Estimation of local water storage change by space

Journal of Applied Geophysics 131 (2016) 23–28

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Estimation of local water storage change by space- and ground-based gravimetry Jiangcun Zhou ⁎, Heping Sun, Jianqiao Xu, Weimin Zhang State Key Laboratory of Geodesy and Earth's Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China

a r t i c l e

i n f o

Article history: Received 24 September 2015 Received in revised form 15 May 2016 Accepted 22 May 2016 Available online 25 May 2016 Keywords: Superconducting gravimeter GRACE Hydrological model Local water storage change Equivalent water height Well water table record

a b s t r a c t We estimated local water storage change by combining space- and ground-based gravimetry in this paper. The gravity change from GRACE was first divided into local and global parts according to potential theory. We then subtracted the GRACE-derived global field from ground gravimeter results to obtain local gravity change which is directly induced by the local water storage. Finally we inferred the local water storage change. We used superconducting gravimeter (SG) data recorded from June 2008 to June 2012 at Wuhan station and GRACE satellite gravimetric data to estimate the local water storage change. To validate the inferred local water storage change, the water table records of a well which is several meters away from SG station were compared. Furthermore, the equivalent water heights from hydrological models and GRACE were used also for comparisons. The comparisons show that the results from combining SG and GRACE data are better than those from either GRACE data alone or hydrological models, which demonstrates the efficiency of the combination method to derive local water storage. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The launch of the twin satellites of gravity recovery and climate experiment mission (GRACE) provided us monthly time-variable gravity field of the Earth with unprecedented precision (Tapley et al., 2004). The products of this mission are now widely applied in geodesy and geophysics (Ramillien et al., 2004; Han et al., 2006; Chen et al., 2007; Pfeffer et al., 2011; Crossley et al., 2012). Specifically, it allows us to evaluate the water transportation. However, due to the large errors in the GRACE-recovered gravity field of the high degrees' spherical harmonics, some special techniques such as Gaussian smoothing should be applied to avoid the south–north stripes. Therefore only a certain resolution of water distribution, e.g. 300 km, can be accurately determined after smoothing. Therefore, GRACE does well in estimating global or regional water storage change but not in estimating local one. On the other hand, the superconducting gravimeter is a type of relative gravimeter with high precision and high stability. It can detect gravity change in 0.1 μgal (1 μgal = 10– 8 m/s2) and has a low drift which is about several μgal per year (Xu et al., 2008). More importantly, the gravity change from SG is measured at a point on the Earth's surface. Consequently, it contains abundant signals due to ⁎ Corresponding author. E-mail address: [email protected] (J. Zhou).

http://dx.doi.org/10.1016/j.jappgeo.2016.05.007 0926-9851/© 2016 Elsevier B.V. All rights reserved.

local environmental effects, including hydrological one. However, we can't obtain the information on the local water storage by using SG data alone. Therefore another data need to be combined, e.g. GRACE data. Nevertheless, the gravity changes obtained from GRACE and SG are not comparable because, unlike SG, GRACE satellites can't sensor the vertical displacement of the Earth's surface. Fortunately, the hydrological effect has a dominant seasonal period while almost only the hydrological effect is left in the GRACE result in this period after removing tidal and atmospheric effects. Therefore, Neumeyer et al. (2006) developed an approach to make the two gravity changes comparable by considering the water loading effect on the vertical displacement of the Earth's surface. However, a factor of 1 + k'n was not taken into account in this term, in which k'n is the load Love number corresponding to additional potential (Farrell, 1972). Hence a correction was made on the formula (de Linage, 2008; Zhou et al., 2009). Now that the gravity changes from GRACE and SG are comparable, we can isolate the local water storage by combining these two kinds of gravimetric data. Section 2 gives the rationale of how to combine GRACE and SG data to estimate local water storage and Appendix A gives the accessory knowledge of the potential theory. Section 3 introduces the data used and the processing procedures of these data and shows an application of the theory and the consequent results and discussions. And finally some conclusions are drawn in Section 4.

