Least Squares Collocation for Estimating Terrestrial Water Storage Variations from GNSS Vertical Displacement on the Island of Haiti

Abstract

Water masses are continuously redistributing across the Earth, so accurately estimating their availability is essential. Global Navigation Satellite Systems (GNSSs) have demonstrated potential for observing vertical deformations, which is partly driven by terrestrial water storage (TWS) variations. This capability has been used in hydrogeodesy to estimate TWS variations. However, GNSS data inversions are often ill-posed, requiring regularization for stable solutions. This study considers the Least Squares Collocation (LSC) statistical method as an alternative. LSC uses covariance functions to characterize observations, parameters, and their interdependence. By incorporating additional physical information into inverse models, LSC allows ill-posed problems stabilization. To assess LSC effectiveness, we apply it to observed and simulated GNSS vertical displacement on Haiti island. Hydrological signals are modeled using Global Land Data Assimilation (GLDAS) data. In sparse GNSS data regions, findings indicate poor agreement between TWS and hydrological input, with a Root-Mean-Square-Error (RMSE) of 115 kg/m², a correlation of 0.3, and a reduction of 73%. However, in dense simulated GNSS areas, TWS and hydrological input show strong agreement, with an RMSE of 41 kg/m², a correlation of 0.83, and a reduction of 92%. The results confirm LSC potentiality for assessing TWS changes and improving water quantification in dense GNSS station region.

1. Introduction

Water is essential for sustaining life, society, and ecosystems. However, this vital resource is under increasing pressure due to rising demand and environmental changes. Monitoring its availability on or near the Earth’s surface is therefore of paramount importance. Terrestrial water storage (TWS), which includes water stored in lakes, rivers, soil moisture, groundwater, snow and ice, and vegetation canopy is a key indicator for assessing water availability. Direct measurement methods, such as piezometers for groundwater or soil moisture sensors, provide valuable insights but are often limited in spatial coverage and are labor-intensive. Hydrological models, which integrate ground-based and satellite observations, offer a broader understanding of TWS’s spatial and temporal patterns. However, these models often involve uncertainties due to incomplete observational data or model limitations. The Gravity Recovery and Climate Experiment (GRACE) satellite mission has successfully estimated TWS variations at regional to global scales using spatial gravity measurements, with a temporal resolution of one month. However, GRACE’s spatial resolution (~350 km) is insufficient for small island nations such as Haiti [1,2]. In 2001, ref. [3] demonstrated the potential of the Global Navigation Satellite System (GNSS) for geophysical analysis. Subsequent studies have revealed a clear correlation between GNSS-derived seasonal vertical displacements and GRACE-detected hydrological loading signals [4,5,6,7,8]. Ref. [9] further demonstrated a strong agreement between GNSSs and the Water Global Assessment and Prognosis (WaterGAP) model [10] on a global scale. Similarly, ref. [11] found a good correlation between GNSSs and the Global Land Data Assimilation System (GLDAS) over the island of Haiti, reinforcing the feasibility of using GNSS data for TWS retrieval. Efforts to retrieve hydrological signals from direct GNSS data inversion have been conducted at global [12], continental [13], and regional scales [14]. These studies have primarily focused on areas with significant hydrological mass variations across different temporal scales, including sub-seasonal, seasonal, inter-annual [15], and even daily variations [16]. A key characteristic of these studies is their reliance on regions with substantial hydrological signal coverage in both spatial and temporal dimensions. However, GNSS-based TWS estimation faces challenges due to the ill-posed nature of the inverse problems, which are often underdetermined and numerically unstable, particularly in areas with sparse GNSS station coverage. To address this, regularization techniques are commonly employed [17,18,19,20]. The Tikhonov regularization method has been widely applied in TWS inversion [14,21,22,23,24,25,26,27], where a regularization parameter is introduced to stabilize the solution. Typically, the L-Curve method is used to optimize this parameter. Recently, ref. [28] applied the Truncated Singular Value Decomposition (TSVD) method to estimate TWS variations in Taiwan. TSVD stabilizes inversion by truncating the ill-conditioned design matrix, discarding its smallest singular values. The truncation parameter is selected using the General Cross Validation method. In 2023, ref. [29] proposed a combined approach, referred to as the TSVD–Tikhonov regularization method, that combines both TSVD and Tikhonov techniques to further stabilize the GNSS-based inversion. Tikhonov, TSVD, and their combined form aim to balance solution accuracy and stability but rely on mathematical optimization rather than physical priors.

