GOCE-Derived Coseismic Gravity Gradient Changes Caused by the 2011 Tohoku-Oki Earthquake

Abstract

In contrast to most of the coseismic gravity change studies, which are generally based on data from the Gravity field Recovery and Climate Experiment (GRACE) satellite mission, we use observations from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) Satellite Gravity Gradient (SGG) mission to estimate the coseismic gravity and gravity gradient changes caused by the 2011 Tohoku-Oki Mw 9.0 earthquake. We first construct two global gravity field models up to degree and order 220, before and after the earthquake, based on the least-squares method, with a bandpass Auto Regression Moving Average (ARMA) filter applied to the SGG data along the orbit. In addition, to reduce the influences of colored noise in the SGG data and the polar gap problem on the recovered model, we propose a tailored spherical harmonic (TSH) approach, which only uses the spherical harmonic (SH) coefficients with the degree range 30–95 to compute the coseismic gravity changes in the spatial domain. Then, both the results from the GOCE observations and the GRACE temporal gravity field models (with the same TSH degrees and orders) are simultaneously compared with the forward-modeled signals that are estimated based on the fault slip model of the earthquake event. Although there are considerable misfits between GOCE-derived and modeled gravity gradient changes (ΔV_xx, ΔV_yy, ΔV_zz, and ΔV_xz), we find analogous spatial patterns and a significant change (greater than 3σ) in gravity gradients before and after the earthquake. Moreover, we estimate the radial gravity gradient changes from the GOCE-derived monthly time-variable gravity field models before and after the earthquake, whose amplitudes are at a level over three times that of their corresponding uncertainties, and are thus significant. Additionally, the results show that the recovered coseismic gravity signals in the west-to-east direction from GOCE are closer to the modeled signals than those from GRACE in the TSH degree range 30–95. This indicates that the GOCE-derived gravity models might be used as additional observations to infer/explain some time-variable geophysical signals of interest.

1. Introduction

Because satellite gravity observations are not limited to Earth’s surface conditions, and can cover the whole Earth quickly, they provide an independent way to detect the coseismic effects of large earthquakes, which is a good complement for other earthquake measurements (e.g., surface deformations), and are of great scientific significance.

A new generation of satellite gravimetry missions, including CHAMP (CHAllenging Minisatellite Payload) [1], GRACE (Gravity field Recovery and Climate Experiment) [2] and GOCE (Gravity field and steady state Ocean Circulation Explorer) [3], have been successfully implemented to detect global static and time-variable gravity signals. A large number of real observations and derived products from the CHAMP, GRACE and GOCE missions have been applied to Earth-science research, especially temporal gravity signals derived from the GRACE mission, which are used to monitor mass transport in the Earth system. The detection of coseismic gravity change signals using GRACE data has been widely studied. Generally, monthly gravity field models from GRACE before and after the earthquake were used to derive coseismic gravity or gravity gradient changes. Long-to-medium-wavelength coseismic and post-seismic gravity changes from large-scale earthquakes (e.g., the 2004 Sumatra-Andaman Mw = 9.1, 2010 Maule Mw = 8.8, 2011 Tohoku-Oki Mw = 9.0) have been adequately detected by the GRACE mission [4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Wang et al. [13] and Li and Shen [17] also determined the coseismic gravity gradient changes caused by the 2011 Japan Tohoku-Oki earthquake. The GOCE mission has been proven successful for constructing regional geoid models combining with the EGM2008 and terrestrial gravity datasets [18,19], and can be used to study the lithospheric modeling, dynamic topography, and glacial isostatic adjustment [20,21,22,23]. However, a few studies have focused on inferring the coseismic gravity change signals from simulated data [24,25,26] or real GOCE observations [27,28,29].