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J. Zhou et al. / Journal of Applied Geophysics 131 (2016) 23–28

2. Rationale

change will be helpful to estimating local water storage. According to the descriptions in Appendix A and Eq. (4), we have

2.1. Space- and ground-based gravity changes The geopotential of the Earth can be represented by the spherical harmonics as ∞  nþ1 X n h i GM X R C nm cosðmλÞ þ Snm sinðmλÞ P nm ð cos θÞ T ðr; θ; λÞ ¼ R n¼0 r m¼0

0 1 0 0 1 ∞ − −ðn þ 1Þkn þ 2hn X GM B C Δg G ¼ −2πGσ þ 2 @ 2 A 0 R n¼0 1 þ kn n h i X C nm cosðmλÞ þ Snm sinðmλÞ P nm ð cos θÞ 

ð5Þ

m¼0

ð1Þ where T is the geopotential of the Earth, r is the distance from the point of interest to the Earth's center, θ and λ are co-latitude and longitude of the point of interest, respectively, G is the gravitational constant, M is the mass of the Earth, R is the radius of the Earth, C nm and Snm are the fully normalized spherical harmonic coefficients, i.e., Stokes coefficients, which are provided by GRACE products, P nm is the fully normalized associated Legendre function, n and m are, the harmonic degree and order, respectively. The gravity change on the Earth's surface can be derived from GRACE products according to (de Linage, 2008; Zhou et al., 2009) Δg S ðR; θ; λÞ ¼ −

GM X∞ R2

n¼0

! Xn

0

n þ 1−

2hn

0

1 þ kn

m¼0

h

i C nm cosðmλÞ þ Snm sinðmλÞ P nm ð cos θÞ

ð2Þ in which Δ gS denotes space-based gravity change on the Earth's surface, hn' and kn' denote the load Love numbers of degree n for vertical displacement and potential, respectively. However, due to the large errors in the high-degree Stokes coefficients provided by GRACE products, a filter is usually applied, for example Gaussian filter (Whar et al., 1998), to obtain a reliable result. On the other hand, the gravity change on the Earth's surface can be measured directly by ground instruments such as superconducting gravimeter. We denote this change by Δ gG. Theoretically, the gravity changes on the Earth's surface derived from space and ground-based techniques are identical. 2.2. Water storage change From the Stokes coefficients, we can derive the equivalent water height which can also be expanded in spherical harmonics, i.e. ∞ X n h i X H H C nm cosðmλÞ þ Snm sinðmλÞ P nm ð cos θÞ H ðθ; λÞ ¼ R

ð3Þ

n¼0 m¼0

The coefficients can be derived from gravity field according to the relation below (Whar et al., 1998) H

C nm H Snm

! ¼

ρE 2n þ 1 0 3ρW 1 þ kn



C nm Snm

 ð4Þ

where ρE and ρW are mean densities of the Earth and water, respectively. Identically, a filter should be applied. The filtered water height represents physically the smoothed and spatially averaged water storage. For a filter with radius of 300 km, this means that the value is spatially weighted globally, in which the weight decreases from 1.0 to 0.5 when angular distance increases from 0 km to 300 km and quickly converges to zero when angular distance becomes larger. It is such a large area that the filtered result may not properly represent the local water storage. As we know, the ground-based gravity change is measured at a point on the Earth's surface. However, the ground-based gravity change alone can't be used to invert the local water storage. It will be shown that combining ground-based gravity change with the space-based gravity

where σ represents the equivalent water surface density at a point just below the instrument. Actually, the water just below the instrument is not isolated but collected with the surrounding water body. Hence the density at this point can represent the surrounding, i.e., local, water storage in a sense. Consequently, σ reflects the local water storage. The gravity change in the left hand side of Eq. (5) can be observed accurately by superconducting gravimeter while the second term in the right hand side can be derived from GRACE results. We can call the second term of right hand side of Eq. (5) global correction term, because it is the term to be corrected to obtain the local gravity change from ground-based gravimetry. It is noted that the effect of vertical displacement on gravity change is considered by the h'n term between the parentheses. As a result, the water surface density σ can be obtained by combining ground- and space-based gravimetric data. Dividing σ by water density, i.e. 1.0 × 103 kg/m3, we can obtain the equivalent local water height. 3. Application 3.1. Data and processing For the ground-based gravimetry, we used the data measured by the superconducting gravimeter at Wuhan national geodetic observatory from June 2008 to June 2012. This station, participated in GGP (Global Geodynamics Project, Crossley et al., 1999), is located at the top of Yanjia Mountain in the suburb of Wuhan. The mountain consists mainly of the Dyas silicalite rocks with thin inter layers of shale, which are the perfect bedrock for observatory because of their great hardness and stability and powerful resistance to weathering. During the processing, the tides were removed by a theoretical model including both Earth's tides and ocean tide loading, computed by tidal factors obtained from harmonic analysis on the SG data (Wenzel, 1996). The atmospheric effect was removed by simultaneously observed air pressure data according to the atmosphere-gravity admittance. And the polar motion effect was removed, according to Wahr's theory (Wahr, 1985), by using Earth's rotation data (http:// www.iers.org). The non-tidal ocean loading is neglected, which is relatively much smaller than the hydrological effect because Wuhan SG station is far away from the ocean. Therefore, the final monthly residual dominantly contains the hydrological signal. For the space-based gravimetry, we used RL05 data released by GFZ (Dahle et al., 2012) and CSR (Bettadpur, 2007) and applied the 300 km fan filter (Zhang et al., 2009). The gravity change at Wuhan station of global part was computed according to Eq. (5). Finally the local water storage change was obtained by subtracting GRACE derived gravity change of global part from the SG derived gravity change. Additionally, for the purpose of validation on the estimation of local water storage from gravimetry, the records of a well water table which is several meters away from the SG station were used. For comparison, the 10-min interval data were averaged to monthly ones. Furthermore, the monthly hydrological models of NOAH-GLDAS (Rodell et al., 2004) and CPC (Fan and van den Dool, 2004) were used. These two models have spatial resolutions of 0.25° × 0.25° and 0.5° × 0.5°, respectively. The water heights at Wuhan station were extracted from the