In this study, we propose an alternative approach, similar to [30], which uses an a priori constraint method that avoids using explicit regularization parameters by leveraging the intrinsic physical characteristics of the data for stabilization. Specifically, we adopt the Least Squares Collocation (LSC) method, a statistical approach based on the covariance function. LSC has been applied successfully in geoid determination [31], interpolation of ground-based data [32], geoid modeling [33], photogrammetry [34], and geodetic transformation [35]. LSC assumes that the spatial variability of the observed field (hydrological or GNSS) can be described by a covariance function, implying that observations at one location provide information about neighboring locations. This spatial dependence diminishes with increasing distance.

This paper assesses the capability of LSC to retrieve TWS variations from GNSS-derived vertical displacements. Firstly, we validate our approach using a synthetic GNSS dataset for Haiti and we then analyze real GNSS time series from operational stations on the island. The article is organized as follows: Section 2 details method and data, Section 3 presents results and discussion, and finally, Section 4 addresses the conclusion.

2. Materials and Methods

This section introduces the basics of the Least Squares Collocation (LSC) method and the GNSS network of the island of Haiti used in this study.

2.1. Least Squares Collocation: Basic Formulation

LSC is a statistical estimation technique used to determine the values of a stochastic process and their associated uncertainties. It is a flexible method allowing direct signal predictions at any location within the study domain by integrating the statistical properties of the observed field, represented by a covariance function. The basic equation for LSC, as described by [36], can be written as follows:

$$d=AX+n$$

(1)

where d is the measurement vector, A is the coefficient matrix of the forward model, X is the vector of unknown parameters, and n represents the measurement error. The covariance matrices C_∈∈ and C_dd characterize the statistical behavior of the noise n and signal d, respectively. The signal at any location $\widehat{P}$, is estimated as follows:

$$\widehat{P}= C_{pd}{\left.\left(C_{\mathsf{\in }\mathsf{\in }} + C_{dd}\right.\right)}^{-1}d$$

(2)

where $C_{pd}$ represents the cross-covariance matrix of the signal and parameters, derived from analytical covariance models. The diagonal elements of $C_{\mathsf{\in }\mathsf{\in }}$ represent the variances of GNSS data uncertainties, while off-diagonal elements are assumed to be zero due to the uncorrelated nature of GNSS errors. The sum $C_{\mathsf{\in }\mathsf{\in }}+C_{dd}$ known as the total covariance of observations is the sum of the GNSS noise covariance matrix and the signal covariance matrix, which is generated from the simulated vertical displacement covariance function. This matrix is symmetric, positive-definite, and thus invertible.

To estimate the parameters, the process starts by calculating the mean values of the data, which are then used to calculate the mean values of the parameters using the forward model. This provides an a priori solution, which is subtracted from the data to yield residuals. These residuals are then inverted, and the a priori solution is subsequently added back to the inverted residuals. Given that the a priori value of the parameters $P_0$ and data $d_0$ are known, Equation (2) can be rewritten as follows:

$$\widehat{P}={\mathrm{P}}_0+{\mathrm{C}}_{\mathrm{pd}}{\left.\left({\mathrm{C}}_{\in \in }+{\mathrm{C}}_{\mathrm{dd}}\right.\right)}^{-1}\left(d-{\mathrm{d}}_0\right)$$

(3)

For GLDAS/NOAH data, the a priori solution corresponds to the monthly mean soil water content over Haiti, averaging $527 \mathrm{k}\mathrm{g}/{\mathrm{m}}^2$ from 2010 to 2019. The a priori values for the data are determined as follows:

$${Gm}_0=d_0$$

(4)

where G represents the Green’s function, and $m_0$ is the a priori solution.