The main scientific objective of the GOCE mission is to recover static gravity field models up to degree and order (d/o) 200 by using SGG (Satellite Gravity Gradient) observations [3]. Because of the lower sensitivity of the gradiometer on board of the GOCE satellite at lower frequencies, the gravity gradient observations are not sensitive to temporal signals, the main power of which is at long wavelengths (commonly, at spatial resolutions higher than 1500 km) [30,31]. However, the coseismic gravity change signals from the 2011 Japan Tohoku-Oki earthquake were still detected by the GOCE mission [27,28]. Garcia et al. [27] showed that GOCE’s gradiometer, acting similarly to a seismometer in orbit, recorded the sound waves from the 2011 Tohoku earthquake. Fuchs et al. [28] combined a new vertical gravity gradient ${V^{\prime }}_{zz}$ from the diagonal components (V_xx, V_yy, and V_zz) along the orbit. The coseismic radial gravity gradient changes ΔV_zz in Fuchs et al. [28] were derived by subtracting the vertical gravity gradients computed by a reference model GOCO03s [32] from ${V^{\prime }}_{zz}$ with an along track and spatial filter applied to deal with the colored noise. Fuchs et al. [28] mentioned that they did not use the spatiospectral localization method like Han and Ditmar [26] to process geopotential coefficients, because there were no released global gravity field models (GGMs) from the post-earthquake GOCE data that could be used to detect coseismic gravity gradient changes.

In this paper, we choose a different approach than that proposed in Fuchs et al. [28]. First, we recover the GGMs (SH coefficients) before and after the earthquake from GOCE gravity gradient observations (V_xx, V_yy, and V_zz) along the orbit, based on the least-squares method. Note that, Fuchs et al. [28] did not estimate the SH coefficients of the pre- and post-earthquake global time-variable gravity field models. Then, we estimate the gravity changes (Δg) and the gravity gradient changes (ΔV_xx, ΔV_yy, ΔV_zz, and ΔV_xz) from the differences between two recovered global gravity models. In order to weaken the influences of colored noise and the polar gap problem from GOCE SGG observations on the recovered model, we use a tailored spherical harmonic (TSH) coefficients to recover the coseismic gravity change. Finally, we compare the results obtained from forward-modeled signals with the GRACE monthly gravity field models. Compared to Fuchs et al. [28], our approach can determine the coseismic gravity changes and the coseismic gravity gradient changes of other gravity gradient tensor (GGT) components in addition to ΔV_zz. The content of this manuscript is organized into four sections, beginning with a description of the methodology of GOCE data processing, forward modeling coseismic gravity gradient changes, and the computation of TSH coefficients in Section 2. The results and analysis are presented in Section 3. A summary and concluding remarks are provided in Section 4.

2. Methodology

2.1. Recovering Gravity Field Model from GOCE SGG Observations

In this paper, we derive the coseismic gravity and gravity gradient changes by evaluating the differences between a post-earthquake gravity field model and a pre-earthquake one from GOCE data; namely, we first recover gravity field models from the Gravity field and steady state Ocean Circulation Explorer Satellite Gravity Gradient (GOCE SGG) observations before and after the 2011 Tohoku-Oki earthquake. Here, we provide a brief description of the data processing strategies of recovering a gravity field model from GOCE SGG observations (more details can be found in Xu et al. [33]).

The component V_ij of the second-order gravity gradient tensor (GGT) is normally defined in the Local North-Oriented Frame (LNOF) with the x-axis pointing north, the y-axis pointing west and the z-axis up, as follows [33]:

$$V_{ij}(r,\theta ,\lambda )={\displaystyle \sum _{m=0}^{\infty }\left[A_m^{ij}(r,\theta )\cos m\lambda +B_m^{ij}(r,\theta )\sin m\lambda \right]},\begin{array}{l}{A}_m^{ij}(r,\theta )\\ B_m^{ij}(r,\theta )\end{array}\}={\displaystyle \sum _{n=m}^{\infty }{H}_{nm}^{ij}(r,\theta )\{\begin{array}{l}{\overline{C}}_{nm}\\ {\overline{S}}_{nm}\end{array}}$$

(1)

where the indices i and j define the gravitational gradient components (xx, yy, zz, xy,…) with respect to the LNOF axes (x, y, z); the indices n and m are the degree and order, respectively, of the spherical expansion; and $(r,\theta ,\lambda )$ are the spherical coordinates in the Earth’s Fixed Reference Frame (EFRF), where r is the geocentric radius; and θ and λ are the spherical colatitude and longitude, respectively. ${\overline{C}}_{nm}$ and ${\overline{S}}_{nm}$ are the (fully normalized) geopotential cosine and sine coefficients.$A_m^{ij}(r,\theta ),B_m^{ij}(r,\theta ),$ and $H_{nm}^{ij}(r,\theta )$ are the Fourier coefficients and transform coefficients, respectively; for more details, please refer to Koop [34].

