Multivariate Prediction of Total Water Storage Changes Over West Africa from Multi-Satellite Data

Abstract

West African countries have been exposed to changes in rainfall patterns over the last decades, including a significant negative trend. This causes adverse effects on water resources of the region, for instance, reduced freshwater availability. Assessing and predicting large-scale total water storage (TWS) variations are necessary for West Africa, due to its environmental, social, and economical impacts. Hydrological models, however, may perform poorly over West Africa due to data scarcity. This study describes a new statistical, data-driven approach for predicting West African TWS changes from (past) gravity data obtained from the gravity recovery and climate experiment (GRACE), and (concurrent) rainfall data from the tropical rainfall measuring mission (TRMM) and sea surface temperature (SST) data over the Atlantic, Pacific, and Indian Oceans. The proposed method, therefore, capitalizes on the availability of remotely sensed observations for predicting monthly TWS, a quantity which is hard to observe in the field but important for measuring regional energy balance, as well as for agricultural, and water resource management. Major teleconnections within these data sets were identified using independent component analysis and linked via low-degree autoregressive models to build a predictive framework. After a learning phase of 72 months, our approach predicted TWS from rainfall and SST data alone that fitted to the observed GRACE-TWS better than that from a global hydrological model. Our results indicated a fit of 79 % and 67 % for the first-year prediction of the two dominant annual and inter-annual modes of TWS variations. This fit reduces to 62 % and 57 % for the second year of projection. The proposed approach, therefore, represents strong potential to predict the TWS over West Africa up to 2 years. It also has the potential to bridge the present GRACE data gaps of 1 month about each 162 days as well as a—hopefully—limited gap between GRACE and the GRACE follow-on mission over West Africa. The method presented could also be used to generate a near-real-time GRACE forecast over the regions that exhibit strong teleconnections.

Appendices

Appendix 1: Computational Details of Total Water Storage Fields

In order to prepare the data sets for analysis, the following processing steps were applied.

• The GRACE Level-2 data that are used here are derived in terms of fully normalized spherical harmonic (SH) coefficients of the geopotential fields (Flechtner 2007). Firstly, the fields were augmented by the degree-1 term from Rietbroek et al. (2009) in order to include the variation of the Earth’s center of mass with respect to a crust-fixed reference system.

• GRACE SHs at higher degrees are affected by correlated noise and are, therefore, smoothed by applying the DDK2 decorrelation filter (Kusche et al. 2009). Werth et al. (2009a) found that the DDK2-filtered GRACE solutions are generally in good agreement with the output of global hydrological models. However, GRACE solutions are also contaminated by errors due to incomplete reduction of short-term mass variations by de-aliasing models (Forootan et al. 2013, 2014). We found that the impact of atmospheric de-aliasing errors on the GRACE-derived TWS over West Africa is negligible (see atmospheric errors over the Niger Basin in Forootan et al. 2014).

• GRACE DDK2 filtered solutions up to degree and order 120 were then used to generate the global TWS values according to the approach of Wahr et al. (1998).

• Similar to the GRACE products above, the DDK2 filter was applied to the gridded WGHM-TWS data set in order to preserve exactly the same spectral content as with the filtered GRACE products.

• After filtering, all data sets were converted to 0.5° × 0.5° grids similar to the WGHM-TWS outputs.

• From each data set, a rectangular region that includes West Africa (latitude between 0° and 25°N and longitude between −20° and 10°E) was selected.