Beyond estimating $\widehat{P}$, the uncertainty of the estimated parameters C_pp is also determined as follows:

$$C_{pp}=C_{pop} - C_{pd}{\left.\left(C_{\mathsf{\in }\mathsf{\in }} + C_{dd}\right.\right)}^{-1}{C}_{dp}$$

(5)

where $C_{pop}$ represents the covariance matrix of the estimated signal $\widehat{P}$.

2.2. Problem Characterization

To evaluate the degree of ill-posedness in our problem, we analyze the rate of decay in the singular value spectrum of the design matrix A (Equation (1)). Singular value decomposition is a commonly used technique for assessing the stability of such problems. Figure 1 (left panel) shows the singular values of the design matrix for the forward problem [11], sorted in descending order, and Figure 1 (right panel) presents the histogram of fitting residuals. The classification of ill-posed problems is based on the decay behavior of singular values as a function of rank K. A problem is considered mildly ill-posed if the decay of the singular values follows a power-law decay of the form $K^{-\alpha }$ with α ≤ 1, moderately ill-posed if α > 1, and severely ill-posed if the decay follows an exponential form such as $\exp (-\alpha K)$ [18]. For the inversion problem over the island of Haiti, the singular value exhibits an exponential decay described by ${\sigma }_k=8.42\times {10}^{-12}\times \exp (-0.52k)$. This exponential decay confirms that the problem is severely ill-posed, necessitating the use of regularization techniques to obtain a stable solution.

2.3. Analytical Covariance Function Determination

Determining an empirical covariance function is inherently challenging, as it requires a complete understanding of the underlying physical processes within the study area. Additionally, a dense distribution of data points is necessary for accurately estimating the empirical covariance function. According to [37], empirical covariance functions were computed, and an analytical covariance function was fitted to construct the covariance matrix, allowing the estimation of unknown values at locations without measurement.

To select the appropriate analytical covariance function, we have tested different functions presented in

Table 1. List of covariance functions.

Covariance Models
Name	Function	Reference
Gaussian	$C_o{e}^{\left({\displaystyle \frac{{-r}^2}{{2\alpha }^2}}\right)}$	[38]
Hirvonen	${\displaystyle \frac{C_o}{{\left(1+{\alpha }^2 r^2\right)}^2}}$	[39]
Exponential	$C_o{e}^{\left({\displaystyle \frac{-rln2}{\alpha }}\right)}$	[40]
Exponential	$C_o{e}^{\left({\displaystyle \frac{-r}{\alpha }}\right)}$	[41]

and plotted in Figure 2.

We tested these 4 functions for each month of the year 2019 with real GNSS data.

Table 2. RMSE, reduction and correlation annual mean for Gaussian, Exponentials, and Hirvonen covariance functions.

	RMSE (kg/m2)	Reduction (%)	Correlation
Gauss [38]	116.5	75.5	0.06
Exponential [41]	104.91	77.9	0.078
Exponential (ln2) [40]	121.48	74.6	0.089
Hirvonen [39]	223.10	53.87	0.15

below showed the annual mean results for RMSE, reduction, and correlation. We observed a slightly similar result for Exponential and Gaussian functions, but both outclass Hirvonen function.

To simplify the analysis, we adopted a Gaussian covariance function with a uniform correlation length for both covariance and cross-covariance functions. The Gaussian function, one of the most widely used covariance functions, is expressed as follows:

$$f(r)=a{e}^{-{k^{2}r}^2}$$

(6)

where the correlation length represents the distance at which the variance $C_0$ decreases to half its value $\left({\displaystyle \frac{C_0}{2}}\right)$ at zero distance [39].

To determine the empirical hydrological covariance function, we used the GLDAS dataset (Figure 3), representing the monthly mean GLDAS/NOAH data from 2010 to 2019 at a spatial resolution of 0.25° × 0.25°. To determine the simulated GNSS vertical displacement covariance function, we generated data for 282 synthetic GNSS stations, evenly distributed across the island (Figure 4). This dense station network ensures good spatial coverage of the island and sufficient information to characterize the covariance function.