Based on Equation (1), the functional and statistical models of the gravitational field recovered from the Satellite Gravity Gradient (SGG) data are defined as a standard Gauss-Markov model. Then, we can determine the geopotential coefficients ${\overline{C}}_{nm}$ and ${\overline{S}}_{nm}$, exploiting the least-squares (LS) method. However, the observed GGT of the GOCE mission is given in the Gradiometer Reference Frame (GRF), so we should transform the observations from the GRF to the LNOF by using the rotation matrix $R_{\mathrm{G}}^{\mathrm{L}}$ [33]. However, the accuracies of the V_xy and V_yz components are lower than those of the other components, so they will contaminate the high-precision components (V_xx, V_yy, V_zz, and V_xz) in the transformation. To avoid this situation, we transform the base functions ($H_{nm}^{ij}(r,\theta )\left\{\begin{array}{c}\cos m\lambda \\ \sin m\lambda \end{array}\right\}$) in Equation (1) instead of transforming the GGT observations in GRF. In particular, we multiply the matrix $R_{\mathrm{G}}^{\mathrm{L}}$ and its transposed matrix $R_{\mathrm{L}}^{\mathrm{G}}$ on both sides of the base functions in Equation (1) at every epoch. Hence, we have

$$V^{\mathrm{GRF}}=R_{\mathrm{L}}^{\mathrm{G}}{V}^{\mathrm{LNOF}}{R}_{\mathrm{G}}^{\mathrm{L}}$$

(2)

where V^GRF and V^LNOF represent the gravitational gradient tensor in the GRF and LNOF, respectively.

Because the power spectral density (PSD) of the trace of the GGT represents the total error of the summation of the diagonal components (V_xx, V_yy, V_zz), the SGG observations are high-precision only within the designed measurement bandwidths (MBW) from 0.005 to 0.1 Hz according to Figure 1 [35]. The noise outside of this MBW increases with the decreasing frequency for frequencies below 0.005 Hz, and the increasing frequency for frequencies above 0.1 Hz, especially for the 1/f behavior at low frequencies, which show the character of the colored noise of the gradiometer [35]. To handle this colored noise in the SGG data, we apply a bandpass Auto Regression Moving Average (ARMA) filter with a pass-band of 0.005–0.041 Hz to both sides of the linear observation equation, these being Equation (2) and Equation (1) [36]. The maximum frequency (0.041 Hz) of the pass-band approximately corresponds to the maximum degree (220) of the recovered gravitational potential model based on the following formula:

$$f_{\max }=N_{\max }/T_r$$

(3)

where T_r = 5383 s is one satellite orbital revolution (cf. [37]).

According to Figure 1, the noise level of the components V_xx and V_yy is about 10 mE/$\sqrt{\mathrm{Hz}}$ (1 mew = 10⁻¹²/s⁻²), that of V_zz is about 20 mE/$\sqrt{\mathrm{Hz}}$, which is consistent with Rummel et al. [38]. Thus, to combine the diagonal components (V_xx, V_yy, and V_zz) in the LS, we set the ratio of the standard deviation factors of V_xx, V_yy, and V_zz to 1:1:2.

