## 1. Introduction

The three-rivers headwater region, known as “China’s Water Tower” located in the eastern Qinghai-Tibet Plateau, represents a fundamental water supply region for the Lancang River, Yangtze River, and Yellow River in China that is sensitive to global climate change [

1]. With the global warming, the shrinkage of lakes size, the decline in runoff flow, and the degradation of grassland, are but a few scenarios consequently posing an adverse effect on the middle and the lower river basins in China and in East Asian countries [

2]. Therefore, large-scale quantification and monitoring is essential for resolving the challenges and conflicts of water supply at both national and international level.

Recently, space-geodetic observations have been widely used to infer terrestrial water storage (TWS) at both a global and regional scale, due to spatially sparse ground-based in-situ observations and insufficient fund to maintain a number of hydrologic monitoring stations. GRACE is one of the space-geodetic missions for monitoring time-variable gravity changes from space, and hence, providing an opportunity to detect the surface mass changes [

3], in which water movement cycle is a substantial component [

4]. GRACE derived TWS has been compared with the estimates from GLDAS NOAH hydrologic model data (e.g., [

5,

6]). Though GRACE is capable of detecting TWS change with some accuracy for regions that are several hundred km or more in scale, it has a low spatial resolution (~300 km) which has limited its application in small regions [

5]. Hydrologic models, like GLDAS, contain only soil moisture, snow, and plant canopy surface water, and they can’t reflect the total TWS change, especially water in the river, reservoir and underground. GPS coordinates (especially in up components) are more sensitive to surface mass change than GRACE, which providing a potential way to detect TWS change with higher resolution. Since this century, seasonal deformation due to surface water transport has been detected by GPS (e.g., [

7,

8,

9,

10]), and by the combination of GPS and GRACE (e.g., [

11,

12,

13,

14,

15]). More recently, Steckler et al. [

16] applied GPS, and GRACE data to model the Earth deformation caused by monsoonal flooding in Bangladesh validated with water level in river gauge stations. Chew and Small [

17] estimated the TWS response to drought in year 2012 from the GPS vertical position anomalies in central United States. Birhanu and Bendick [

18] used the GPS displacement time series to investigate the hydrological loading caused by precipitation in Ethiopia and Eritrea. All the aforementioned research studies convert GRACE gravity field to vertical deformation comparable to that measured by GPS.

The technique for obtaining the change in water thickness from GPS vertical displacements through a well-known Green’s function [

19], has been newly developed to investigate the TWS in California. The obtained TWS is comparable to that from GRACE and hydrologic models [

20]. These pioneer studies have been making an advancement on the application of GPS to hydrology. Seasonal vertical oscillations of 922 GPS sites are successfully used to invert the seasonal variation in total water storage in California [

20] which are also compared with GRACE results and hydrological models.

The CMONOC network, with access to more than 200 continuous stations since 2009, is a network for monitoring the crustal deformation, with stations less dense when compared to the Plate Boundary Observatory (PBO) network in United States. This study aims to investigate the potential usage of CMONOC GPS stations for inverting the seasonal variation of TWS in Yunnan Province in China, where it is connected to Lancang River (also called Mekong River system outside the boundaries of China), with the Yangtze River and Zhujiang River nearby. Its annual total precipitation can reach more than 1000 mm, but ~83% of that happened from May to October and the other 17% happened from November to next April. The seasonal change of precipitation in this region is so large, which will enlarge the seasonal signal of TWS, making it a good experimental region for this study. Two algorithms are employed to obtain the resulting EWH, followed by an accuracy assessment and validation with GPS and GRACE and GLDAS NOAH hydrologic model.

## 3. Determination of Water Thickness from GPS

Since the limited number of GPS stations within the study region, the annual amplitude (i.e., seasonal vertical oscillations) cannot be inverted to infer EWH at a high spatial resolution. Thus, the study region (latitude: 21.5°N–29.5°N, longitude: 97.5°E–105°E) was subdivided into 1° × 1° grids. To take into account the hydrologic loading outside the region, the region is extended by an extra 5° (

Figure 5), where the estimation of EWH in the core areas are of fundamental interest.

