The effects of leakage error on terrestrial water storage variations in the Yangtze River Basin measured by GRACE

Journal of Applied Geophysics 160 (2019) 264–272

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The effects of leakage error on terrestrial water storage variations in the Yangtze River Basin measured by GRACE Wanqiu Li a,b, Chuanyin Zhang b, Hanjiang Wen b,⁎, Wei Feng c, Wei Wang b, Yulong Zhong d, Zhen Li e, Cai Yin f, Yuanqi Lu f a

College of Geomatics, Shandong University of Science and Technology, Qingdao 266590, China Chinese Academy of Surveying and Mapping, Beijing 100830, China Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China d Faculty of Information Engineering, China University of Geosciences (Wuhan), Wuhan 430074, China e Jinan Investigation Surveying and Mapping Institute, Jinan 250101, China f Qingdao Geotechnical Investigation and Surveying Research Institute, Qingdao 266032,China b c

a r t i c l e

i n f o

Article history: Received 22 January 2018 Received in revised form 23 October 2018 Accepted 2 December 2018 Available online 07 December 2018 Keywords: GRACE TWS Leakage error Scale factor Third-filter

a b s t r a c t The Yangtze River Basin is an important economic area in China. Monitoring TWS change in this basin is strongly meaningful for exploitation of water resources and economic development. Filtering is necessary to derive the TWS from GRACE time-variable gravity field models, but the signal is distorted. Therefore, we subtract the leakage error from the filtered TWS change to acquire the attenuation signal. We use a scale factor derived from the basin characteristic function to recover the attenuation signal. We propose a so-called third-filter method to gain the temporal distribution of the leakage error. In addition, we compare the TWS obtained using the third-filter method with that from other approaches, e.g., the scaling-factor approach, addictive correction approach, and multiplicative correction approach. In the spatial distribution, we compare the results from these three approaches with that from the second-filter method. Furthermore, we validate the effectiveness of our method in simulation studies. By using the GRACE data, we further analyze the effect of leakage error on TWS change in the Yangtze River Basin, and we compare the restored signal after correcting the leakage error with the WGHM and in-situ measurements of TGR impoundment. The results indicate: (1) From the third-filter method, the annual amplitude of TWS change in the Yangtze River Basin corresponds to a 4.4 ± 0.5 cm equivalent water thickness, while the semi-annual amplitude is up to an equivalent water thickness of 0.7 ± 0.5 cm. These results are more consistent with the WGHM model after leakage error is corrected. (2) Additionally, the GRACE and model results show coincident annual fluctuation of the TWS in the basin, with the largest increase in August every year. The whole trend of third-filter method result in TGR area is closer to yearly change rate of TGR volume. (3) The third-filter method effectively reconstructs the temporal distribution of the TWS from the GRACE filtering data. And the multiplicative correction method may achieve a higher precision of spatial TWS change in the basin. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Water resource change is one of the key issues in national economic and social development. Accurate estimation of the terrestrial water storage (TWS) change is important for studying and forecasting climate change, agricultural production, floods and other natural disasters (Zhong et al., 2009). Methods for estimating TWS are mainly based on satellite remote sensing and hydrological observations (Cazenave and Nerem, 2002). The emergence of Gravity Recovery and Climate Experiment (GRACE) mission has provided an effective technical means to ⁎ Corresponding author. E-mail address: [email protected] (H. Wen).

https://doi.org/10.1016/j.jappgeo.2018.12.001 0926-9851/© 2018 Elsevier B.V. All rights reserved.

study TWS changes at the global and basin scales (Tapley et al., 2004; Wahr et al., 2004;). The inversion method based on the GRACE timevariable gravity field has been widely used to investigate global TWS change (Seo et al., 2006; Velicogna and Wahr, 2006; Yang et al., 2009; Chen et al., 2010). TWS changes essentially correspond to mass transport and redistribution in shallow ground surface, which cause the earth gravity field to change with time. However, prominent north-south stripes exist using GRACE gravity field spherical harmonic coefficients (SHCs) to estimate global TWS variations (Wahr et al., 2006; Swenson and Wahr, 2006). The stripe noise is typically suppressed using a combined filter (Han et al., 2005; Zhang et al., 2009; Zhan et al., 2011; Li et al., 2017a,b). Nevertheless, filtering evidently causes the GRACE results to deviate from the