J. Zhou et al. / Journal of Applied Geophysics 131 (2016) 23–28

hydrological models by bilinear interpolation in terms of the four nearby grid values. 3.2. Results and discussions We computed the gravity changes at Wuhan station by using the GRACE-recovered Stokes coefficients through Eq. (2), in which 300 km fan filter was used. The water loading effect on gravity was also computed by using the hydrological model and the Green's function method according to Farrell's theory (Farrell, 1972) for comparison. Meanwhile, to intuitively see how large the global correction terms are, these terms were computed by GFZ and CSR products as well. Fig. 1 superimposes these results as well as the gravity residual observed by SG. From Fig. 1, we find that the hydrological-effect-induced seasonal variations appear in the three curves, but the magnitude of gravity residual in last half of 2008 and first half of 2009 is obviously larger than either that from GRACE or that from water loading. However, the phases of the curves are in good agreement in this time band. The results from GFZ and CSR are nearly identical. Furthermore there is a good agreement between GRACE results and hydrological model results in both amplitude and phase. The hydrological models have spatial resolutions of 0.25° × 0.25° and 0.5° × 0.5°, which means that the water distribution at Wuhan station is averaged in a grid with a certain size of about 28 km × 24 km or about 56 km × 48 km. Similarly, the gravity change from GRACE is smoothed with a radius of 300 km. As a result, both GRACE and hydrological models give a spatial mean of the regional water storage changes. This may be the reason why the results of GRACE and hydrological models agree well. However, they are all slightly different from the gravity residual obtained by SG. Fig. 1 also shows the global parts derived from GRACE results (blue circles in upper plate of Fig. 1). The magnitude of the variations from GFZ and CSR are both about 2.5 μgal which corresponds to 6 cm equivalent water height according to infinite plate approximation. Due to its high precision, absolute gravimetry is commonly used to calibrate SG (e.g. Francis et al., 1998) and to determine the drift of SG. There were no sufficiently available absolute gravity observations at the SG station to determine the drift of SG. Fortunately, there is another station in Wuhan which is about 2 km away. There were some absolute

Fig. 1. Comparison of gravity changes derived from GRACE, SG and water loading. (upper plate: the black circles show the gravity residual; the red circles show the gravity change derived from GFZ (solid) and CSR (hollow) GRACE-recovered Stokes coefficients, respectively, and; blue circles show the global correction terms correspondingly from GFZ and CSR solutions. Lower plate: the black circles show the gravity residual; the blue circles show the water loading effect on gravity derived from GLDAS-NOAH (solid) and CPC (hollow) hydrological models, respectively.) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