Lake Volta (see Fig. 7) is one of the largest man-made reservoirs in the world, created by the Akosombo Dam, which holds back the water for generating hydroelectric power (for details see Speth et al. 2011). Satellite altimetry observations indicate a sharp increase of water level since mid 2007, where much of the excess water resulted from heavy rainfall within the catchment (Crétaux et al. 2011). This introduces an artificial TWS anomaly located over the lake, which is removed to avoid its misinterpretation as a part of subsurface TWS changes. The equivalent water height (EWH) change of Volta was computed by assuming a grid mask representing a unit change in EWH of 1 mm over the entire lake surface and zero elsewhere. The grid mask has been converted into a set of spherical harmonic coefficients up to degree 120 and subsequently filtered using the same DDK2 filter used for filtering the original GRACE-TWS data. Then, each field was scaled using the lake height time series (in mm) derived from the results of Crétaux et al. (2011). The averaged storage changes derived from GRACE-TWS (from GFZ) and altimetry are shown in Fig. 7. Both altimetry and GRACE GFZ-TWS indicate an increase of water storage within the lake. The amplitude of the signal derived from GRACE GFZ-TWS is larger than that of the altimetry likely since GRACE-TWS also reflects the groundwater signal of the surrounding area of the lake. For the lake area, we estimate a TWS increase of 2.95 ± 1.32 km³ year⁻¹, during 2003 to 2010. The time series of Lake Volta water storage changes were then removed from GRACE-TWS fields (including both GFZ and ITG2010).

Fig. 7

Overview of water storage changes of Volta Lake. The graph on top shows the location of the lake, while that of the bottom-left compares the averaged contribution of Volta Lake level changes (derived from altimetry in black) with the averaged TWS variations derived from GRACE (GFZ-TWS products, in red). The bottom-right graph shows the GRACE GFZ-TWS signal after removing the water storage signal of Volta Lake

In order to compare the signal strength over the region, the signal root mean square (RMS) value and the linear trend of the three mentioned TWS data sets (GFZ, ITG2010, and WGHM) are computed for the period January 2003–August 2009, in which the three data sets were available (see Fig. 8). From the RMS, one concludes that all the three data sets show a strong variability over the tropical and the Gulf of Guinea coastal regions. The computed linear trends, however, are not consistent. Particularly, GRACE-derived TWS changes show a mass gain over Volta Lake, which we remove from the GRACE-TWS fields before performing decomposition. Removing such artificial anomaly is necessary, since otherwise the amplitude of TWS forecast over the lake will be overestimated.

Fig. 8

Comparing the signal variability (RMS) and linear trends of TWS data used in this study after smoothing using the Kusche et al. (2009)’s DDK2 filter. a) TWS data of GRACE GFZ RL04, b TWS data of GRACE ITG2010, and c TWS of WGHM

Appendix 2: Details of ICA and ARX Methods

This appendix provides details of computations regarding to the methodology described in Sect. 3.

1.1 The ICA Computations

ICA decomposition is performed here by applying a 2-step algorithm (Forootan and Kusche 2012) on the available data sets, where step 1 consists of data decorrelation using principal component analysis (PCA). In step 2, the $j$-dominant components of PCA are rotated to be as independent from each other as possible. Storing the available data in a $n \times p$ data matrix $\mathbf{X}$, after removing their temporal mean, where $n$ is the number of months and $p$ is the number of grid points, ICA decomposes $\mathbf{X}$ as

$$\begin{aligned} \mathbf{X}\simeq \mathbf{X}_j{= {\bar{\mathbf{P}}_j \mathbf{R}_j \mathbf{\Lambda }_j \mathbf{R}_j^T\bar{\mathbf{E}}_j^T}}. \end{aligned}$$

(4)

In Eq. (4), $\bar{\mathbf{P}}_j \mathbf{\Lambda }_j \bar{\mathbf{E}}_j^T$ is derived from the PCA decomposition of $\mathbf{X}$ in step 1. Therefore, $\mathbf{\Lambda }_j$ is an $j \times j$ diagonal matrix that stores the singular values arranged with respect to the magnitude, $\bar{\mathbf{E}}_j$ ($j \times p$) contains the corresponding unit-length spatial eigenvectors, $\bar{\mathbf{P}}_j$ ($n \times j$) contains the associated normalized temporal components, and $j<n$ is the number of retained dominant modes (Preisendorfer 1988). The orthogonal rotation matrix $\hat{\mathbf{R}}_j$ ($j \times j$) is defined in step 2, so that it rotates PCs and make them as statistically independent as possible. The method equals to temporal ICA (Forootan and Kusche 2012), which is simply called ICA in the paper. Considering Eqs. (1) and (2), $\mathbf{Y}$ and $\mathbf{U}$ are equivalent to ${\bar{\mathbf{P}}}\mathbf{R}$, while $\mathbf{A}$ and $\mathbf{B}$ are equivalent to $\mathbf{\Lambda } {\bar{\mathbf{E}}} \mathbf{R}$. An optimum $\mathbf{R}$ was found by digonalization of the fourth-order cross cumulants of the dominant temporal components $\bar{\mathbf{P}}$ (see details in Forootan and Kusche 2012).

