Inversion of regional terrestrial water storage changes using GPS vertical displacements based on TSVD-Tikhonov regularization method
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摘要:
利用GPS垂直位移反演区域陆地水储量变化（TWSC）属于典型的病态问题，其关键是如何进行稳定求解并提高反演结果的精度和可靠性.本文引入TSVD-Tikhonov组合正则化方法对利用GPS垂直位移反演区域TWSC的病态问题进行求解，并以四川省TWSC反演为例进行分析与验证.首先，通过数值模拟对TSVD、Tikhonov和TSVD-Tikhonov正则化方法采用不同正则化参数选取策略（RMSE最小准则、GCV法和L-curve法）进行反演，结果显示基于TSVD-Tikhonov正则化反演的TWSC比单独使用TSVD或Tikhonov正则化反演结果的精度和可靠性更高，这三种正则化方法反演2005年1月至12月的TWSC差值的平均STD分别为14.97 mm、7.03 mm和5.04 mm.其次，利用中国地壳运动观测网络（CMONOC）的72个GPS测站的垂直位移数据，基于TSVD-Tikhonov正则化反演了四川省2010年12月至2021年2月的TWSC时间序列，结果表明GPS反演的TWSC与GRACE/GFO Mascon模型（JPL、CSR和GSFC）的空间分布特征及季节性变化符合较好，但其TWSC信号的振幅比GRACE/GFO Mascon模型更强.最后，采用广义三角帽方法（GTCH）融合不同类型的降水、蒸散发和径流数据，并根据水量平衡方程计算的dTWSC/dt序列（PER-dS/dt）对GPS反演的dTWSC/dt序列（GPS-dS/dt）和GRACE/GFO Mascon模型融合的dTWSC/dt序列（GRACE/GFO-dS/dt）进行验证，结果表明这三类dTWSC/dt序列的季节性变化符合较好，平滑后GPS-dS/dt和GRACE/GFO-dS/dt序列与PER-dS/dt序列的相关系数分别为0.78和0.87，但GPS相比GRACE/GFO对降水变化的响应更为敏感.本文研究证明了TSVD-Tikhonov组合正则化方法能够提高GPS垂直位移反演区域TWSC的精度和可靠性，同时也表明GPS观测数据对局部水质量负荷变化更为敏感，可作为GRACE/GFO反演区域TWSC的有益补充.
Abstract:Inversion of GPS vertical displacements for regional Terrestrial Water Storage Change (TWSC) is a typical ill-conditioned problem, and the key issue is how to solve it stably and improve the accuracy and reliability of the inversion results. In this study, we introduced the TSVD-Tikhonov regularization method to solve the ill-conditioned problem of inverting GPS vertical displacements for regional TWSC, and the GPS-inferred TWSC in Sichuan Province was analyzed and verified. First, the TSVD, Tikhonov, and TSVD-Tikhonov regularization methods based on different regularization parameter selection strategies (i.e., Root Mean Square Errors (RMSE) minimum criterion, Generalized Cross Validation (GCV) method, and L-curve method) were investigated through numerical simulation. The results demonstrate that the TWSC solved by TSVD-Tikhonov regularization is more accurate and reliable than that solved by TSVD or Tikhonov regularization, and the average Standard Deviation (STD) of TWSC differences calculated by TSVD, Tikhonov, and TSVD-Tikhonov regularization methods from January to December 2005 are 14.97 mm, 7.03 mm, and 5.04 mm, respectively. Second, the TWSC time series in Sichuan Province from December 2010 to February 2021 were inferred from the vertical displacements of 72 GPS stations of the China Crustal Movement Observation Network (CMONOC) based on TSVD-Tikhonov regularization. The results show that the spatial patterns and seasonal changes of GPS-inferred TWSC series are in good agreement with the GRACE/GFO Mascon models (JPL, CSR, and GSFC). However, the amplitude of GPS-inferred TWSC is stronger than those of GRACE/GFO Mascon models. Finally, the Generalized Three-Cornered Hat (GTCH) method was used to fuse different types of precipitation, evapotranspiration, and runoff data, and then the dTWSC/dt time series (PER-dS/dt) calculated by the water balance equation were used to validate the GPS-inferred dTWSC/dt time series (GPS-dS/dt) and GRACE/GFO Mascons fused dTWSC/dt time series (GRACE/GFO-dS/dt). The results show that the seasonal changes of the three types of dTWSC/dt time series are in good agreement, and the correlation coefficients between GPS-dS/dt, GRACE/GFO-dS/dt and PER-dS/dt after smoothing are 0.78 and 0.87, respectively, but GPS is more sensitive to the precipitation change compared with GRACE/GFO. Our study verifies that the TSVD-Tikhonov regularization method can improve the accuracy and reliability of GPS-inferred regional TWSC, and demonstrates that GPS has stronger sensibility to the change of local water mass load, which can be a useful complement to the GRACE/GFO-estimated regional TWSC.
