1. Introduction
The Gravity Recovery And Climate Experiment (
grace) satellite mission has provided valuable information towards understanding the continental to regional scale hydrology [
1,
2]. The time-variable gravity field of the Earth from the
grace satellite mission can be obtained from various organizations, at various levels of complexity and time resolution. Monthly, weekly and even daily fields are provided either in terms of band-limited spherical harmonic coefficients or global grids of mass change from Center for Space Research (
csr), GeoForschungZentrum (
gfz), Jet Propulsion Laboratory (
jpl), and several other processing centers. The most commonly used
grace products are the monthly fields in terms of spherical harmonic coefficients up to a certain degree and order [
3,
4], which can then be processed to obtain global grids of Equivalent Water Height (
ewh) at a desired grid size (for example a half degree, 1 degree, 2 degrees). These global fields are noisy; therefore, we first filter them and then perform spatial integration over a region of interest to obtain meaningful information [
5,
6,
7]. The limits of spatial integration are decided by our region of interest (e.g., catchment boundary).
Wahr et al. [
8] demonstrated that filtering of
grace products is inevitable and will cause signal leakage between ocean and land. Therefore, after the launch of
grace satellites, a number of filtering methods were developed with an aim to reduce signal leakage [
9,
10,
11,
12,
13,
14]. However, soon it was realized that sophisticated filtering algorithms may reduce signal leakage but would introduce biases that vary in space and in time [
11,
15,
16]. Furthermore, filtering affects the spatial resolution [
17] and damages the signal via leakage and attenuation [
8,
10,
18,
19,
20,
21]. Therefore, many research contributions developed dedicated correction methods for repairing the signal damage due to filtering [
14,
22,
23]. Signal damage and decay in the spatial resolution are related. Therefore, ideally, correction methods are supposed to improve the effective spatial resolution of
grace products. Although there have been several studies that compare the efficacy of correction methods [
22,
24,
25], the impact on the spatial resolution has not been investigated at a global scale.
The correction methods can be classified in terms of the application for which they were developed (for example: for ice sheets, for hydrology, and for land-ocean signal leakage), or in terms of the source of the correction quantity (for example: model-dependent and data-driven) [
6,
19,
20,
21,
24,
25,
26,
27,
28,
29]. Since in this article we are focusing on land-hydrology from
grace, we choose the correction methods relevant for hydrological investigations only. Furthermore, we would like to classify the selected correction approaches based on their source of correction terms: model-dependent and data-driven. Since our aim is to understand how the application of correction methods affects the spatial resolution, we first need to discuss the idea behind the spatial resolution of
grace products.
The mass change fields from
grace satellite mission are outcomes of geophysical inversion of the satellite observations, which is in stark contrast to the products of optical/microwave remote sensing. Therefore, the idea of spatial resolution for
grace fields is strongly tied to the geophysical inversion process. In fact, at the level of satellite observations (satellite to satellite tracking data in
grace parlance), the concept of
gravimetric resolution is more relevant. It is defined as the capability of
grace satellites to detect a given mass change of any size [
30]. However, for scientific applications, the
grace spherical harmonic coefficients are processed and synthesized as maps of a preferred grid size in the spatial domain. Therefore, while communicating
grace products to the hydrology and the remote sensing community, it is more relevant and practical to discuss the
spatial resolution of
grace products and how it is affected due to post-processing [
17]. It is to be noted here that users may prefer half degree gridded fields over two-degree gridded fields by synthesizing a given set of
grace spherical harmonic coefficients, but this will not improve the information content because the former is akin to an interpolated version of the latter. Therefore, one must be careful when carrying out point-based analysis over finely gridded
grace datasets and usually we prefer catchment scale analysis. To this end, it is intuitive to comprehend that the catchment size should not be less than the spatial resolution of the
grace products.
This brings us to questions such as, what resolution in the spatial domain is appropriate so that it corresponds to the band-limited information in the spectral domain? What is the limit on the minimum catchment size that can be observed effectively with filtered grace fields? How is the spatial resolution affected by the correction approaches that are used to negate the impact of filtering on the signal? Or one may sum up all these questions to ask, what is the current spatial resolution of the grace satellite products for hydrological investigations?
