Constrained Data Assimilation Filtering

Abstract

Assimilating Gravity Recovery And Climate Experiment (GRACE) data into land hydrological models provides a valuable opportunity to improve the models’ forecasts and increases our knowledge of terrestrial water storages (TWS). The assimilation, however, may harm the consistency between hydrological water fluxes, namely precipitation, evaporation, discharge, and water storage changes. To address this issue, we propose a weak constrained ensemble Kalman filter (WCEnKF) that maintains estimated water budgets in balance with other water fluxes. Therefore, in this study, GRACE terrestrial water storages data are assimilated into the World-Wide Water Resources Assessment (W3RA) hydrological model over the Earth’s land areas covering 2002–2012. Multi-mission remotely sensed precipitation measurements from the Tropical Rainfall Measuring Mission (TRMM) and evaporation products from the Moderate Resolution Imaging Spectroradiometer (MODIS), as well as ground-based water discharge measurements are applied to close the water balance equation. The proposed WCEnKF contains two update steps; first, it incorporates observations from GRACE to improve model simulations of water storages, and second, uses the additional observations of precipitation, evaporation, and water discharge to establish the water budget closure. These steps are designed to account for error information associated with the included observation sets during the assimilation process. In order to evaluate the assimilation results, in addition to monitoring the water budget closure errors, in-situ groundwater measurements over the Mississippi River Basin in the US and the Murray-Darling Basin in Australia are used. Our results indicate approximately 24% improvement in the WCEnKF groundwater estimates over both basins compared to the use of (constraint-free) EnKF. WCEnKF also further reduces imbalance errors by approximately 82.53% (on average) and at the same time increases the correlations between the assimilation solutions and the water fluxes.

Models water storage states are in balance since model structure, e.g., its equations, governs variations in the water state changes due to the incoming and outgoing hydrological water fluxes. An assimilation of water storage states (e.g., GRACE data) does not constraint the assimilated state to be balanced.

—M. Khaki (This chapter is presented following Khaki et al. (2017c), “A two-update ensemble Kalman filter for land hydrological data assimilation with an uncertain constraint”)

Appendices

Appendix A. Some Useful Properties of Random Sampling

Property 1

(Hierarchical sampling; Robert et al. 2006). Assuming that one can sample from $p(\mathbf{x}_1)$ and $p(\mathbf{x}_{2}|\mathbf{x}_{1})$, then a sample, $\mathbf{x}_2^{*}$, from $p(\mathbf{x}_2)$ can be generated by drawing $\mathbf{x}_1^*$ from $p(\mathbf{x}_1)$ and then $\mathbf{x}_2^*$ from $p(\mathbf{x}_2 | \mathbf{x}_1^*)$.

Property 2

(Conditional sampling; Hoffman et al. 1991). Consider a Gaussian pdf, $p(\mathbf{x} , \mathbf{y})$, with $\mathbf{P}_{xy}$ and $\mathbf{P}_{y}$ denoting the cross-covariance of $\mathbf{x}$ and $\mathbf{y}$ and the covariance of $\mathbf{y}$, respectively. Then a sample, $\mathbf{x}^{*}$, from $p(\mathbf{x}|\mathbf{y})$, can be generated as, $\mathbf{x}^{*} = \widetilde{\mathbf{x}} + \mathbf{P}_{xy} \mathbf{P}_y^{-1} [\mathbf{y} - \widetilde{\mathbf{y}}]$, where $(\widetilde{\mathbf{x}} , \widetilde{\mathbf{y}}) \sim p(\mathbf{x} , \mathbf{y})$.

Appendix B. Derivation of the WCEnKF Algorithm

Equation (7.12), which computes the forecast ensemble ${\{ \mathbf{x}_{t}^{f,(i)} \}}_{i=1}^n$ from the previous analysis one, is obtained by applying Property 1 above to the forecast step (7.6). Regarding the first update step (with $\mathbf{y}_t$), one first applies Property 1 on the following formula,

$$\begin{aligned} \nonumber p(\mathbf{y}_t | \mathbf{r}_{0:t-1}) = \int \underbrace{ p(\mathbf{y}_t | \mathbf{x}_t) }_{\mathcal{N}(\mathbf{H}_t \mathbf{x}_t , \mathbf{R}_t)} p(\mathbf{x}_t | \mathbf{r}_{0:t-1}) d\mathbf{x}_t , \end{aligned}$$

to sample the observation forecast ensemble, ${\{ \mathbf{y}_{t}^{f,(i)} \}}_{i=1}^n$, as in Eq. (7.13). Property 2 is then used in Eqs. (7.7) to obtain the ensembles ${\{ \tilde{\mathbf{x}}_{t}^{a,(i)} \}}_{i=1}^n$ (Eq. (7.14)) and ${\{ \mathbf{x}_{t-1}^{s,(i)} \}}_{i=1}^n$, respectively. For the second update step (with $\mathbf{z}_t$), one first uses Property 1 in Eq. (7.11), with $p(\mathbf{z}_t |\mathbf{x}_t , \mathbf{x}_{t-1}) {\mathop {=}\limits ^{(7.4)}} \mathcal{N}(\mathbf{G} \mathbf{x}_t + \mathbf{L}{} \mathbf{x}_{t-1} , \varvec{\Sigma }_t)$, to obtain the pseudo-observation forecast ensemble ${\{ \mathbf{z}_{t}^{f,(i)} \}}_{i=1}^n$ (Eq. (7.18)), then Property 2 in Eq. (7.10) to compute the state analysis ensemble ${\{ \mathbf{x}_{t}^{a,(i)} \}}_{i=1}^n$ (Eq. (7.19)).