W. Li et al. / Journal of Applied Geophysics 160 (2019) 264–272

real signal (Wahr et al., 1998a,b; Landerer and Swenson, 2012), which is regarded as signal distortion arising from leakage error. During filtering the influence of signal in interested area to surrounding area is called “leakage-out” error, which can weaken the amplitude of signal in interested area. The influence of signal in surroundings to interested area is called “leakage-in” error, which can attenuate signal in interested area. To recover the amplitude of attenuation signal, the scale factor based on the hydrological model is usually used such as frequency grid scale factor as described by Landerer and Swenson (2012), the multi-time scale factor as described by Klees et al. (2006), the single scale factor as described by Feng et al. (2012), and the spatial grid scale factor (Long et al., 2015a,b). To reduce the uncertainty from the hydrological model, we adopt a basin-related scale factor to minimize the “leakage-out” error. Considering the leakage error, most studies use the hydrological model to get the temporal distribution (Klees et al., 2007; Chen et al., 2005a,b; Swenson and Wahr, 2007). Different approaches are used to minimize the filtering effect, include the optimized Gaussian averaging function (Wahr et al., 1998a,b), the optimizing averaging kernel technique (Swenson and Wahr, 2002), and the forward modeling method (Jin and Zou, 2015), etc. In their results, “leakage-in” and “leakageout” error is not distinguished, and the estimation of the leakage error is not considered. The Yangtze River Basin is an important economic basin in China. Therefore, an accurate estimation of TWS change in this basin is crucial to achieve sustainable water resource management. Relevant monitoring research in recent years has focused on the spatial-temporal analysis of TWS change obtained from the GRACE gravity field models (Hu et al., 2006; Zhai et al., 2009; Xu et al., 2013; Ni et al., 2014; Li et al., 2017a,b; Long et al., 2015b), less attention is paid on the leakage error estimation. In this paper, considering the leakage error from the surroundings of the Yangtze River Basin (i.e., “leakage-in” error), we extract the basin TWS change and estimate a scale factor using a characteristic function to reconstruct the amplitude of attenuated signal. We combine the third-filter method with the multiplicative correction method to accurately estimate the spatial-temporal distribution of the leakage error. We consider the GLDAS model as true signal, and simulate GRACE data to evaluate the precision of TWS change using different methods. Then we analyze the effects of leakage error on TWS variations in this basin using GRACE data. 2. Data and method 2.1. Data

265

Research (CSR).These data are normalized coefficients, with some effects being removed, such as the solid earth tide, ocean tide, pole tide, non-tidal atmosphere and oceans, and gravity disturbances due to objects such as the sun and moon (Bettadpur, 2012). The C20 is replaced by that from the Satellite Laser Ranging (SLR) (Chen et al., 2005a,b), and the coefficients of degree 1 are set to zero (Chen et al., 2005a,b). For any month without data, the coefficients are derived by averaging that from two adjacent months. 2.2. Method We use a filter to suppress the short wavelength noise in the mass change in shallow ground surface, which weakens the amplitude of signal and change of spatial distribution. The effect of filtering corresponds to the effect of leakage error on the real signal. Therefore, we must accurately estimate the leakage error (i.e., “leakage-in” error. And in the following section, the “leakage-in” error is abbreviated as leakage error) to separate the attenuation signal from the filtered signal. The mass change can be derived from the time-variable SHCs, as follows: Δσ ðθ; λÞ ¼

Rρe N 2n þ 1 n ∑n¼0 W n ∑m¼0 ½ΔC nm cosmλ þ ΔSnm sinmλ 1 þ kn 3  W m  P nm ð cosθÞ ð1Þ

where θ, λ are the colatitude and longitude of the calculation point, respectively;(ΔCnm, ΔSnm) is the change of normalized SHCs with degree n and morder m;P nm ðÞ is the fully normalized associated Legendre function with degree n and morder m; kn′is the load love number of degree n; ρe ≈ 5.5 × 103kg/m3 is the average density of solid earth; R is the average radius of the earth; Wn and Wm are the degreedependent and order-dependent smoothing functions, respectively (Zhang et al., 2009). For the Yangtze River Basin, we convolve the global grid with the characteristic function to extract TWS change in this basin from the GRACE filtered data. The characteristic function is defined 1 inside the basin and 0 otherwise. We assume that the stripe noise can be completely removed by filtering. And propose a third-filter method for recovering signal from the attenuated signal. In this approach a scale factor and leakage error is estimated. Assuming homogenous spatial distribution of the TWS, we may compute a filtering scale factor s using the following formula: (Long et al., 2015a; Vishwakarma et al., 2016; Vishwakarma et al., 2017): Z