For properly selecting the subspace dimension $j$ or $j'$, we used a Monte Carlo approach which simulates data from a random distribution ${\mathbf{N}(\mathbf 0,\mathbf \Sigma )}$, with $\mathbf{\Sigma }$ containing the column variance of $\mathbf{X}$. The null hypothesis is that $\mathbf{X}$ is drawn from such a distribution (see also Preisendorfer 1988, pp. 199–205). To apply the rule, 100 time series realizations of ${\mathbf{N}(\mathbf 0,\mathbf \Sigma )}$ are generated, their eigenvalues computed and placed in decreasing order. The 95th and 5th percentile of the cumulative distribution are then plotted (red lines in Fig. 9). Eigenvalues from the actual data sets that are above the derived confidence boundaries are unlikely to result from a data set consisting of only noise. To estimate the uncertainties of the eigenvalues, we randomly selected a subsample of $\mathbf{X}$ and applied PCA, then selected another subsample and repeated this operation 200 times. This approach follows the ‘bootstrap**’ method as presented, e.g., in Efron (1979) and yields uncertainty estimates (see error bars in Fig. 9). The repeat number of 200 is chosen experimentally to be sure that the distribution of the estimated eigenvalues is independent from the selections of the subsamples.

To illustrate what we describe above, Fig. 9 shows the eigenvalue spectrum of the centered time series of GRACE GFZ-derived TWS, SST and rainfall computed using PCA. The significance levels are shown by red lines and the error bars show the uncertainties of eigenvalues. The eigenvalues above the red lines are statistically significant. The significant eigenvalues along with their orthogonal components are rotated toward independence using Eq. (4) and interpreted in Sect. 4.

Fig. 9

Eigenvalue results derived from implementing the PCA method on a time series of GRACE GFZ-TWS maps, b maps of SST over the Atlantic, c maps of SST over the Pacific, d maps of SST over the Indian Ocean basins, and e West African rainfall maps from TRMM. Uncertainties are shown by error bars around each eigenvalues. The red lines correspond to the significant test, described in ‘Appendix 2.’ The variance fractions of the dominant eigenvalues are represented in each graph

Based on the uncertainties of the PCA results (Fig. 9), in order to estimate the uncertainty of the ICs [Eq. (4)], we generated 100 realizations of $\mathbf{X}$, reconstructed by $\bar{\mathbf{P}}_j, \mathbf{\Lambda }_j$, and ${\bar{\mathbf{E}}_j}$ along with 100 realizations of their errors. Then, applying Eq. (4) to the realizations allows the estimation of uncertainties (see, e.g., error bars in Figs. 1, 2, 3).

The projection of the data $\mathbf{X}$ onto the i’th spatial pattern of the ICA $\hat{\mathbf{p}}_i=\mathbf{X}\hat{\mathbf{e}}_i$, provides its corresponding temporal evolution

$$\begin{aligned} \hat{\mathbf{p}}_i(t) = \sum _{s=1}^{p} {x}(t,s)\;\hat{\mathbf{e}}_i(s), \end{aligned}$$

(5)

where $t$ is time ($1,\ldots ,n$) and $s$ is the number of grid points ($1,\ldots ,p$).