Several studies put a limit on the minimum area of a region that can be effectively investigated with
grace products, but these numbers vary from one study to the other, which increases the ambiguity. For example, Longuevergne et al. [
19] suggested that the spatial resolution of
grace fields is ≈200,000 km
and we should develop superior methods for observing mass changes at finer spatial scales. Rowlands et al. [
31] proposed a limit of ≈150,000 km
. On the other hand, Lorenz et al. [
7] demonstrated that many catchments smaller than these limits were well observed by
grace provided they have a strong seasonal cycle in terms of water storage changes. Tourian et al. [
32] showed that
grace was able to capture the water loss in Urmia basin, which has an area of ≈52,000 km
. In a recent study, Khaki et al. [
33] proposed a two-step Kernel Fourier Integration (KeFIn) filter to reduce errors in high-frequency mass changes and also to decrease spatial leakage errors. While authors show that the filter reduces mass estimation errors in many small and medium size river basins, they do not evaluate the spatial resolution of
grace data.
In this contribution, we discuss the ideal spatial resolution of the
grace product and how it changes with the post-processing strategy we choose. We use popular methods and tools developed in the past decade by fellow researchers to help us set a limit to the catchment size that can be observed with an accuracy better than a certain limit. We carry out the investigation in a closed-loop simulation environment that emulates
grace satellite products. In order to be comprehensive in our analysis, we analyze the error behaviour with respect to the catchment size for four popular repairing strategies over 255 catchments. In general, we find that the error increases as the catchment size decreases, but we observe that the amount of error varies from one correction method to the other. We also find that one can obtain better
grace time-series over smaller catchments with the data-driven correction scheme by Vishwakarma et al. [
25].
3. Data and Method
In order to use the relation between the accuracy and catchment size for identifying the spatial resolution of
grace products in a comprehensive manner, we analyze more than 250 catchments shown in
Figure 3. The largest catchment is the Amazon in South America with an area of ≈4,672,876 km
and the smallest is Cavally catchment in West Africa with an area of ≈30,744 km
. The study is carried out in a closed-loop simulation environment used by Vishwakarma et al. [
21] and Vishwakarma et al. [
25]. It consists of total water storage anomaly from the Global Land Data Assimilation System (
gldas) Noah Land Surface Model as the background truth [
42]. These global hydrological fields are contaminated with noise extracted from
gfz grace fields [
3]. In order to extract noise, we first filter the
gfz grace fields with a destriping and a
Gaussian filter [
10], then we subtract the filtered fields from the corresponding noisy
grace fields to derive a realistic
grace-type noise. The
grace-type noise is then added to the
gldas model fields for 72 months to obtain
grace-type noisy fields with known truth. Further details about the simulated field can be found in Vishwakarma et al. [
21]. We can process the
grace-type fields to obtain time-series for a catchment and it can be validated against the time-series from
gldas fields. In this setup, we can assess the magnitude of error accurately because we have control over each component.
We filter the simulated
grace-type noisy fields
with a Gaussian
filter and then compute catchment averages
for 255 catchments to obtain the respective time-series. Let us denote the corrected time-series by
. The true catchment average
from the band-limited field is not known, but the regional average from the filtered field
is known. One may approach true catchment average, which is approximately equal to the catchment average from the band-limited field, from catchment average of the filtered field
with the help of correction methods. Out of many available methods, we choose four methods: the
Multiplicative method by Longuevergne et al. [
19], the
scaling method by Landerer et al. [
24], the
additive method by Klees et al. [
20], and the
data-driven method by Vishwakarma et al. [
25]. The mathematical relation that helps you correct for the signal damage due to filtering is given below:
Multiplicative: | | [19] |
Additive: | | [20] |
Scaling: | | [24] |
Data-driven: | | [25]. |
The multiplicative method approaches corrected time-series
by first removing leakage
from catchment aggregate of filtered
grace field
, and then amplifying it by a scale factor
s. Leakage is obtained from a hydrological model and the scale factor
s is obtained from catchment mask and filtered catchment mask:
where
is the catchment mask: 1 inside and 0 outside,
is the filtered catchment mask,
is the domain of the surface of the Earth, and
is the infinitesimal surface element
.