2.1.1. GLDAS hydrology model The GLDAS global hydrology model was released by NASA and the National Center for Environmental Prediction (Rodell et al., 2004). The model is constrained by near-real-time earth surface and spatial data. Data assimilation is used to output the parameters of the terrestrial surface. These data reflect change in soil water and snow on terrestrial surfaces with a spatial resolution of 1° × 1°from January 2003 to December 2014. The GLDAS includes four models, i.e., Noah, VIC, Mosaic, and CLM. We choose the Noah model for simulation, and the VIC model for leakage error and bias effect estimation through multiplicative correction and addictive correction approach. 2.1.2. WGHM hydrology model WGHM (WaterGAP Global Hydrology Model) V2.2c was developed by the Institute of Physical and Geography (IPG, Goethe University of Frankfurt, Germany). The model is a combination of monthly parameters with climatic and geographic data from 2003 to 2013 with a spatial resolution of 0.5° × 0.5° (Döll et al., 2014). 2.1.3. GRACE data. The GRACE level-2 (RL05) products, i.e., monthly gravity field models from 2003 to 2014 are provided by the Center for Space

ηðθ; λÞdΩ s¼Z

Ω

ηðθ; λÞ  ηðθ; λÞdΩ

ð2Þ

Ω

where Ω is the entire earth surface, η(θ, λ) is the basin characteristic function, andηðθ; λÞ is the filtered result using basin characteristic function. The signal can be reconstructed by multiplying attenuated signal with the scale factor s; the attenuation signal is then obtained by subtracting the leakage error from the result. In addition, we define η∗(θ,λ) as the complement of the characteristic function η(θ, λ). We adopt the third-filter method to estimate the temporal distribution of the leakage error in the following steps: ①to calculate Δσ ðθ; λÞ by filtering SHCs according to Formula (1), and convolve Δσðθ; λÞ with η∗(θ, λ), whereas, the mass change inside basin is called FirstFilter leakage error Is(θ, λ). ②to expand convolution result into corresponding gravity field SHCs. ③to filter the expanded SHCs from step ②, and to calculate the mass change in shallow ground surface, whereas, the mass change inside basin is called second-filter leakage error Is ðθ; λÞ. ④to filter again the SHCs that were filtered in step③, and calculate mass change over the earth surface, mass change inside

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Table 1 Mathematical relations among various approaches used to restore signal after filtering. Approach Scaling- Multiplicative factor correction

Addictive correction

Second-filter (spatial)

Third-filter (temporal)

Relation

Δσ −Im þ bm

sðΔσ−I s Þ

sðΔσ−^Is Þ

sðΔσ−Im Þ

kΔσ

basin is called third-filter leakage error Is ðθ; λÞ. ⑤For each month, we derive the mass change grid by space averaging in the basin, and then obtain a time series of the leakage error Is, I s , Is . The leakage error derived from the first-filter, second-filter and third-filter, can be written as (Vishwakarma et al., 2016): Z

    Δσ θ0 ; λ0 η ðθ; λÞW θ; λ; θ0 ; λ0 dΩ0

I s ðθ; λÞ ¼ ηðθ; λÞ Is ¼

1 As

Z

ZΩ

0

ηðθ; λÞ Ω0

Ω

Z

1 Is ¼ As

ZΩ

Z

0

ηðθ; λÞ Ω0

Ω

Z Z

ZΩ

0

ηðθ; λÞ Ω

Ω0

9 > > > > =

    > Δσ θ0 ; λ0 η ðθ; λÞW θ; λ; θ0 ; λ0 dΩ0 dΩ > > > ;     Δσ θ0 ; λ0 η ðθ; λÞW θ; λ; θ0 ; λ0 dΩ0

I s ðθ; λÞ ¼ ηðθ; λÞ 1 Is ¼ As

    > Δσ θ0 ; λ0 η ðθ; λÞW θ; λ; θ0 ; λ0 dΩ0 dΩ > > > ;     Δσ θ0 ; λ0 η ðθ; λÞW θ; λ; θ0 ; λ0 dΩ0