1.2 The ARX Computations

Considering Eq. (3) as the ARX model, the ARX forecast requires two steps: (i) The coefficients $(a_1,\ldots ,a_{n_a})$ and $(b_{q,1},b_{q,2},\ldots ,{b_{q,{n_b}}}), q=1,\ldots , m$ are estimated, e.g., using a least squares approach. This step is usually referred to as ‘simulation’ or ‘training step’ in the literature (see, e.g., Ljung 1987). Step (i) is performed under the assumption that the output and inputs up to the time $t=t_n-1$ are known. Furthermore, the outputs and exogenous values on the right-hand side of Eq. (3) are not stochastic. To avoid negative indices, one might consider the observations $\mathbf{y}(t) = [{y}(t),{y}(t-1),\ldots ,{y}(c)]^T$, where $c= {\rm max}(n_a,n_b)+ {\rm max}(k_q)+1$. Eq. (3) is expanded as

$$\begin{aligned} \mathbf{y} =\left[ {\begin{array}{cccccc} -y(t-1)&\cdots&-y(t-n_a)&{u}_q(t-k_q)&\cdots&{u}_q(t-k_q-n_b+1)\\ -y(t-2)&\cdots&-y(t-n_a-1)&{u}_q(t-k_q-1)&\cdots&{u}_q(t-k_q-n_b)\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots \\ -y(c-1)&\cdots&-y(c-n_a)&{u}_q(c-k_q)&\cdots&{u}_q(c-k_q-n_b+1)\\ \end{array}}\right]\left[ \begin{array}{c} a_1 \\ \vdots \\ a_{n_a}\\ b_{q,1}\\ \vdots \\ b_{q,n_b} \end{array}\right]+\varvec{\** }(t), \end{aligned}$$

(6)

where $q=1,\ldots ,m$ and $\varvec{\** }(t) = [{\xi }(t), {\xi }(t-1),\ldots , {\xi }(c)]^T$. Eq. (6) can be rewritten compactly as

$$\begin{aligned} {\mathbf{y}(t)} = \mathbf{\Phi }(t) \varvec{\Theta }+\varvec{\** }(t). \end{aligned}$$

(7)

The least squares estimation of the unknown coefficients is derived from

$$\begin{aligned} \hat{\varvec{\Theta }} = {\left( {\mathbf{\Phi }(t)}^T{\mathbf{\Phi }(t)}\right) }^{-1}{\mathbf{\Phi }(t)}^T\; {\mathbf{y}(t)}. \end{aligned}$$

(8)

The quality of the fit ($\eta$) can be assessed by computing the signal-to-noise ratio as

$$\begin{aligned} \eta = 1-\frac{{\mathbf{y}(t)}^T\mathbf{\Phi }(t)\hat{\varvec{\Theta }}}{{\mathbf{y}(t)}^T\mathbf{y}(t)}. \end{aligned}$$

(9)

The residual of the ARX model ($\hat{\varvec{\** }}(t) = [\hat{\xi }(t),\hat{\xi }(t-1),\ldots ,\hat{\xi }(c)]^T$) can be estimated as

$$\begin{aligned} \hat{\varvec{\** }}(t) = \mathbf{y}(t)-{\mathbf{\Phi }(t)}\;\hat{\varvec{\Theta }}. \end{aligned}$$

(10)

In step (ii), based on $\hat{\varvec{\Theta }} = \left[ \hat{a}_1 \; \ldots \; \hat{a}_{n_a} \; \hat{b}_{q,1} \; \hat{b}_{q,2} \; \ldots \; \hat{b}_{q,{n_b}} \right] ^T$, when the inputs ${u}_q(t)$ are known, one can forecast the output $\hat{y}(t_n)$ at time $t_n$ using

$$\begin{aligned} \hat{y}(t_n)=-\sum _{i=1}^{n_a}a_{i}y(t_n-i)+\sum _{q=1}^{m} \sum _{l=1}^{n_b}{b_{q,l}{u}_q(t_n-k_q-(l-1))}. \end{aligned}$$

(11)

To estimate the uncertainty of the ARX simulation, using Monte Carlo sampling, we numerically generate several realizations of the ICs (described before). By inserting them into Eq. (3) and fitting ARX models, we are able to perform an error assessment of the fitted model up to the time $t_n$. For error estimation of the forecast (the ARX value at the time $t_n+1$ and later), however, one should compute an accumulated error, since there is no observed value for the output $y$ at time $t_n+1$ and later.