The additive method approaches corrected time-series by subtracting leakage from the catchment aggregate of filtered grace field and adding bias , where both leakage and bias are obtained from a hydrological model. The scaling method approaches corrected time-series by multiplying grace products by a scale factor k that is obtained by estimating a multiplicative factor, between a hydrological model and its filtered version, via least squares estimation.
The data-driven method corrects time-series by removing leakage
and deviation integral
from the catchment average of filtered
grace fields
. To compute the leakage and the deviation integral, we do not rely on hydrological models, but use the
grace fields only. Here, one may argue that the
grace fields are noisy, thus the leakage and the deviation integral computed from noisy fields will not be accurate. This is the reason they are computed from filtered
grace fields. Since we know that the filtered
grace fields are not equal to the truth, the leakage and the deviation integral from these fields are also not accurate. However, Vishwakarma et al. [
25] demonstrated that if we multiply the leakage and the deviation integral from filtered
grace fields, by a scalar ratio between the leakage and the deviation integral from once filtered
grace fields and twice filtered
grace fields, we can approach near-truth leakage and deviation integral. Although this approximation works for most of the catchments, it has limitations over arid regions and the estimated leakage and deviation integral for these regions is less accurate. Nevertheless, the accuracy of data-driven methods is still either at par or better than what we can achieve from other methods [
25]. Please refer to Vishwakarma et al. [
25] and Vishwakarma et al. [
21] for more information. Since the multiplicative, the additive, and the scaling approach use a hydrological model to estimate corrected time-series, we call them model-dependent approaches, while the data-driven approach uses
grace fields only and do not depend on models, we can also call it model-independent.
Global hydrological models have huge uncertainties that vary in space and in time, which is responsible for their differences with
grace fields [
43]. Using these models for correcting
grace products brings their uncertainty to the final
grace product. In order to imitate the impact of using a model for correcting
grace, we prefer not to use the
gldas model that is also the background truth; instead, we use the WaterGAP hydrological model (
wghm model [
44]) to compute model-dependent correction terms employed by model-dependent methods. The data-driven method computes its correction terms, leakage
and the deviation integral
, from once filtered and twice filtered
grace-type noisy fields. The MATLAB scripts for the data-driven method can be downloaded from
https://www.researchgate.net/publication/324804360MatlabscriptsDatadriven or from Institute of Geodesy, Stuttgart web page
http://www.gis.uni-stuttgart.de/research/projects/DataDrivenCorrection/.
The corrected time-series from the above-mentioned four methods is compared to the truth obtained from gldas model fields for 255 catchments. The following statistical measures are computed to characterize the behavior of difference (error) between the corrected time-series and the true time-series with respect to the catchment size:
Root Mean Square of Error (RMSE):
Cyclostationary Nash–Sutcliffe Efficiency (
) [
45,
46]:
where
represents the true value obtained from
gldas fields,
is the mean annual behaviour (mean monthly values) and
m is the number of epochs. RMSE can attain any positive value, a RMSE close to zero represents excellent agreement between
and
.
can attain any value between
and 1. A positive
value indicates that the difference between the true signal (
) and the corrected time-series (
) is well below the non-seasonal variations in the time-series. Typically, hydrological signals have a clear seasonal signal, i.e., change in the amplitude and/or sign of the signal from the summer months to the winter months. However, every summer/winter is not the same and there is a random variation in the amplitude of the signal from year-to-year, which is the non-seasonal variation (the natural variability of the signal). When the error in corrected time-series is smaller than this natural variability, then we can conclude that the signal is very well restored. If we were to use the
as proposed by Nash and Sutcliffe [
47], without accounting for the cyclostationary seasonal signal, the differences will be compared to the amplitudes of the annual signal. This will not provide a proper indicator for the efficacy of the repairing schemes.