I s ðθ; λÞ ¼ ηðθ; λÞ

9 > > > > =

9 > > > > =

    > Δσ θ0 ; λ0 η ðθ; λÞW θ; λ; θ0 ; λ0 dΩ0 dΩ > > > ;

3. Results and accuracy assessment ð3Þ

ð4Þ

ð5Þ

We use the equation ηðθ; λÞ  η ðθ; λÞ ¼ τðθ; λÞ to further reduce Eqs. (3)–(5) to a simpler form: 9 > Δσ ðθ; λÞτðθ; λÞdΩ > > > > > > > > ZΩ > = 1 Is ¼ Δσ ðθ; λÞτðθ; λÞdΩ As > > > ZΩ > > > 1 > Is ¼ Δσ ðθ; λÞτðθ; λÞdΩ > > > As ;

Is ¼

1 As

Z

ð6Þ

Ω

If the signal intensity from the surroundings is approximately equal to that from the area of interest, we obtain a proportional relation among the time series of leakage error as follows (Vishwakarma et al., 2016, 2017): Is =Is ≈ Is =Is

For the temporal distribution, we compare the third-filter with previous approaches, such as: scaling-factor approach (Landerer and Swenson, 2012), addictive correction approach (Klees et al., 2007), and multiplicative correction approach (Chen et al., 2005a,b; Swenson and Wahr, 2007). The relations among these approaches is shown in Table 1 (Long et al., 2015a). For the spatial distribution, we compare the three approaches with the second-filter method. In the following simulation, we evaluate the precision of recovering the filtering signal using different methods,and analyze the effect of leakage error on TWS change in the Yangtze River Basin from the GRACE data.

ð7Þ

Eq. (7) makes leakage error estimation ^Is closer to the actual error Is. We firstly compute the time series of the leakage errorIs andIs, and compute their phase difference. Secondly, we shift Is by the phase difference towards Is , and compute the mean ratio at each time, after removing outliers. The time series is shifted in the frequency domain. We first take the Fast Fourier Transform (FFT) of the time series to be shifted, and multiply each frequency component by the complex exponential of the phase difference, and then make an Inverse Fast Fourier Transform (IFFT). So we may compute the ratio mean. We shift Is by the phase difference and multiply it by the ratio mean to obtain ^Is (Vishwakarma et al., 2016). Thirdly, we remove leakage error from the filtering signal to get the attenuation signal. We then multiply the attenuation signal by the scale factor to restore TWS change in the Yangtze River Basin in the temporal distribution.

3.1. Simulation We simulate the leakage error from the surroundings of the Yangtze River Basin based on the GLDAS-Noah model, and compare the effect of leakage error on the GRACE filtering, in spatial-temporal distribution. We consider consistent degree SHCs from the GLDAS-Noah model with the GRACE gravity model. Thus, we expand the GLDAS-Noah model into SHCs up to degree 60. Then, we estimate the equivalent water height based on the SHCs up to degree 60, which is considered the true TWS change. In addition, we can compare this result with our simulation results. 3.1.1. Simulation of GRACE data We deduct the average of monthly GLDAS-Noah model SHCs and consider the time-variable SHCs as the true gravity field model. To simulate the GRACE data, we firstly perform post-processing for the GRACE SHCs, such as replacing C20, truncating at degree 60, removing the average for all the monthly data, and filtering combined with the P3M15 and Fan. Secondly, we compute the difference between the SHCs with filtering and without filtering. Thirdly, we consider the difference of the SHCs as stripe noise, and add this noise to the original gravity field. So we may estimate global TWS change from simulated GRACE data. Furthermore, we use the original gravity field model to estimate the global TWS change. The results are shown in Fig. 1. No filtering is applied. 3.1.2. Spatial distribution of the TWS We show the effectiveness of Eq. (7) for estimating the leakage error with the simulated data. We get leakage error using the GLDAS-Noah model without noise according to Eq. (3), and then estimate the time series of the leakage error from the second-filter and third-filter according to Eqs. (4) and (5), respectively, as shown in Fig. 2. In addition, we don't remove the mean from the monthly GLDAS-Noah model. No significant differences appear as shown in Fig. 2. The ratio Is =Is ranges from 1.03 to 1.07, and the ratio I s =I s ranges from 1.02 to 1.04. From this result, we can see that Eq. (7) is valid for the leakage error estimation. For the Yangtze River Basin, we analyze the effect of the leakage error on the spatial distribution of the GRACE result. We fit the rate of TWS change based on least squares and obtain the spatial distribution. In addition, we obtain a scale of s = 1.53 from Eq. (2), which is independent of the hydrology model. The true rate of TWS change is shown in Fig. 3(a). We use four methods to minimize the leakage error effect, e.g. scaling-factor, multiplicative correction, addictive correction, and spatial second-filter, as shown in Table 1. We obtain different spatial distributions shown in Fig. 3(b)-(e). To compare their capability of recovering signal loss, we use the true value in Fig. 3(a). It is worth mentioning that the factor k in the scaling-factor method can be computed using the GLDAS-Noah or filtered GLDAS-Noah time series from 2003 to 2014, and we get a factor k of 1.28 from least squares. Furthermore,

