Truncated Singular Value Decomposition Regularization for Estimating Terrestrial Water Storage Changes Using GPS: A Case Study over Taiwan

Abstract

It is a typical ill-conditioned problem to invert GPS-measured loading deformations for terrestrial water storage (TWS) changes. While previous studies commonly applied the 2nd-order Tikhonov regularization, we demonstrate the truncated singular value decomposition (TSVD) regularization can also be applied to solve the inversion problem. Given the fact that a regularized estimate is always biased, it is valuable to obtain estimates with different methods for better assessing the uncertainty in the solution. We also show the general cross validation (GCV) can be applied to select the truncation term for the TSVD regularization, producing a solution that minimizes predictive mean-square errors. Analyzing decade-long GPS position time series over Taiwan, we apply the TSVD regularization to estimate mean annual TWS variations for Taiwan. Our results show that the TSVD estimates can sufficiently fit the GPS-measured annual displacements, resulting in randomly distributed displacement residuals with a zero mean and small standard deviation (around 0.1 cm). On the island-wide scale, the GPS-inferred annual TWS variation is consistent with the general seasonal cycle of precipitations. However, on smaller spatial scales, we observe significant differences between the TWS changes estimated by GPS and simulated by GLDAS land surface models in terms of spatiotemporal pattern and magnitude. Based on the results, we discuss some challenges in the characterization of TWS variations using GPS observations over Taiwan.

Inversion of GNSS Vertical Displacements for Terrestrial Water Storage Changes Using Slepian Basis Functions

“…By establishing the relationship between the mass loads and vertical displacements through Green's function, the surface mass changes (e.g., TWS changes) can be recovered using the least squares adjustment. It should be noted that inversion of TWS changes from GNSS observations using Green's function method is a serious ill‐posed problem, and the regularization method like Tikhonov and Truncated Singular Value Decomposition methods (Argus et al., 2014; Lai et al., 2020) should be introduced to stabilize the inversion results. The objective function of Green's function method based on the Tikhonov regularization is as follows (Argus et al., 2014; Fu et al., 2015): $$$\mathbf{min}\left\{{\Vert \frac{(\boldsymbol{G}\boldsymbol{x}-\boldsymbol{u})}{\boldsymbol{\sigma }}\Vert }^{\mathbf{2}}+{\boldsymbol{\beta }}^{\mathbf{2}}{\Vert \boldsymbol{K}\boldsymbol{x}\Vert }^{\mathbf{2}}\right\}$$$ where $$\boldsymbol{G}$$ is the design matrix associated with Green's function; $$\boldsymbol{x}$$ is the unknown TWS change vector that needs to be estimated; $$\boldsymbol{u}$$ is the GNSS observation vector, $$\boldsymbol{\sigma }$$ is the noise standard deviation (STD) of the GNSS observations; $$\boldsymbol{K}$$ is the Laplacian matrix; and $${\boldsymbol{\beta }}^{\mathbf{2}}$$ is the regularization parameter.…”

“…Unlike the SBF method, the Green's function method represents the surface deformation by the convolution of the mass loads and Green's function in the space domain (Farrell, 1972) as follows: By establishing the relationship between the mass loads and vertical displacements through Green's function, the surface mass changes (e.g., TWS changes) can be recovered using the least squares adjustment. It should be noted that inversion of TWS changes from GNSS observations using Green's function method is a serious ill-posed problem, and the regularization method like Tikhonov and Truncated Singular Value Decomposition methods (Argus et al, 2014;Lai et al, 2020) should be introduced to stabilize the inversion results. The objective function of Green's function method based on the Tikhonov regularization is as follows (Argus et al, 2014;Fu et al, 2015):…”

“…(2014) first inverted the GNSS vertical displacements for regional TWS changes using the Green's function method in California, and the spatial resolution of GNSS‐derived TWS changes could reach ∼50 km. Subsequently, the Green's function method has been widely used to invert GNSS observations for regional TWS changes in different regions, such as the United States (Argus et al., 2017; Borsa et al., 2014; Enzminger et al., 2018; Fu et al., 2015; Jin & Zhang, 2016; Shen et al., 2020), China (Fok & Liu, 2019; Hsu et al., 2020; Jiang, Hsu, Yuan, & Huang, 2021; Lai et al., 2020; Li et al., 2022; B. Liu et al., 2022; Shen et al., 2021; Zhang et al., 2016; Zhong et al., 2020) and South America (Ferreira et al., 2019). For example, Borsa et al.…”

Estimation of Terrestrial Water Storage Variations in Sichuan-Yunnan Region from GPS Observations Using Independent Component Analysis

“…To improve the TWS inversion results in the boundary regions, Shen et al [20] developed a boundary-included inversion model that accounts for the mass change effect from the near but exterior region. Lai et al [21] applied truncated singular value decomposition regularization to more stably solve the inversion problem.…”

BER analysis of truncated SVD‐based MU‐MIMO ZF detection scheme under correlated Rayleigh fading channel

“…To bridge this gap, truncated singular value decomposition (TSVD) is deployed in ZF detection scheme. Moreover, TSVD technique is used in wide range of applications to obtain a smooth solution for ill‐posed problems 28–30 . For example, a detection algorithm for spectrally efficient frequency division multiplexing (SEFDM) system is proposed by Isam et al 31 using TSVD in order to decrease the condition number of channel matrix.…”