25

gravity measurements at that station in the same time band. Due to a small distance between these two stations, the two stations have identical tectonic background and meteorological environment. As a consequence the absolute gravity change at that station can be used as a reference to assess the SG drift although the SG and AG data were not measured at the same station. Fig. 2 shows the gravity changes obtained by SG and AG as well as the hydrological effect derived from CPC model. AG data in Fig. 2 have the following acquisition procedures (Chen and Liu, 2015): for each measurement, twenty five groups had been conducted in about 1 h interval. Each group has more than 100 drops. The data were corrected according to the standard processing requirement of absolute gravity. The final result is the mean value of the 25 groups. It is intuitive that they agree well, in particular between AG and CPC results. Although we can't determine the SG drift by AG data because they were measured at two different stations, however we can safely say that the SG drift is small due to the good agreement and the small distance between the two stations. According to the theory in Section 2, we computed the equivalent water heights from GRACE, hydrological model and combination of SG and GRACE, respectively. And the results are shown in Fig. 3. In Fig. 3, the upper plate shows the comparison between GRACErecovered equivalent water height and the well table record. We find that GRACE captured well the time variation pattern, i.e. seasonal change, of the water storage change. It is noted that the two scales (left for well water table record and right for equivalent water height) are different. We artificially keep the two curves fitted (the same for the other two plates). Then the ratio between the two scales means the porosity (or equivalently water content) of the rock beneath Wuhan station. The water level of the well changed about 8 m while the equivalent water height changed about 0.3 m. This means that the porosity of the rock over Wuhan station inferred from GRACE is about 3.75%. The rocks at Wuhan station are comprised of mainly silicalite and shale (Xu et al., 2008), of which the typical porosity is about 5%– 10%. The porosity derived from GRACE is beyond the typical range. Consequently, the water storage derived from GRACE did not reflect the true water storage change. The middle plate shows the comparison between well water table record and the equivalent water height extracted from the hydrological model. Similarly, the time variation pattern of the curves is in a good agreement. However the porosities inferred from hydrological models are identical to that from GRACE and also smaller than the typical value. The lower plate shows the comparison between well water table record and the equivalent water height derived by combining GRACE and SG data. It is seen that there is a better agreement between them than those in the two upper plates. The two curves obtained by using GFZ and CSR products to correct global terms are nearly identical. As mentioned above, the effect of global terms on equivalent water height is about 6 cm which is about 1/9 of the total water height change. The consequent porosity is about 6.25% which is more reasonable than

Fig. 2. Comparison of gravity changes obtained from SG (purple), AG (blue with error bar) and CPC (black) hydrological model at Wuhan. The distance between the two stations where gravity changes measured by AG and SG is about 2 km. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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J. Zhou et al. / Journal of Applied Geophysics 131 (2016) 23–28

Fig. 3. Comparison between well water table record and the equivalent water heights derived from GRACE alone (upper), hydrological model (middle) and combination of GRACE and SG (lower).

those inferred from both GRACE gravimetry alone and hydrological models. This shows that combining GRACE and SG gravimetric data can better estimate the local water storage change than using either GRACE data alone or hydrological models. To quantitatively assess the differences among the curves in Fig. 3, their cross correlation coefficients were computed and listed in Table 1. According to the correlations between derived equivalent water heights from GRACE, hydrological models and combination of SG and GRACE. It is found that the correlations between well table record and the result of combination of SG and GRACE are the largest, which are 0.74 for SG-GFZ and 0.73 for SG-CSR, respectively. Although the correlations between GFZ or CSR GRACE results and well table record are relatively large, 0.69 and 0.58 respectively, the magnitude of equivalent water heights are smaller than that of combination of SG and GRACE, from Fig. 3. The correlations between well table record and hydrological models are the smallest. As a consequence, combining SG and GRACE data to estimate local water storage change is useful.

Table 1 The cross-correlations of the water heights derived from different means.

well GFZ CSR GLDAS CPC SG-GFZ SG-CSR

well

GFZ

CSR

GLDAS

CPC

SG-GFZ

SG-CSR

1.00 0.69 0.58 0.39 0.47 0.74 0.73

– 1.00 0.85 0.72 0.76 0.41 0.40

– – 1.00 0.75 0.71 0.26 0.22

– – –

– – – – 1.00 0.24 0.22

– – – – – 1.00 1.00

– – – – – – 1.00

1.00 0.86 −0.02 −0.04

4. Conclusions We applied an effective method which is derived according to potential theory to estimate the local water storage change by gravimetry. The method combines GRACE and SG gravimetric data by using the advantages of the two kinds of gravimetric means, i.e., good local information from SG and good global information from GRACE. As a validation, we compared the well water table record with the equivalent water heights derived from GRACE data alone, hydrological models and combination of GRACE and SG, respectively. We inferred that both GRACE and hydrological models give the local water storage with relatively smaller magnitude at Wuhan station due to their relatively crude resolutions while combination of GRACE and SG can well estimate the local water storage. The main reason is that SG compensates for the GRACE's resolution problem due to filtering. Using GRACE data to correct the global term can contribute about 6 cm of equivalent water height in estimating water storage change. This value is about 1/9 of the total water storage change at Wuhan station. Therefore, the combination of SG and GRACE data has improved the result. Although we didn't make an experiment to determine the porosity of rocks beneath Wuhan station but used a typical value, the result is still sound because we know exactly the composition of the rocks. Furthermore, the inferred porosity from combination of GRACE and SG data is more reasonable than that either from GRACE alone or hydrological models. As a result, one can derive the local water storage change from ground- and space-based gravimetry provided that the composition or the porosity of the rock is known.