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267

Fig. 1. Real and simulated global TWS change. The left is from the GLDAS-Noah model. The right is froam the simulated GRACE data

the leakage error Im in multiplicative correction method can be derived using the GLDAS-VIC model, as shown in Fig. 3(f), the bias bm in the addictive correction method depends on the GLDAS-VIC model. Fig. 3 shows that the spatial distribution of TWS change (mm/a) in the Yangtze River Basin based on the scaling-factor method is quite different from the true signal, possibly because that the stronger leakage error shown in Fig. 3(f) is aliased into the filtered results, which is then scaled by a factor of 1.28 and increases the deviation from the true signal. In addition, after deducting the leakage error from the filtered results, we find that TWS change rate from the multiplicative method is closer to the true value by comparing Fig. 3(e) and of Fig. 3 (a). A more consistent decreasing trend appears in the upper and lower reaches of the Yangtze River Basin, with a consistent increasing trend in the middle reaches. In addition, there is a more prominent increasing trend in the middle reaches estimated using the addictive method comparing Fig. 3(d) with Fig. 3(a). There is a lower increasing trend in the middle reaches estimated using the second-filter method as can be seen in Fig. 3(c), and we observe a large spatial distribution discrepancy relative to the true TWS change from addictive correction method and second-filter method. We use RMS to evaluate influence of the leakage error on the spatial distribution of TWS change based on simulated GRACE data, as shown in Table 2. As shown in Table 1, the addictive correction approach, multiplicative correction approach, the second-filter method, remove the spatial distribution of the leakage error and effectively improve the accuracy of the scaling-factor approach by approximately 61%, 70%, 51%, respectively. The reason is that the processing leakage error spatial change is beneficial to recover the spatial distribution of the distorted TWS change due to filtering. Better results are from the multiplicative correction approach, which is used to study the leakage error effect on TWS spatial change in the Yangtze River Basin. 3.1.3. Temporal distribution of the TWS To validate the effect of leakage error on the time-variable TWS in the Yangtze River Basin, we compute the space average for the grid value of the monthly TWS change. We firstly compute the time series of the leakage error from the second-filter and third-filter by Eq. (6), and estimate actual time series of leakage error by Eq. (7). Secondly, we remove the actual time series of the leakage error from the filtering results to determine the attenuation signal, and then multiply the attenuation signal by a scale factor of 1.53.Thirdly, we estimate the TWS change in the temporal distribution from the third-filter method, as shown in magenta in Fig. 4. In addition, we use the addictive correction and multiplicative correction approach to estimate the TWS change time series, as shown in orange and red, respectively, in Fig. 4. The blue curve in Fig. 4 presents the true signal from GLDAS-Noah model. The leakage error estimated from third-filter method, shown in grey in Fig. 4, is considered in third-filter, addictive correction and multiplicative correction approaches, and its amplitude is approximately half

of the true value. We use the scaling-factor method to recover the temporal change of the TWS, considering only the amplitude damping and ignoring the leakage error. Hence, we directly multiply the filtered signal by a scale factor of 1.28, as shown in green in Fig. 4. Fig. 4 shows that the amplitude from the scaling-factor method is bigger than the true. However, the time series estimated by the other methods present a similar TWS change as the true signal. In particular, results from the third-filter method agree well with the true signal. Because large leakage error aliases into the filtered signal, after the filtered signal is multiplied with a scale factor 1.28, the amplitude becomes bigger. After removing the leakage error, we can get more accurate time series from the multiplicative approach, the addictive approach or third-filter method. We use phase difference, NSE (Nash-Sutcliffe Efficiency) and RMS to analyze the ability to recover the filtering signal using different methods, as shown in Table 3. We still consider that from GLDASNoah model as true value. We derive phase difference using the Hilbert transform and obtain the NSE with the following formula that expresses the proximity of two time series: Pm  NSE ¼ 1‐

