Regularization and error characterization of GRACE mascons

Abstract

We present a new global time-variable gravity mascon solution derived from Gravity Recovery and Climate Experiment (GRACE) Level 1B data. The new product from the NASA Goddard Space Flight Center (GSFC) results from a novel approach that combines an iterative solution strategy with geographical binning of inter-satellite range-acceleration residuals in the construction of time-dependent regularization matrices applied in the inversion of mascon parameters. This estimation strategy is intentionally conservative as it seeks to maximize the role of the GRACE measurements on the final solution while minimizing the influence of the regularization design process. We fully reprocess the Level 1B data in the presence of the final mascon solution to generate true post-fit inter-satellite residuals, which are utilized to confirm solution convergence and to validate the mascon noise uncertainties. We also present the mathematical case that regularized mascon solutions are biased, and that this bias, or leakage, must be combined with the estimated noise variance to accurately assess total mascon uncertainties. The estimated leakage errors are determined from the monthly resolution operators. We present a simple approach to compute the total uncertainty for both individual mascon and regional analysis of the GSFC mascon product, and validate the results in comparison with independent mascon solutions and calibrated Stokes uncertainties. Lastly, we present the new solution and uncertainties with global analyses of the mass trends and annual amplitudes, and compute updated trends for the global ocean, and the respective contributions of the Greenland Ice Sheet, Antarctic Ice Sheet, Gulf of Alaska, and terrestrial water storage. This analysis highlights the successful closure of the global mean sea level budget, that is, the sum of global ocean mass from the GSFC mascons and the steric component from Argo floats agrees well with the total determined from sea surface altimetry.

Least-squares and statistical assumptions

1.1 Introduction

Linear least-squares parameter estimation and error assessment typically rely on a set of statistical assumptions that may not be valid. Here, we examine the effect of the assumed statistical properties of the linear system of equations on the bias and covariance of the least-squares state estimator, $\hat{\mathbf {x}}$. We are specifically interested in the properties of the a priori state equation, which we show defines the bias of the estimator, and must be accounted for when assessing the error of the mascon parameters.

We begin by defining linear system of equations for the assumed and truth cases:

$$\begin{aligned} \begin{array}{ll} \mathrm {cost} \mathrm {function:} &{} J(\mathbf {x}) = {\nu }^\mathrm {T}\mathbf {W}{\nu } + {\eta }^\mathrm {T}\mathbf {P}{\eta } \\ \mathrm {assumed:} &{} \left\{ \begin{array}{lll} \mathbf {d} &{}= \mathbf {A}\mathbf {x} + {\nu } &{} , {\nu } \sim \mathcal {N}(0, \mathbf {W}^{-1}) \\ \mathbf {x}_a &{}= \mathbf {x} + {\eta } &{} , {\eta } \sim \mathcal {N}(0, \mathbf {P}^{-1}) \end{array} \right. \\ \mathrm {truth:} &{} \left\{ \begin{array}{lll} \mathbf {d} &{}= \mathbf {A}\mathbf {x} + {\zeta } &{} , {\zeta } \sim \mathcal {N}(0, \mathbf {R}^{-1}) \\ \mathbf {x}_m &{}= \mathbf {x} + {\epsilon } &{} , {\epsilon } \sim \mathcal {N}(0, \mathbf {Q}^{-1}), \end{array} \right. \end{array} \end{aligned}$$

(A.1)

where J is the least-squares cost function to be minimized by the parameters of interest $\mathbf {x}$, $\mathbf {d}$ is the data vector, $\mathbf {A}$ is the design matrix, $\mathbf {x}_a$ is the a priori best estimate of the true mean state $\mathbf {x}_m$ of $\mathbf {x}$, and $\mathbf {W}^{-1}$, $\mathbf {R}^{-1}$, $\mathbf {P}^{-1}$, and $\mathbf {Q}^{-1}$ are the covariance matrices of the various zero-mean errors ${\nu }$, ${\zeta }$, ${\eta }$, and ${\epsilon }$, respectively. The errors in statistical information considered here arise in the a priori information as misspecifications in the mean and covariance of the distribution of x, that is, the difference between $\mathbf {x}_a$ and $\mathbf {x}_m$ and the difference between $\mathbf {P}$ and $\mathbf {Q}$, respectively, and misspecification in the covariance of the data noise, that is, the difference between $\mathbf {W}$ and $\mathbf {R}$. The least-squares minimizer of J for the assumed statistical information is given by:

