## 1. Introduction

[2] The NASA/DLR (Deutsches Zentrum fur Luft und Raumfahrt) satellite mission GRACE (Gravity Recovery and Climate Experiment), launched in March, 2002, is intended to provide highly accurate monthly solutions for the Earth's gravity field at scales of a few hundred km and larger. The mission, which has a 5-year lifetime, consists of two identical satellites in identical Earth orbits, one following the other at a distance of about 220 km. The satellites use microwaves to continually monitor their separation distance. As the satellites pass through gravity highs or lows, that distance changes. Thus, after removing the effects of nongravitational accelerations as detected by on-board accelerometers, the distance measurements can be used to solve for the gravity field.

[3] The month-to-month gravity variations obtained by differencing the GRACE gravity fields provide information about changes in the distribution of mass within the Earth and at its surface. There are a surprisingly large number of Earth-based processes likely to produce a time-variable gravity signal detectable by GRACE, including mass variations in the oceans, the atmosphere, the polar ice sheets, and the solid Earth. In general, the largest time-variable gravity signals observable in the GRACE data are expected to come from changes in the distribution of water and snow stored on land [*Wahr et al.*, 1998]. As a result, GRACE promises to provide a wealth of new and useful hydrologic information: namely, estimates of monthly changes in continental water storage, averaged over scales of several hundred km or larger anywhere in the world [*Dickey et al.*, 1997; *Wahr et al.*, 1998; *Rodell and Famiglietti*, 1999, 2001, 2002; *Swenson and Wahr*, 2002]. The primary purpose of this paper is to describe optimal methods for constructing those estimates, and to determine their probable accuracy.

[4] To determine whether GRACE is able to detect water storage variability within a particular region, two issues must be addressed. One is whether the gravitational effects are greater than the GRACE observational errors. Because of the decreased sensitivity of GRACE at short wavelengths, accurate estimates can be obtained only for regions that are several hundred km or more in scale. The other issue is whether the gravitational signal from the region of interest can be separated from all the other time-varying gravitational signals acting on GRACE.

[5] There are two types of separation problems. One is when the contaminating signal comes from a different region on the surface, such as from the water storage variability in a neighboring region, or from the ocean. In principle, GRACE could remove the effects of any such signals if it had perfect spatial resolution. One of the main concerns raised in this paper, and in an earlier paper [*Swenson and Wahr*, 2002] extended here, is to find a method of analysis that can optimally resolve the competing criteria of needing short scales to minimize the separation problem, but preferring longer scales to reduce the effects of GRACE measurement errors.

[6] The other separation problem arises when there are mass variations either in the atmosphere above the region, or in the solid Earth below it. Gravity has no vertical resolving power. If there is unmodeled mass variability in the atmosphere or solid Earth, its effects will be mapped directly into the water storage estimates. The GRACE Project is removing the effects of the atmospheric signal from the GRACE measurements prior to constructing each gravity field, using 6-hourly global surface pressure and geopotential height fields provided by the European Center for Medium Range Weather Forecasts (ECMWF). However, there are errors in the ECMWF fields, and those errors will affect the water storage estimates. The atmospheric errors are likely to be the limiting error source at scales of about 700 km and larger, but their effects on the regionally averaged water storage recovery are generally not apt to be larger than a few mm of equivalent water thickness [*Velicogna et al.*, 2001].

[7] In the solid Earth, by far the largest inadequately modeled contributions come from postglacial rebound (PGR), the ongoing, viscoelastic response of the solid Earth to the deglaciation at the end of the last ice age (for recent reviews, see *Wu* [1998] and *Mitrovica and Vermeersen* [2002]). This signal is linear in time, and is concentrated under Canada, Scandinavia, Antarctica, and Greenland. Thus it will contaminate GRACE estimates of the linearly varying water storage signals in those regions. Its effects will spill over somewhat into neighboring regions, and this issue is addressed in section 5.2. PGR should have no effect on the recovery of nonlinear temporal variability.

[8] The remainder of the gravity anomaly signal can then be attributed to changes in continental water storage, snow, and ice. Because water storage variability occurs in a thin layer at the surface of the Earth, GRACE is insensitive to the vertical distribution of mass [*Swenson and Wahr*, 2002]. GRACE thus senses changes in vertically integrated water storage, i.e., the total change in groundwater, soil moisture, surface water, snow and ice.

[9] The GRACE data set, which is provided to users every month, is not a set of point measurements. Instead, it is a finite set of spherical harmonic (Stokes) coefficients. These coefficients can be used to construct water storage averages over regions of any size and shape anywhere on the globe. However, the problem of constructing these averages is complicated by the competing requirements of reducing the effects of GRACE measurement errors (caused by such things as system-noise error in the inter-satellite range-rate, accelerometer error, error in the ultrastable oscillator, and error in the orbits) while minimizing the contaminating signal from adjacent regions.

[10] *Swenson and Wahr* [2002] developed a method of estimating water storage variability in an arbitrary region that minimizes the combined effects of satellite measurement errors and unwanted signals from nearby regions (referred to as “leakage”). This method uses measurement error covariance matrices provided by the GRACE Project to construct averaging kernels for the region of interest. The errors in these water storage results due to leakage from surrounding regions can be reliably estimated only if there is a priori knowledge of certain characteristics of mass variability within those regions, such as the amplitude of monthly variability, the temporal correlation between the region of interest and neighboring regions, and the rate at which temporal correlations change with distance.

[11] One way of estimating the uncertainty in GRACE water storage estimates is to simulate GRACE data by combining large-scale (preferably global), gridded models of the expected monthly water storage signal with models of errors in the meteorological fields used to remove atmospheric effects and GRACE measurement errors. Regional water storage estimates can be constructed from those simulated data, and those estimates can be compared with the true regional averages obtained from the hydrology model alone. The differences are a measure of the error in the GRACE estimates. We construct such simulations in this paper.

[12] Our objectives are as follows: (1) to demonstrate how the methods developed by *Swenson and Wahr* [2002] can be applied and optimized when dealing with the signals expected in actual GRACE data and (2) to illustrate how simulations can be used to estimate the uncertainties in the water storage estimates. Our simulations assume a specific global hydrology model and adopt the prelaunch estimates of the GRACE measurement errors. Although users will certainly use other hydrology models, and the error covariance matrices for real GRACE data are likely to differ, perhaps substantially, from the prelaunch estimates, the simulation methods described here will still be appropriate. In addition, preliminary predictions are presented of GRACE water storage accuracies for specific North American river basins.