J. Zhou et al. / Journal of Applied Geophysics 131 (2016) 23–28

Acknowledgments

27

Considering the deformation due to the loading effect of the surface layer, i.e. elastic part, we obtain the final gravity change on the ground

This study was financially supported by the NSFC projects (Grant Nos. 41374025 and 41321063) and the “973” project (Grant No. 2014CB845902).

Δg ¼

0 0 ∞ n h i 3g R X −ðn þ 1Þ−ðn þ 1Þkn þ 2hn X σ σ ̅ ð cosθÞ C nm cosðmλÞ þ Snm sinðmλÞ P nm ρE R n¼0 2n þ 1 m¼0

ðA8Þ Appendix A. Derivation of local water storage from gravimetry According to the potential theory (Heiskanen and Moritz, 1967), the gravitational potential V at a point in space induced by a spherical water surface layer can be derived by Z V ðθ; λÞ ¼ f

  σ θ0 ; λ0 R2 sin θ0 dθ0 dλ0   l θ; λ; θ0 ; λ0

ðA2Þ

and    Z   ∂ 1 2 ∂V  ¼ 2πf σ ðθ; λÞ þ f σ θ0 ; λ0 R sin θ0 dθ0 dλ0  ∂n R− ∂n l

Δg G ¼

! 0 0 ∞ n h i GM X −ðn þ 1Þ−ðn þ 1Þkn þ 2hn X C nm cosðmλÞ þ Snm sinðmλÞ P nm ð cosθÞ 0 2 R n¼0 1 þ kn m¼0

ðA9Þ

ðA1Þ

where θ and λ are, respectively, the colatitude and longitude of the point of interest in space, f is the gravitational constant, σ is the surface density of the surface layer, θ' and λ' are, respectively, the colatitude and longitude of the point on the surface layer, R is the radius of the spherical surface, l is the distance between the two points. The potential is continuous everywhere but its derivative normal to the surface is not. As the surface is approached from outside and inside, the out normal derivatives of the potential are, respectively,    Z  0 0 ∂ 1 2 ∂V  ð Þ ¼ −2πf σ θ; λ þ f σ θ ; λ R sin θ0 dθ0 dλ0 ∂n Rþ ∂n l

If the surface layer is derived from GRACE gravimetry and Eq. (5) and σ = ρwH are both considered, then we obtain

ðA3Þ

as derived by Zhou et al. (2009), where C nm and Snm are the Stokes coefficients provided by GRACE products as mentioned in Section 2.1. Alternatively, if we do not apply Eq. (A6), we will directly derive that 0 1 0 0 1 ∞ − −ðn þ 1Þkn þ 2hn fM X B 2 C Δg ¼ −2πf σ þ 2 @ A 0 R n¼0 1 þ kn n h i X C nm cosðmλÞ þ Snm sinðmλÞ P nm ð cos θÞ  G

ðA10Þ

m¼0

The advantage of Eq. (A10) is that the summation term in the right hand side converges more rapidly than that in Eq. (A9). This is the reason why Merriam (1980) used Eq. (A10) to compute ocean tide loading, in which the first term in the right hand side was dropped for land stations where there is no ocean mass load. For our application, on the contrary, this term is that we expected. References

It is clear that if the gravimetry is located on land then Eq. (A2) should be used; if the gravimeter is located in basement then Eq. (A3) should be used. For our application, the superconducting gravimetry is installed on land. Hence we adopt Eq. (A2). If we expand the surface density into spherical harmonics, that is σ¼

∞ X n h X

i σ σ C nm cosðmλÞ þ Snm sinðmλÞ P nm ð cosθÞ

ðA4Þ

n¼0 m¼0

where ρw is the density of water. We can derive that (Heiskanen and Moritz, 1967)   ∂ 1 2 R sin θ0 dθ0 dλ0 ∂n l ∞ n h i 3g R X 1 X σ σ ¼− C nm cosðmλÞ þ Snm sinðmλÞ P nm ð cos θÞ ðA5Þ 2ρE R n¼0 2n þ 1 m¼0

Z f

σ

According to Munk and MacDonald (1960) we have

2πf σ ¼

∞ X n h i 3gR X σ σ C cosðmλÞ þ Snm sinðmλÞ P nm ð cosθÞ 2ρE R n¼0 m¼0 nm

ðA6Þ

Adding Eqs. (A5) and (A6) together, we obtain  ∞ n h i ∂V  3g X nþ1 X σ σ ¼− R C cosðmλÞ þ Snm sinðmλÞ P nm ð cos θÞ ρE R n¼0 2n þ 1 m¼0 nm ∂n Rþ ðA7Þ

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