4. Results and Discussion
In
Figure 4 and
Figure 5, we have plotted the RMSE and the
for four corrected time-series and time-series from filtered fields over 255 catchments. The catchments are sorted by their area. Since the scatter of these statistical measures is wide, we fit a smooth line, obtained by using locally weighted scatter-plot smoothing (LOESS), to identify the general pattern. LOESS is a non-parametric method that uses iterative locally weighted regression to fit a second degree polynomial through points in a scatter plot [
48]. Since the number of large catchments is smaller than the number of small catchments, the data points on the
x-axis, i.e., catchment area, suffer from unequal intervals. Therefore, we have defined bins to counter this irregular data distribution. The first bin is for catchments from 30,000 km
to 100,000 km
with sample points every 1000 km
, the next bin is for catchments from 100,000 km
to 1,000,000 km
with sample point every 10,000 km
, and the last bin is for catchment from 1,000,000 km
to 4,000,000 km
with sample points every 100,000 km
. A window size of 51 is used for smoothing. We have chosen these parameters to obtain a smooth fit that would represent the general behaviour of the scatter.
We can observe that the error, in general, increases as the area of the catchment decreases, irrespective of the correction method used. However, the rate at which the error increases varies from one method to the other. The multiplicative approach, due to a larger scale factor for smaller catchment, experiences a steep decay in performance. The additive approach, the scaling approach and the time-series from filtered fields are competitive. However, the data-driven method is able to provide better mass change estimates at all catchment scales.
Although, in general, the disagreement between the corrected and the true time-series increases as the catchment size decreases, several small catchments exhibit less error in comparison to a few relatively large catchments. The magnitude of error corresponding to a catchment is pertinent to this simulation setup, and it will change a little bit if we change the background models and simulation setup. Nevertheless, the general pattern that error increases as catchment size decreases will hold.
Ideally, with no noise and no approximations, we expect to recover the full band-limited signal after applying the data-driven method (see
Figure 3, Vishwakarma et al. [
25]). However, in reality, we lose some accuracy and the error is not zero even for the largest catchment Amazon. The error in the corrected time-series, for catchments smaller than the resolution of the filtered field and larger than the ideal resolution of band-limited
grace data, should be close to the error in time-series from filtered fields for larger catchments. We can use the trade-off between accuracy and the catchment size to discuss the potential
grace resolution: those catchments with an error less than a defined threshold can be categorized as resolvable.
Let us say we can accept the
grace products for all the catchments with an RMSE better than a certain value, say
or
. We leave it to the user to define the maximum tolerable error for their application and then decide whether the catchment or region of interest is suitable for analysis with
grace products corrected with a certain repair scheme. For example, if we choose
as the error limit, then filtered products can be used to monitor catchments larger than ≈750,000 km
, a multiplicative approach can be used for catchments larger than ≈1,563,000 km
, an additive approach for catchments larger than ≈1,103,000 km
, a scaling approach for catchments larger than ≈563,000 km
and the data-driven method for catchments larger than ≈250,000 km
. If the tolerable error is
, then the limit is ≈152,000 km
for filtered and additive, ≈810,000 km
for multiplicative and ≈63,000 km
for both scaling and the data-driven method. In
Table 2, we have summarized the resolvable catchment corresponding to an error level.
Since
accounts for both the difference in the magnitude and the correlation between the corrected and the true time-series, it is a powerful indicator and popularly used in time-series analysis. One may choose an acceptable
value for their application to determine the corresponding spatial resolution. For example, if we choose 0.8 as an acceptable limit, then filtered products can be used to monitor catchments larger than ≈1,100,000 km
, a multiplicative approach can be used for catchments larger than ≈1,750,500 km
, an additive approach for catchments larger than ≈1,100,500 km
, a scaling approach for catchments larger than ≈600,000 km
and the data-driven method for catchments larger than ≈200,000 km
. In
Table 3, we have provided the approximate spatial resolution corresponding to a
value. It is to be noted that the LOESS fit for the multiplicative approach and for the additive approach never attain
value of 0.9. Furthermore, the spatial resolution improves with increasing
values, while the corresponding RMSE decreases.