$$\begin{aligned} \hat{\mathbf {x}} = \left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P}\right) ^{-1} \left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {d} + \mathbf {P} \mathbf {x}_a \right) . \end{aligned}$$

(A.2)

1.2 The dispersion matrix and mean squared error

The stochastic processes considered here are assumed second-order stationary, and thus, are completely explained by their mean and covariance. For a vector stochastic process z, these are related by considering the expected value of the total variation, or dispersion, between all pairs of elements, which can be assembled into a dispersion matrix $\mathbf {D}_z$ that can decomposed as follows:

$$\begin{aligned} \mathbf {D}_z= & {} \mathbb {E} \left[ \mathbf {z} \mathbf {z}^\mathrm {T} \right] \nonumber \\= & {} \mathbb {E} \left[ (\mathbf {z} - \mathbb {E}[\mathbf {z}] + \mathbb {E}[\mathbf {z}]) (\mathbf {z} - \mathbb {E}[\mathbf {z}] + \mathbb {E}[\mathbf {z}])^\mathrm {T} \right] \nonumber \\= & {} \mathbb {E} \left[ (\mathbf {z} - \mathbb {E}[\mathbf {z}]) (\mathbf {z} - \mathbb {E}[\mathbf {z}])^\mathrm {T} \right] + \mathbb {E} \left[ \mathbb {E}[\mathbf {z}] \mathbb {E}[\mathbf {z}]^\mathrm {T} \right] \nonumber \\= & {} \mathbf {C}_z + \mathbf {z}_m \mathbf {z}_m^\mathrm {T}, \end{aligned}$$

(A.3)

where $\mathbb {E}[\cdot ]$ is the expectation operator, $\mathbf {C}_z$ is the covariance matrix of z, and $\mathbf {z}_m=\mathbb {E}[\mathbf {z}]$ is the mean vector.

The mean squared error (MSE) of the estimate is then defined as follows:

$$\begin{aligned} MSE \left( \hat{\mathbf {x}} \right)= & {} \mathbb {E} \left[ (\hat{\mathbf {x}}-\mathbf {x})^\mathrm {T} (\hat{\mathbf {x}}-\mathbf {x})\right] \nonumber \\= & {} Tr\left[ \mathbb {E} \left[ (\hat{\mathbf {x}}-\mathbf {x}) (\hat{\mathbf {x}}-\mathbf {x})^\mathrm {T} \right] \right] \nonumber \\= & {} Tr[\mathbf {D}_{\hat{x}-x}] \nonumber \\= & {} Tr[\mathbf {C}_{\hat{x}-x}] + \mathbf {b}^\mathrm {T}\mathbf {b}, \end{aligned}$$

(A.4)

where $Tr[\cdot ]$ is the trace operator and $\mathbf {b}=\mathbb {E}[\hat{\mathbf {x}}-\mathbf {x}]$ is the estimate bias vector. Therefore, we see that the error-covariance matrix $\mathbf {C}_{\hat{x}-x}$ and bias vector b completely define the discrepancy between $\hat{\mathbf {x}}$ and $\mathbf {x}$.