Using the constructed GLDAS covariance function (Equation (7a)) and the GLDAS monthly mean data (Figure 3), we interpolated the GLDAS data at the synthetic GNSS station locations (Figure 4), resulting in a denser hydrological dataset (Figure 5, left panel). This dataset was used to calculate the simulated loading at these stations using the Green’s function approach [42]. The empirical covariance function for the vertical displacement of the Earth’s surface was then derived from the simulated loadings.

Figure 5 (right panel) above displays the inversion results using the cross-covariance function (Equation (7c)). A good agreement between the input data and the inverted solution validates the covariance function’s appropriateness. The observed differences include a maximum of 66 kg/m², a minimum of −86 kg/m², a correlation of 0.83, a reduction of 92%, and an RMSE of 41 kg/m² for a signal with a mean amplitude of 536.3 kg/m².

The empirical covariance and cross-covariance functions were fitted into the appropriate analytical functions (Figure 6). First, we fixed the variance for the simulated covariance and cross-covariance functions. Through simulations, we determined that a correlation length of 0.37° provided the best inversion results. While simulations showed discrepancies between analytical and empirical covariance functions, such deviations are expected. As noted by [43], these inconsistencies can arise due to dependencies and biases in the empirical covariance function.

The analytical expressions for GLDAS covariance function (Equation (7a)), the simulated vertical displacement covariance function (Equation (7b)), and GLDAS and simulated vertical displacement cross-covariance function (Equation (7c)) are as follows:

$$f(r)=4483\times e^{({\displaystyle \frac{-r^2}{2\times L{C}^2}})}$$

(7a)

$$f(r)=3.18\times {10}^{-07}{\times e}^{({\displaystyle \frac{-r^2}{2\times L{C}^2}})}$$

(7b)

$$f(r)=-0.06\times e^{({\displaystyle \frac{{-r}^2}{2\times L{C}^2}})}$$

(7c)

2.4. GNSS Network

This section focuses on the GNSS network data available over the island, as shown in Figure 7. The network consists of 21 stations in Haiti and 26 in the Dominican Republic [44]. For detailed information about the data processing methods applied to the GNSS network, readers are referred to [11,45]. Figure 8 shows the data availability for the GNSS network during the study period spanning from July 2011 to February 2022. Unlike other regions with dense operational GNSS stations [14], we have a relatively limited number of operational stations in the island.

3. Results and Discussion

This section presents the results of the simulated and hydrological signals based on data from the GNSS network in Haiti.

3.1. Simulated Data Result

To evaluate the effectiveness of the LSC method in retrieving the hydrological signal, two types of synthetic tests were conducted: a data resolution test and an error estimation test. The data resolution test examines the impact of the number of stations on the inversion results, while the error estimation test assesses the influence of data noise.

Data Resolution Test

Given the limited number of operational GNSS stations on the island, we performed a data resolution test to assess the sensitivity of the LSC inversion method to the number of available stations. We considered 13 sets of simulated stations, comprising, respectively, 1, 10, 20, 40, 60, 80, 100, 120, 140, 160, 180, and 200 stations. The number of stations progressively increased from 1 to 200, with selected examples illustrated by red dots in Figure 9, showing the inverted solutions.

For each station set, we estimated the simulated vertical displacement at the station locations. These simulated displacements were then inverted to retrieve the original GLDAS data (Figure 3). The results demonstrate that increasing the number of stations enhances the retrieval accuracy of the initial hydrological signal. Specifically, as the station count rises, the correlation coefficient shows a significant increase (Figure 10a), the reduction coefficient, which represents the average contribution of the inverted hydrological data to the original hydrological data, is calculated according to [46,47], exhibits a slight improvement (Figure 10b), and the RMSE decreases substantially (Figure 10c). These trends are expected, as the original hydrological data contain inherent spatial variability with localized peaks.