2 F s −Δσ true 2 Pm  ~ i¼1 F s − F s i¼1

where Fsis the time series from the four methods, ~F s is the average of time series, Δσ true is the true time series, m is total months(144 in our case), i(i = 1, 2, 3⋯, 144) is every month. The NSE ranges from -∞ to 1, and the closer the NSE is to 1, the more consistent is Fs with Δσ true . From Table 3, one can see that the smallest RMS and largest NSE are from the third-filter approach. The time series RMS of the third-filter method is significantly enhanced by 15%, 37%, and 35%, relative to the other approaches, e.g. the addictive approach, the multiplicative approach, the scaling-factor approach, respectively, which shows that

Fig. 2. Leakage error from the surroundings derived from each filtering process for the simulated data.

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Fig. 3. Effect of the leakage error on the spatial distribution of the long-term rate of TWS change estimated from the simulated GRACE data.

temporal changes of the TWS using the third-filter method is closer to the true signal, except for the phase delay. Our simulation shows that the third-filter approach is more reliable and provides an effective means to derive TWS temporal change using the GRACE gravity model in the Yangtze River Basin. In addition, more accurate spatial changes of the TWS may be derived from the multiplicative method. Therefore, in the following GRACE data experiment, we estimate the spatial change of TWS using the multiplicative method in the Yangtze River Basin. 3.2. GRACE data experiment

mass change. Then, we convolve the global mass change with η∗(θ, λ) to obtain the filtered TWS change in this basin using the GRACE gravity models. We use the scaling-factor method to minimize the filtering effect, and get the TWS change in the two months as shown in Fig. 5 (c) and Fig. 5(d), respectively. To study the leakage error effect on TWS change estimation in the Yangtze River Basin, we compute the leakage error using the GLDAS-VIC model, as shown in Fig. 5(e) and Fig. 5(f). Because WGHM model has a groundwater compartment including natural recharge and discharge, we compare the results from Fig. 5 (c) to Fig. 5(f) with that from WGHM models, and the WGHM model is expanded into the SHCs of degree 60. Furthermore, the WGHM

Based on the reliable methods for recovering signal, we apply different methods in the spatial and temporal distribution to the GRACE data experiment to gain more accurate TWS change in the Yangtze River Basin. In addition, we evaluate the effect of the leakage error on TWS change in this basin by comparing with scaling-factor method. 3.2.1. Spatial change of the TWS We analyze and compare the spatial distribution of the leakage error effect on TWS change. Taking the gravity SHCs of GRACE data from May 2010 and December 2011 as an example, we conduct the same post-processing as in the simulation to estimate the global Table 2 Effect of the leakage error on the spatial distribution of the long-term change of the TWS from the simulated GRACE gravity model. Spatial distribution influence

kΔσ

Δσ −Im þ bm

sðΔσ −I m Þ

sðΔσ−It Þ

RMS (mm/a)

0.209

0.082

0.062

0.103

Fig. 4. Effect of leakage error on TWS temporal change from 2003 to 2014 based on the simulated GRACE data.

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269

Table 3 Comparisons of phase difference, NSE, and RMS of TWS change from different methods with true signal. Comparison with true signal

Result

Δσ−Im þ bm

kΔσ

sðΔσ−^Is Þ

sðΔσ −Im Þ

Phase diff.

NSE/RMS

Phase diff.

NSE/RMS

Phase diff.

NSE/RMS

Phase diff.

NSE/RMS

−6.89°

0.89/1.31 cm

6.10°

0.85/1.00 cm

−5.23°

0.84/1.34 cm

−10.10°

0.91/0.85 cm

(V2.2) simulates the surface water, soil layers, aquifers, and accommodates human impacts on the SWS and GWS changes. In our research, the WGHM TWS change is considered as a reference for comparing with the GRACE TWS change with and without leakage error correction (Long et al., 2015a). Fig. 5 shows that the TWS in the Yangtze River Basin in late spring increases markedly compared to that in winter, which may be related to heavy rainfall. For the spatial distribution trend, the results from the GRACE gravity model and WGHM model in May 2010 show a gradual increase of the TWS from the middle and upper reaches to the lower reaches of the Yangtze River Basin; similar results can be found in Xu et al. (2013). However, the spatial distribution of the TWS change estimated from the GRACE data in December 2011 undergoes a significant transformation, and the TWS turns to ascend from the south to the north, but remains in good consistent with the WGHM model. In the local spatial distribution, an increasing signal of the TWS in May 2010 appears in the upper basin after using the multiplicative method,