In this set, we have 324 EWH parameters (64 in the core area) to be estimated, with only 29 annual amplitudes of the GPS vertical positions as observations. By using the Green’s functions [

19], one can relate the elastic response (i.e., GPS vertical observations) to Earth surface loading (i.e., the unknown EWH parameters). For a point load, the vertical displacement it caused can be expressed as:

where

u is the vertical displacement (in meters),

m is the mass of load (in kilograms) and

G is the Green’s function of θ, the angular distance between the point load and the station. Theoretically, we should integrate vertical displacements caused by all point loads on the Earth surface to equal the total vertical displacement of the station, but this is impractical. In fact, the elastic vertical motion decreases rapidly with the distance from a load.

A large integrating area means more unknown parameters to be estimated that makes the ill-posed problem even worse. Thus, the additional area with a width of 5° is defined so that hydrological loads in the additional and core area are considered together. Since the number of unknown EWH parameters are larger than GPS observations, additional constraints are required to invert the EWH parameters. Here, the Laplacian operator,

B, are served as the additional constraints that smooth the solution. The equations are as follows:

where

$A=\left[\begin{array}{cccc}{\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{1}^{1}}\rho G\cdot dS}& {\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{2}^{1}}\rho G\cdot dS}& \cdots & {\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{324}^{1}}\rho G\cdot dS}\\ {\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{1}^{2}}\rho G\cdot dS}& {\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{2}^{2}}\rho G\cdot dS}& \cdots & {\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{324}^{2}}\rho G\cdot dS}\\ \vdots & \vdots & \ddots & \vdots \\ {\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{1}^{29}}\rho G\cdot dS}& {\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{2}^{29}}\rho G\cdot dS}& \cdots & {\displaystyle \sum _{\varphi ,\lambda}^{{\mathrm{\Omega}}_{324}^{29}}\rho G\cdot dS}\end{array}\right]$,

$L=\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\\ \vdots \\ {u}_{29}\end{array}\right]$,

$X=\left[\begin{array}{c}{H}_{1}\\ {H}_{2}\\ \vdots \\ {H}_{324}\end{array}\right]$.

where

A is the design matrix of observation equations consisting of integrated vertical motion caused by 1 m height water change. ϕ and λ are latitude and longitude. During integration, each 0.025° × 0.025° area is treated as a point load.

L is the observations consisting of the annual amplitudes of the vertical positions at the 29 sites.

X is the vector of the 324 EWH parameters.

ρ is the density of liquid water,

dS is the area of the integral element.

G is Green’s functions which are consistent with those calculated by Guo et al. [

41] for the preliminary reference Earth model (PREM) of Dziewonski and Anderson [

42]. The solution for Equation (2) can be written as:

where

P is the weight matrix of

L,

k is an unknown nonnegative parameter. This becomes the classic regularization problem. If

k is set to zero,

$\widehat{X}$ is the least squares (LS) solution, but here we cannot get a reliable LS solution. The regularized solution (3) depends on the choice of a proper regularization parameter

k. Many methods have been developed to determine the

k, including L-curve method [

43,

44,

45], MMSE criterion method [

46], the principal components estimator [

47], Akaike’s Bayesian information criterion [

48] and the HVCE method [

49,

50], etc. Here, we choose the MMSE and HVCE methods to determine

k, respectively. The MMSE method is performed using the same procedures as in [

46] and will not be further illustrated here. The equations for HVCE in this study can be written as:

where

$S=\left[\begin{array}{cc}{n}_{1}-2tr({N}^{-1}{N}_{1})+tr({N}^{-1}{N}_{1}{N}^{-1}{N}_{1})& tr({N}^{-1}{N}_{1}{N}^{-1}{N}_{2})\\ tr({N}^{-1}{N}_{1}{N}^{-1}{N}_{2})& {n}_{2}-2tr({N}^{-1}{N}_{2})+tr({N}^{-1}{N}_{2}{N}^{-1}{N}_{2})\end{array}\right]$,

$\widehat{\theta}={\left[\begin{array}{cc}{\widehat{\sigma}}_{01}^{2}& {\widehat{\sigma}}_{02}^{2}\end{array}\right]}^{\text{\hspace{0.05em}}T}$,

$W=\left[\begin{array}{l}{V}_{1}^{T}P{V}_{1}\\ k{V}_{2}^{T}{V}_{2}\end{array}\right]$,

$N={N}_{1}+{N}_{2}$,

${N}_{1}={A}^{T}PA$,

${N}_{2}=k{B}^{T}B$,

${V}_{1}=AX-L$,

${V}_{2}=BX$.

n_{1},

n_{2} are the number of observation equations and constraint equations, respectively.