Further analysis in Figure 11 reveals that as the number of stations increases, the standard deviation of the inversion decreases. In regions with higher station density, such as the central part of the island, the standard deviation is lower compared to coastal areas, where station coverage remains sparse. This highlights the importance of station density in achieving more accurate and stable inversion results.

In conclusion, these simulations demonstrate that the LSC method can effectively reproduce TWS in regions with a dense GNSS network. The results indicate a strong agreement between the input values (Figure 3) and the inversion results (Figure 9), validating the method’s effectiveness. Furthermore, the number of stations used in the data resolution test for Haiti represents a technically realistic scenario. However, the findings also highlight the challenge of achieving reliable spatiotemporal inversions across the entire island due to the limited availability of operational GNSS stations. Expanding the GNSS network could significantly improve the accuracy and robustness of hydrological signal retrieval in this region.

3.2. Error Estimate Test

To assess the sensitivity of the inversion method to errors in the observation data, we conducted two specific tests: the stability test and the robustness test.

3.2.1. Stability Test

The stability test examined how the inversion method responds to varying levels of noise in the observation data. We simulated different noise amplitudes using a configuration of 160 synthetic stations on the island (Figure 12). Initially, we applied an error to the simulated displacements with an amplitude equal to that of the simulated vertical displacement (1/1). Subsequently, we progressively reduced the noise amplitude by considering error levels of 1/3, 1/6, and 1/9 of the simulated vertical displacement amplitude.

The results, summarized in

Table 3. Summary of the stability test results.

Error (Fraction svd)	Max Difference (kg/m2)	Min Difference (kg/m2)	RMSE (kg/m2)	Correlation (%)	Reduction (%)
1/1	149	−121	48	75.26	91
1/3	131	−101	42	81.9	92
1/6	117	−99	39	83.76	93
1/9	112	−97	38	84	93

, indicate that as the error decreases, the solution improves. Small modifications in the data error resulted in small adjustments to the inversion solution, indicating that the inversion method is stable [48]. The inversion solutions for the different noise levels are shown in Figure 13.

3.2.2. Robustness Test

A numerical method is considered robust if it remains stable despite the presence of a small number of large errors [17,18,49]. To assess the robustness of the inversion, we used 160 synthetic stations, as shown in Figure 12. Initially, we applied a relative error of 20% to the simulated vertical displacement values at 157 stations (represented by black dots). Then, we introduced errors 100 times larger (considered outliers) at three stations (represented by red dots). Despite these large errors at the three stations, the inversion solution remained stable (see

Table 4. Summary of the robustness test results.

Error (Fraction svd)	Max Difference (kg/m2)	Min Difference (kg/m2)	RMSE (kg/m2)	Correlation (%)	Reduction (%)
100	120	−96	40	83	93
1/5	120	−100	40	83	92

and Figure 14). This outcome demonstrates the robustness of the inversion method. The left panel of Figure 14 displays the inversion results, while the right panel shows the corresponding standard deviation.

3.3. Real GNSS Data Result

For the comparison, we used monthly GNSS and GLDAS data. With Tsoft software version 2.2.15 [50], the monthly time series for both datasets were detrended and filtered using a low-pass filter with a cut-off frequency of 1/25 cycle per day [11]. The first day of each month was selected from the filtered signals. The LSC method was then applied to invert each filtered monthly GNSS dataset on a specific grid, while the monthly GLDAS data were interpolated onto the same grid. Both datasets utilized a common correlation length of 0.37°. The inversion process was conducted for stations with continuous data throughout the year and the fewest data gaps. Due to the limited number of stations operating simultaneously during the study period, the comparison focused on GLDAS data from 2019, 2020, and 2021. For these years, 19, 27, and 21 stations were used, respectively, with their locations shown as red dots in Figure 15. These stations were unevenly distributed across the island. Monthly GLDAS/NOAH data served as the reference for the spatiotemporal comparison (Figure 15, left column).