These limits on the spatial resolution are obtained, from
Figure 4 and
Figure 5, by observing the point at which the fit crosses a RMSE value or a
value for the first time. Thus, the values in
Table 2 and
Table 3 are only meaningful if the LOESS fit is smooth; as soon as the fit seems noisy, we should be careful in interpreting the resolvable catchment size. Therefore, in this analysis, we have ignored all the catchments for which the fit tends to be noisy—for example, the fit for the data-driven method becomes noisy for catchments below ≈600,000 km
. We can find a lower resolvable catchment area for the data-driven method corresponding to an RMSE level of
, but this part of the fit is noisy and we avoid any interpretation. Please note that, within these suggested spatial resolution limits, where one can choose from many methods, one method may perform better than the others as per the fit. Such a general rule can be used but with the understanding that individual catchments may deviate from the defined error behaviour. This is shown by the disagreement between the scatter and the fit.
5. Conclusions
The grace satellite mission has helped us observe continental scale hydrology. Moreover, with improved data processing skills, we have been able to use grace products for monitoring catchment scale hydrology. Although the spatial resolution of grace is accepted to be coarse, it was necessary to put a number to the spatial resolution of the band-limited grace fields and of the filtered grace fields. It was established by previous contributions that ideal spatial resolution can be used to define their spatial resolutions.
Notwithstanding this, there have been many attempts to identify the potential spatial resolution of the grace products in terms of catchment size, but every attempt concluded with a different number. The difference in the research findings can be attributed to usage of different processing methods and selective regional analysis. For example, one can choose a filter out of many available ones and then a corrective method for repairing the signal damage due to the filtering. Thus, the end product is influenced by combined effects of different processes and it is hard to quantify whether the final product is able to resolve a catchment efficiently. With developments in post-processing algorithms for grace data, it was necessary to understand the contemporary spatial resolving power of grace products. This study provides a comprehensive analysis of grace products corrected with different repairing schemes to demarcate the catchment size that can be effectively observed. We have carried out such an analysis for 255 catchments of small to very large size in a closed-loop simulation environment, in order to answer the question of what is the minimum size of catchment that can be observed with the help of improved grace products.
We found that in general the error increases as the catchment size decreases, but the error level varies from one repair method to the other. grace time-series corrected with multiplicative approach shows the highest amount of error and the grace time-series from the data-driven approach has the lowest amount of error. We discussed the effective resolution of corrected grace products with respect to an acceptable error threshold. We found that the data-driven method and scaling method were able to approach the ideal grace resolution with an acceptable error RMSE level of . The data-driven method has minimum divergence in terms of the spatial resolution with respect to the performance indicators (RMSE and ).
Therefore, based on our investigation and its findings, we recommend the following:
- 1.
The spatial resolution of the band-limited grace spherical harmonics is not the half-wavelength at the equator, but the ideal spatial resolution. The spatial resolution of filtered grace data can also be described by the ideal spatial resolution.
- 2.
The users have to be wary that the spatial resolution of the corrected dataset is dependent on the adopted method. Furthermore, the spatial resolution is associated with the error tolerance required by the application and it has to be defined by the user.
- 3.
Given the fact that with enhanced processing techniques grace is able to see some catchments smaller than the spatial resolution and many catchments close to the band-limit resolution, it is worthwhile to provide spherical harmonics up to a maximum degree of 120 or higher.
The impending launch of the
grace-Follow On mission, which is a near replica of the
grace mission with only the Laser Ranging Instrument as the additional instrument, raises the question of the relevance of this study for
grace-fo data. Recently, a simulation study by Flechtner et al. [
49] indicated that the expected improvement from
grace-fo is rather moderate, in which case we expect our quantitative results to hold even for
grace-fo data. Having said that, the mathematical foundation and the understanding of the idea of spatial resolution developed in this study will always be relevant for data disseminated in terms of spherical harmonic coefficients.