data assimilation algorithm. These measurements reflect the effect of random model and measurement errors. Performance can be measured in terms of estimation error.

Synthetic Experiment (OSSE) based on SGP97 Field Campaign Synthetic experiment uses real soil, landcover, and precipitation data from SGP97 (Oklahoma). Radiobrightness measurements are generated from our land surface and radiative transfer models, with space/time correlated model error (process noise) and measurement error added. SGP97 study area, showing principal inputs to data assimilation algorithm:

Window configurations Effects of Smoothing Window Configuration Position and length of variational smoothing window affect estimation accuracy. Estimation error is less for longer windows that are reinitialized just after (rather than just before) measurement times.

Variational algorithm performs well even without precipitation information. In this case, soil moisture is inferred only from microwave measurements. Effects of Precipitation Information

Estimation of Model Error Representer-based variational algorithm is able to estimate a smoothed version of time-dependent model error:

1. Developed and tested an efficient variational smoothing algorithm based on an indirect representer solution technique. Method is able to accommodate time-dependent model errors. 2. Developed and applied an approach for assessing accuracy of soil moisture and temperature estimates (computation of radiobrightness prediction error variances). 3. Used variational method to study soil moisture mission design issues, including spatial resolution/downscaling, length of smoothing interval, and effects of precipitation withholding. 4. Developed and tested an ensemble Kalman filter (EnKF) which is able to handle highly nonlinear models. 5. Compared the performance of the variational and EnKF approaches. Summary of Recent Progress Publications: Reichle, R. H., 2000: Variational Assimilation of Remote Sensing Data for Land Surface Hydrologic Applications, PhD dissertation, Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, Cambridge, MA 02139, USA. Reichle, R., D. Entekhabi, and D. McLaughlin, Downscaling of Radiobrightness Measurements for Soil Moisture Estimation: A Four-Dimensional Variational Data Assimilation Approach, Water Resources Research, in press. Reichle, R., D. McLaughlin, and D. Entekhabi, Variational data assimilation of microwave radiobrightnes observations for land surface hydrologic applications, IEEE Transactions on Geoscience and Remote Sensing, in press. Reichle, R., McLaughlin, D., and D. Entekhabi, Hydrologic data assimilation with the ensemble Kalman filter, Monthly Weather Review, in press.