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图 1
实验区的地形及GPS测站位置分布
Figure 1.
Topography of the test area and distribution of GPS stations
图 2
基于TSVD和Tikhonov正则化由不同正则化参数选取方法反演的TWSC(a和b)及其与原始信号的差值标准差(c和d)和相应的最优正则化参数与截断阶数(e和f)
Figure 2.
TWSC solved by Tikhonov and TSVD regularizations through different regularization parameter selection methods (a and b), STD of differences between the estimated TWSC and original signal (c and d), and the corresponding optimal regularization parameters and truncation orders (e and f)
图 3
设计矩阵奇异值(a)和不同截断阶数对应的累积贡献率(b)
Figure 3.
Singular values of design matrix (a) and their cumulative contribution ratio (b) with different truncation orders
图 4
Tikhonov正则化和不同截断阶数K的TSVD-Tikhonov正则化反演结果的误差STD(a)和月平均误差STD(b)
Figure 4.
STD (a) and monthly mean STD (b) of the errors of the inversion results based on Tikhonov regularization and TSVD-Tikhonov regularization with different truncation order K
图 5
TSVD (a)、Tikhonov (b)和TSVD-Tikhonov (c)正则化反演的TWSC序列与原始信号差值STD的空间分布
Figure 5.
Spatial distributions of STD of the differences between the estimated TWSC series and original signal based on TSVD (a), Tikhonov (b), and TSVD-Tikhonov (c) regularizations
图 6
SCBZ和SCYY测站的垂直位移时间序列滤波前后比较
Figure 6.
Comparisons of the vertical displacement time series at SCBZ and SCYY stations before and after filtering
图 7
72个GPS测站恢复EWH信号的棋盘测试
Figure 7.
Checkerboard test of EWH signal recovered from 72 GPS stations
图 8
GRACE/GFO Mascon模型(JPL、CSR和GSFC)与GPS垂直位移反演的TWSC周年振幅空间分布
Figure 8.
Spatial patterns of annual amplitudes for TWSC derived from GRACE/GFO Mascons (JPL, CSR, and GSFC) and GPS vertical displacements
图 9
GRACE/GFO Mascon模型(JPL、CSR和GSFC)及其融合模型(GTCH-TWSC)与GPS垂直位移反演的TWSC时间序列比较
Figure 9.
Comparisons of TWSC time series derived from GRACE/GFO Mascons (JPL, CSR, and GSFC) and their fusion model (GTCH-TWSC), and GPS vertical displacements
图 10
不同类型水文气象数据计算的四川省月平均降水、蒸散发和径流变化(2011—2021年)
Figure 10.
Monthly average precipitation, evapotranspiration, and runoff changes of Sichuan Province derived from different hydrometeorology datasets (from 2011 to 2021)
图 11
(a) GTCH方法融合后的降水、蒸散发和径流月平均时间序列；(b)和(c) 分别为平滑前后的PER-dS/dt、GPS-dS/dt和GRACE/GFO-dS/dt时间序列
Figure 11.