which is more consistent with the WGHM model. Additionally, prominent TWS signal loss in the south of middle reaches can be observed in May 2010, where the maximum decrease is a -10 cm equivalent water thickness.The largest signal increase is concentrated in the lower reaches area, and it is about 10 cm equivalent water thickness, which are nearly consistent with the WGHM model. In addition, the multiplicative method slightly reduces the TWS change amplitude of the middle and lower reaches from December 2011 relative to the scaling-factor method. In addition, a slight TWS decrease in the southearn basin and a faint increase in the northern basin can be observed. In the middle and upper reaches, positive and negtive change of the TWS in the northern and southern area of the Yangtze River Basin can also be seen, which agree with WGHM model. Differences between the multiplicative method and the scaling-factor method are not evident, possibly because the hydrology signal from the surroundings is weak during certain months, leading to a slight leakage error.

Fig. 5. TWS change with and without leakage error correction using the GRACE gravity model for May 2010 and December 2011.

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Fig. 6. Comparison of the TWS between WGHM and GRACE gravity models. (a) Time series of TWS change with and without leakage error correction in the Yangtze River Basin and time series of the models. (b) Seasonal changes of the TWS with and without leakage error correction, including seasonal changes of the models.

3.2.2. Temporal change of the TWS After the leakage error is corrected by the multiplicative method, the spatial distribution becomes more similar to the model. We apply the third-filter to GRACE SHCs that were previously processed using the second-Filter to derive time series of leakage error according to Eq. (3) to (5). We then use Eq. (6) to compute the actual leakage error from filtering the GRACE SHCs. We compare various methods for the temporal change of the TWS from 2003 to 2014 using the GRACE gravity models, as shown in Fig. 6 (a). We directly multiply the filtered TWS change by a scale factor of 1.28 to recover the time series, as shown in green in Fig. 6(a). After deducting the leakage error from the filter results, we reconstruct the TWS change by multiplying with a scaling factor of 1.53, as shown in red in Fig. 6(a). In addition, we compute the space average for the monthly TWS change grid from the WGHM model, as shown in blue in Fig. 6(a). To analyze the leakage error effect on the time series of TWS change in this basin, we fit the seasonal change of the time series based on least squares, together with, including annual change and semi-annual change, as shown in Fig. 6 (b). Moreover, we estimate the phase and amplitude for the seasonal signal, as shown in Table 4. As shown in Fig. 6, the three types of time series (as estimated by the scaling-factor method, third-filter method and WGHM model) are generally similar to each other. There appears a peak in each August, which is consistent with that in Hu et al. (2006). The annual amplitude of TWS change is 5.9 ± 0.6 cm before correcting leakage error; which reduces to 4.4 ± 0.5 cm after correction. The semi-annual amplitude decreases from 1.0 ± 0.6 cm to 0.7 ± 0.5 cm. For comparison, the annual and semi-annual amplitude from WGHM model are 4.9 ± 0.4 cm and 0.5 ± 0.4 cm, respectively. It can be seen that TWS in the Yangtze River Basin is dominated by annual changes, with weaker semi-annual signal. The amplitude of signal decreases after leakage error is corrected and becomes more consistent with model. The annual amplitude after correction is not much different from that of Hu et al. (2006), Zhai et al. (2009) and Ni et al. (2014). There is a phase advance in annual change from third-filter method relative to other two results. However, the phase of semi-annual change from