1.3 Error covariance and bias

We now rewrite the true statistics of the underlying true state x as:

$$\begin{aligned} \mathbf {x}_m= & {} \mathbf {x} + {\epsilon } \nonumber \\ \mathbf {x}_a= & {} \mathbf {x} + \mathbf {x}_b + {\epsilon }, \end{aligned}$$

(A.5)

where $\mathbf {x}_b = \mathbf {x}_a - \mathbf {x}_m$, and substitute this and $\mathbf {d} = \mathbf {A} \mathbf {x} + {\zeta }$ into A.2 such that

$$\begin{aligned} \hat{\mathbf {x}} =&\left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P}\right) ^{-1} \nonumber \\&\left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} \mathbf {x} + \mathbf {A}^\mathrm {T} \mathbf {W} {\zeta } + \mathbf {P} \mathbf {x} + \mathbf {P} \mathbf {x}_b + \mathbf {P} {\epsilon }\right) . \end{aligned}$$

(A.6)

The error may now be written as

$$\begin{aligned} \hat{\mathbf {x}} - \mathbf {x} = \left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P} \right) ^{-1} \left( \mathbf {A}^\mathrm {T} \mathbf {W} {\zeta } + \mathbf {P} \mathbf {x}_b + \mathbf {P} {\epsilon } \right) , \end{aligned}$$

(A.7)

whose expected value, or bias $\mathbf {b}$, is

$$\begin{aligned} \mathbb {E} \left[ \hat{\mathbf {x}} - \mathbf {x} \right] = \left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P} \right) ^{-1} \mathbf {P} \mathbf {x}_b. \end{aligned}$$

(A.8)

If we define the a priori value of $\mathbf {x}_a$ to be zero, as is the case for GSFC mascon estimation, then this can be rewritten as

$$\begin{aligned} \mathbb {E} \left[ \hat{\mathbf {x}} - \mathbf {x} \right]&= -\left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P} \right) ^{-1} \mathbf {P} \mathbf {x}_m \nonumber \\&= \left( \mathbf {R} - \mathbf {I} \right) \mathbf {x}_m, \end{aligned}$$

(A.9)

where $\mathbf {R}$ is the resolution operator defined in Eq. 18. As previously noted, the solution bias of Eq. A.9 matches the ridge regression bias presented by (Hoerl and Kennard 1970). To derive the covariance, we note that

$$\begin{aligned} \hat{\mathbf {x}} - \mathbf {x} - \mathbb {E} \left[ \hat{\mathbf {x}} - \mathbf {x} \right] = \left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P} \right) ^{-1} \left( \mathbf {A}^\mathrm {T} \mathbf {W} {\zeta } + \mathbf {P} {\epsilon } \right) , \end{aligned}$$

(A.10)

which leads to

$$\begin{aligned} \mathbf {C}^\prime _{\hat{\mathbf {x}} - \mathbf {x}}= & {} \left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P} \right) ^{-1} \left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {R}^{-1} \mathbf {W} \mathbf {A} + \mathbf {P} \mathbf {Q}^{-1} \mathbf {P} \right) \nonumber \\&\left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P} \right) ^{-1}, \end{aligned}$$

(A.11)

where it has been assumed that $\mathbb {E}[{\zeta \epsilon }^\mathrm {T}]=0$.

To conclude, we observe in Eqs. A.9 and A.11 the effect of invalid statistical assumptions on the bias and covariance, respectively. If the covariance statistics are assumed to be valid, that is, $\mathbf {W}=\mathbf {R}$ and $\mathbf {P}=\mathbf {Q}$, then Eq. A.11 reduces to the familiar form:

$$\begin{aligned} \mathbf {C}_{\hat{\mathbf {x}} - \mathbf {x}}= \left( \mathbf {A}^\mathrm {T} \mathbf {W} \mathbf {A} + \mathbf {P} \right) ^{-1}. \end{aligned}$$

(A.12)

It can be shown that $\mathbf {C}^\prime _{\hat{\mathbf {x}} - \mathbf {x}} > \mathbf {C}_{\hat{\mathbf {x}} - \mathbf {x}}$, which indicates that this difference in symmetric positive-definite matrices is itself a symmetric positive-definite matrix, that is, it has positive eigenvalues. This means that any variance produced through covariance propagation will be larger for $\mathbf {C}^\prime _{\hat{\mathbf {x}} - \mathbf {x}}$ than the corresponding variance produced by $\mathbf {C}_{\hat{\mathbf {x}} - \mathbf {x}}$.