(a) Monthly average precipitation, evapotranspiration, and runoff time series fused by GTCH method; (b) and (c) are the PER-dS/dt, GPS-dS/dt, and GRACE/GFO-dS/dt time series before and after smoothing
图 12
平滑前后GPS-dS/dt和GRACE/GFO-dS/dt序列与PER-dS/dt序列之间的散点图及相关系数(CC)和差值STD比较
Figure 12.
Comparisons of scatter plot, correlation coefficient (CC), and STD of the differences between GPS-dS/dt, GRACE/GFO-dS/dt and PER-dS/dt series before and after smoothing
表 1
GRACE/GFO Mascon模型(JPL、CSR和GSFC)及其融合模型(GTCH-TWSC)与GPS反演的TWSC时间序列去除线性趋势后的相关系数和差值STD统计
Table 1.
Correlation coefficients and STD of the differences between TWSC time series derived from GPS and GRACE/GFO Mascons (JPL, CSR, and GSFC) and their fusion model (GTCH-TWSC) after removing linear trends
数据 相关系数 STD (mm) JPL 0.69 45.07 CSR 0.66 46.79 GSFC 0.67 45.85 GTCH-TWSC 0.69 45.04 
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表 2
GPS、GRACE/GFO Mascon模型(JPL、CSR和GSFC)及其融合模型(GTCH-TWSC)估计的TWSC时间序列的周年振幅、相位和线性趋势统计
Table 2.
Annual amplitudes, phases, and linear trends of TWSC time series estimated by GPS, GRACE/GFO Mascons (JPL, CSR, and GSFC) and their fusion model (GTCH-TWSC)
数据 时间范围 振幅(mm) 相位(months) 趋势(mm·a-1) GPS 2010-12—2021-02 70.98±8.19 9.16±0.22 -0.88±1.95 2010-12—2017-06 63.65±9.61 8.87±0.28 -1.95±3.55 2017-07—2021-02 88.44±12.08 9.58±0.26 -0.67±8.05 JPL 2010-12—2021-02 47.87±5.65 9.48±0.21 7.38±1.23 2010-12—2017-06 45.33±7.22 9.36±0.29 5.13±2.60 2018-06—2021-02 51.78±8.73 9.61±0.30 11.54±7.92 CSR 2010-12—2021-02 46.75±5.84 9.33±0.23 4.17±1.27 2010-12—2017-06 42.24±6.29 9.15±0.27 0.39±2.26 2018-06—2021-02 54.27±10.61 9.50±0.35 7.53±9.60 GSFC 2010-12—2021-02 57.85±6.53 9.00±0.20 5.84±1.42 2010-12—2017-06 55.62±8.05 8.92±0.26 2.75±2.89 2018-06—2021-02 61.30±11.15 9.08±0.32 6.87±10.05 GTCH-TWSC 2010-12—2021-02 49.41±5.42 9.29±0.20 5.67±1.18 2010-12—2017-06 46.15±6.39 9.16±0.25 2.58±2.29 2018-06—2021-02 54.65±9.17 9.43±0.30 8.79±8.29
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表 3
不同类型时间序列数据的周年振幅和相位统计
Table 3.
Statistics of annual amplitudes and phases for the different types of time series datasets
数据集 序列名称 振幅(mm·month-1) 相位(months) GTCH融合数据 GTCH-P 98.34±6.45 7.11±0.12 GTCH-ET 37.77±1.33 6.77±0.07 GTCH-R 35.53±4.42 8.10±0.24 TWSC GPS-TWSC 70.46±8.32 9.18±0.23 GRACE/GFO-TWSC 45.71±5.72 9.21±0.24 dTWSC/dt PER-dS/dt 29.23±2.37 6.43±0.16 Smooth PER-dS/dt 21.80±1.13 6.42±0.10 GPS-dS/dt 36.49±8.78 5.69±0.46 Smooth GPS-dS/dt 28.08±2.80 5.68±0.20 GRACE/GFO-dS/dt 24.45±5.30 5.75±0.39 Smooth GRACE/ GFO-dS/dt 19.15±2.21 5.79±0.20
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