3.3.1. Spatiotemporal Result

Figure 15 (middle column) presents the inverted solutions for April 2019, 2020, and 2021. April was selected due to its highest correlation 0.3 between the inversion and the interpolated GLDAS model, which reduced by 73% and showed an RMSE of 115 kg/m² (Figure 16). The hydrological signal is successfully retrieved in regions near GNSS stations. In areas without GNSS stations, the inverted and a priori solutions are similar, as expected. Overall, the solution does not always align perfectly with the GLDAS model. For some months, the solutions and GLDAS model exhibit similar trends, while for others, the inversion either overestimates or underestimates the GLDAS model. The results demonstrate that as the number of stations involved in the inversion increases, the standard deviation of the solution decreases (Figure 15, right column). Additionally, areas with denser GNSS station coverage show lower standard deviations. Figure 16 presents the correlation, reduction, and RMSE monthly results for 2019.

3.3.2. Temporal Inversion by Cell

To assess the impact of the number of stations on the amplitude of the inverted signal, we performed a temporal inversion on a cell-by-cell basis, as the number of available GNSS stations was insufficient for reliable spatiotemporal verification. We present the inversion results for cells 15 and 25, considering the influence of stations (JME2, CN05) on the inversion signal for cell 15 and (JME2, CN05, and CRLR) for cell 25. Figure 17 illustrates the GLDAS grid cells, represented by blue dots and numbered from cell 1 to cell 101. The figure also includes a selection of arbitrary operational GNSS stations (shown as red dots) for the period between 2011 and 2016.

The inversion results shown in Figure 18 (top panel) highlight the influence of GNSS station operations on the hydrological inverted signal at cell 15. During the period when station JME2 was not operational from January 2011 to June 2013 (Figure 17, top panels and Figure 18, bottom panel), the hydrological inverted signal at cell 15 remained flat (Figure 18, top panel). After JME2 resumed operations in June 2013 (Figure 17, middle panels and Figure 18, bottom panel), the amplitude of the hydrological inverted signal at cell 15 increased (Figure 18, top panel). In contrast, when JME2 was again non-operational from February 2015 to June 2016 (Figure 17, bottom-left panel and Figure 18, bottom panel), the amplitude of the inverted signal decreased (Figure 18, top panel). The reactivation of JME2 in June 2016 (Figure 17, bottom-right panel, and Figure 18, bottom panel) resulted in a subsequent increase in the amplitude of the inverted signal (Figure 18, top panel).

A similar trend is observed for cell 25 (Figure 19). Between January 2011 and February 2014, when station CN05 was non-operational, the hydrological inverted signal remained flat (Figure 19, top panel). Once CN05 resumed operations in February 2014 (Figure 19, middle panel), the amplitude of the signal in cell 25 increased (Figure 19, top panel). Similarly, when the CRLR station became operational in December 2015 (Figure 19, bottom panel), there was a notable increase in the hydrological inverted signal amplitude at cell 25 (Figure 19, top panel). This pattern suggests that as the number of operational GNSS stations increases, the hydrological signal retrieval improves, leading to a more accurate inverted solution.

In summary, real GNSS stations can effectively generate hydrological signals in areas where GNSS data are available. The greater the number of stations, the more hydrological signals can be retrieved. However, the spatiotemporal comparison with GLDAS/NOAH data at both daily and monthly scales was not entirely successful. This result is not unexpected, as prior simulations indicated that, due to the limited number of stations across the island, achieving a reliable spatiotemporal inversion for the entire region would be challenging.

4. Conclusions

This study aimed to assess the feasibility of the Least Squares Collocation (LSC) method for estimating terrestrial water storage variations using GNSS-derived vertical displacements in Haiti. The synthetic tests demonstrated that the LSC method, when applied with a Gaussian covariance function, effectively estimates terrestrial water storage from a dense and homogeneous GNSS network. The retrieval of the hydrological signal improves as the number of stations and their spatial coverage increase. Due to the limited number of operational GNSS stations on the island, it was not possible to accurately reproduce the spatiotemporal hydrological signal from the GLDAS/NOAH model. Nevertheless, the LSC approach shows considerable potential for estimating hydrological signals from a dense GNSS network. Future research should focus on testing this method in regions with a higher density of operational GNSS stations and comparing LSC with alternative statistical regularization methods.