third-filter method is 1.5 ± 0.7°, which is closer to that from WGHM model. In addition, we compute RMS (relative to WGHM model) of time series from scaling-factor method and third-filter method, as shown in Table 4. Therefore, the temporal TWS change with leakage correction is closer to that from WGHM model. To validate our methods, we first compute TWS change from GRACE in Three Gorges Reservoir (TGR) area, then we compute TWS change from WGHM simulation in this area. Then we compute the difference, denoted by GRACE-WGHM, which can reflect the reservoir impoundment effect. Finally, we compare GRACEWGHM change from June 2003 to May 2010 with in-situ observations of TGR impoundment from Wang et al. (2011). The results (in cyan and blue) and water impoundment of TGR (in black) are shown in Fig. 7. Water impoundment of TGR since 2003 mainly caused a longterm variation of TWS in the Yangtze River Basin, which can be observed by GRACE (Li et al., 2018). Therefore, we only compare trend with that from in-situ measurements. In the long-term trend, as shown in Fig. 7, the change rate from scaling-factor method is 1.67 km 3 ·a −1 , while that from third-filter method is 1.93 km 3 ·a −1 . So the rate with leakage correction is closer to that from Wang et al. (2011) (2.24 km3·a−1). 4. Discussions The GRACE signal restoration is challenging for hydrological studies and applications, although it is sometimes not important for large river basins. If the leakage error from the surroundings of the Yangtze River Basin is not considered in the signal restoration, spatial-temporal change of the TWS in this basin can deviate significantly from the true value, as shown in our simulation. The removal of leakage error is critical. In simulation, there is true value, which makes validation of combining third-filter with multiplicative correction method for minimizing leakage error effect. The TWS change in the Yangtze River Basin from combination method is more consistent with WGHM model. However, the inconsistency between them could be caused by uncertainty of

Table 4 Amplitude, and phase for the annual change and semi-annual change from the WGHM and GRACE with and without leakage error correction. Comparison

Scaling-factor Third-filter WGHM

Annual change

Semi-annual change

RMS

Amplitude/cm

Phase/(°)

Amplitude/cm

Phase/(°)

/cm (Relative to WGHM)

5.9 ± 0.6 4.4 ± 0.5 4.9 ± 0.4

−2.6 ± 0.1 0.72 ± 0.1 −2.6 ± 0.1

1.0 ± 0.6 0.7 ± 0.5 0.5 ± 0.4

1.3 ± 0.6 1.5 ± 0.7 1.8 ± 0.8

2.48 2.31

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of our methods. Our methods may be used to derive accurate TWS spatial-temporal change in other similar regions. Acknowledgements

Fig. 7. Comparison of GRACE-WGHM water storage change with and without correction from June 2003 to May 2010 with in-situ observations of TGR impoundment.

WGHM model and GRACE models. The contribution of GRACE uncertainty may attribute to GRACE satellite itself observation error and orbit design error (Zheng et al., 2012), and SHCs post-processing error. In GRACE data experiment, spatial difference between the multiplicative correction and scaling-factor method is not evident, it may be due to faint hydrology signal from surroundings in a certain month. Results from many more months may fully compare with well. We use the RMS and NSE to evaluate the effect of leakage error on the temporal change of regional TWS based on the simulated GRACE data. Our third-filter method may achieve higher precision especially for temporal change. However, there are still phase delay in simulation and annual phase advance in GRACE experiment, respectively, both of which may be related to the ratio computation between the time series of leakage error from second-filter and third-filter, but the specific reasons requires further study. In addition, we adopt in-situ TGR measurements to validate our methods. In the whole trend, third-filter method is closer to yearly change rate of TGR volume. However, in some time period, e.g., from June 2006 to May 2008, the time series from our method deviate from water impoundment change of TGR, which may attribute to WGHM model, but the reason needs further research.

5. Conclusions Based on the effect of leakage error on TWS change in the Yangtze River Basin estimated from the GRACE data in this paper, we propose the third-filter method to estimate the leakage error in the temporal distribution, and multiplicative method for the leakage error in the spatial distribution. We deduct leakage error from filtered signal to gain attenuation signal, which is then multiplied by a scale factor to restore signal. The specific conclusions are as follows: (1) We estimate a filtering scale factor of 1.53 for the Yangtze River Basin using characteristic function, which is independent of hydrology model. (2) We explicitly present the theoretical method of calculating leakage error and recovering filtering signal. We adopt the GLDASNoah model to verify the equivalence relation among different filtering. (3) The leakage error is important for estimating TWS change in the Yangtze River Basin using GRACE data. Our simulation shows that by combining the third-filter with the multiplicative correction method one can effectively recover temporal-spatial distribution of TWS change in this basin. In addition, TWS change with leakage error correction agree well with that of WGHM model especially in temporal distribution. Comparisons with insitu measurements of TGR impoundment shows good reliability

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