${\widehat{\sigma}}_{01}^{2}$ and

${\widehat{\sigma}}_{02}^{2}$ are the variances of unit weight for observation equations and constraint equations, respectively.

V_{1} and

V_{2} are the residual vectors for observation equations and constraint equations. Give a small value to

k as initial weights for constraint equations and the weights for observations are assumed to be unchangeable,

${\widehat{\sigma}}_{01}^{2}$ and

${\widehat{\sigma}}_{02}^{2}$ can be estimated by Equations (3) and (4) in an iterative manner. During iterations, the

jth new

k is reset to:

The

${\widehat{\sigma}}_{01}^{2}$ and

${\widehat{\sigma}}_{02}^{2}$ are repeatedly estimated until 0.999 <

${\widehat{\sigma}}_{01}^{2}/{\widehat{\sigma}}_{02}^{2}$ < 0.001. Note that the initial

k cannot be too far away from the final

k as the number of redundant observations is small in this case that the above procedures will have limited capacity to adjust the weights automatically. If the

k is too large, the iteration may not converge. The values of

k and variance of united weight estimated by MMSE and HVCE are shown in

Table 2. As two methods use different criteria, the estimated

k and

${\widehat{\sigma}}_{0}^{2}$ are different, and hence, different estimated EWHs.

## 4. Equivalent Water Height from GRACE and GLDAS Hydrological Model

In this section, TWS derived from GRACE data and Global Land Data Assimilation System (GLDAS) [

51] are used to validate the GPS-derived TSW. Comparisons among the three data sets are conducted.

GRACE is a joint scientific mission of the National Aeronautics and Space Administration (NASA) and the German Aerospace Center that has been measuring the time-variable Earth gravity field since its launch on 17 March 2002 [

52]. The degree-60 GRACE Level-2 Release05 (RL05) GSM monthly gravity data product, in the form of spherical harmonic coefficients (SHC), allows us to compute the EWH at a regular grid. These data are accessible solutions calculated from the Center for Space Research (CSR) at University of Texas. The data, spanning between January 2004 and December 2012 are used. Before deriving EWH, the C

_{20} term is replaced in the GRACE GSM data by the Satellite Laser Ranging (SLR) results [

53]. The degree one terms are added by the results from Swenson et al. [

54] to take into account the geocenter motion. To remove spatially correlated errors resulting from the accumulated sum of instrument, orbit, and model errors that generated high uncertainties of stokes coefficients at a higher degree [

55], Gaussian smoothing with 350 km averaging radius has been applied [

56]. Standard practices are followed to compute EWH. After obtaining the monthly EWH, we use the same temporal model to fit the data as that of GPS. Finally, we obtain the annual amplitudes of the GRACE-derived EWH at 1° × 1° grid.

The Global Land Data Assimilation System (GLDAS) [

51], currently has four land surface models: Mosaic, Noah, the Community Land Model (CLM), and the Variable Infiltration Capacity (VIC). The hydrological data used in this study is from the GLDAS-1 Noah model. GLDAS could provide the 3-hourly or monthly 0.25 and 1.0° products globally. In this study, the monthly 0.25° products are employed. In GLDAS model, TWS is the sum of soil moisture in all layers, accumulated snow, and plant canopy surface water, so the groundwater is not included. In Noah model, there are four layers of soil moisture with the maximum depth of 2.0 m, so we sum up the four layers of soil moisture, snow water equivalent, and total canopy water storage to obtain TWS. TWS data from January 2010 to December 2014 are calculated. And then a temporal model consisting of constant, linear and annually varying terms are fitted to the TWS time series at a regular grid. Finally, we obtain the annual amplitude of the TWS from GLDAS-1 Noah model.