2.2. Tailored Spherical Harmonic Coefficients

Based on the data processing strategies in Section 2.1, the post-earthquake and pre-earthquake gravity field models can be recovered from GOCE SGG data. However, the lower and higher frequency bands of the recovered models are heavily influenced by the 0.005–0.041 Hz band-pass ARMA filter. Moreover, the polar gap problem of GOCE’s orbit mainly affects the low-order spherical harmonic (SH) coefficients. We perform a numerical simulation to show how the recovered model’s SH coefficients are affected by the band-pass filter and the polar gap problem. First, a 1 s sampling of 61-day GOCE satellite orbits are produced from the released reduced-dynamic orbit data (SST_PRD_2), with 10 s sampling by polynomial interpolation. Then, we simulate the GGT observations along the orbit by using the EGM2008 model up to d/o 220. We add simulated colored noise according to the prior PSD of the GGT’s trace, as given by Cesare [35], to the simulated GGT data. Finally, by using these simulated observations, we recover the gravity field model based on the LS approach with the 0.005–0.041 Hz band-pass filter applied. The absolute errors of the recovered model’s SH coefficients are obtained by calculating the difference between the model and the EGM2008 model. Figure 2 displays the error spectra of SH coefficients in log10 scale. According to this figure, SH coefficients up to d/o 160 with large errors are mainly located in the arrow-shaped region, which is framed by the red lines. Referring to the conclusions by Sneeuw [39], the errors of the near-zonal coefficients (m ≤ 10) are mainly caused by the ill-posed problem occasioned by the 96.7° inclination of the GOCE satellite orbit. The other coefficients with large errors in the arrow-shaped region are due to the lower frequency limit of the band-pass filter. The coefficients outside the arrow region are called the tailored spherical harmonic (TSH) coefficients here, which correspond to two symmetrical quadrilaterals. These TSH coefficients will be used to show the coseismic gravity and gravity gradient changes.

As shown in Figure 2, the degree of the middle corner coefficients in the arrow-shaped region is approximately 30, which is very close to the degree 27 estimated by Equation (3), according to the minimum frequency of the 0.005–0.041 Hz band-pass filter. Thus, the minimum degree of the TSH coefficients is set to 30 herein. The vertices of the lower-left and lower-right corners of the arrow region approximately correspond to a degree of 80. Fuchs et al. [28] presented their results according to three bandwidths (0.005–0.05, 0.0039–0.03, and 0.00475–0.0175 Hz), and the bandwidth of 0.00475–0.0175 Hz was determined by the matched filter approach, which maximizes the expected signal. The degree range 30–95 approximately corresponds to the bandwidth of 0.005–0.0175 Hz [28]. So, we also choose 95 (see the red dashed line in Figure 2) for the maximum degree of the TSH coefficients. Moreover, the maximum degree of 95 is very close to the maximum d/o of the SH coefficients of Gravity field Recovery and Climate Experiment (GRACE)’s time-variable gravity field model, so we can compare the results from GRACE with those from GOCE at the same frequency band.

2.3. Post-earthquake Gravity Changes from GOCE

Based on the theory and data processing strategies described above, we estimated two GGMs (the pre-earthquake model and the post-earthquake model) up to d/o 220 from the released GOCE’s calibrated and corrected gravity gradients in the GRF (EGG_NOM_2 products) (GGT and IAQ) and precise science orbits with quality report (SST_PSO_2 products) (PRD and PRM) [40] before and after the 2011 Japan Tohoku-Oki earthquake. The IAQ products are the GRF to IRF attitude quaternions, and the PRM products are the Earth Fixed Reference Frame (EFRF) to Inertial Reference Frame (IRF) quaternions. We selected the data period from the 1st of November 2009 until the 28th of February 2011 for the pre-earthquake time span, and the period from the 15th of March 2011 until the 31st of May 2012 for the post-earthquake time span. Only the high-precision diagonal components (V_xx, V_yy, and V_zz) of the GGT are selected for modeling. Before forming the observation equation, some data preprocessing tasks were performed, such as data interpolation and outlier detection [33]. For a more detailed description of GGT data processing, please refer to Xu et al. [33]. To reduce the influence of high-frequency errors, a Gaussian filter with the smoothing radius of 210 km is also applied to the TSH coefficients. A radius of 210 km for the Gaussian filter is approximately determined with the formula 20000 km/N_max, corresponding to a maximum degree N_max of 95.

2.4. Post-earthquake Gravity Changes from GRACE

For comparison with GOCE, we also derived the coseismic gravity changes and gravity gradient changes from the Release05 (RL05) GRACE time-variable monthly gravity field models from the Center for Space Research (CSR), which were downloaded from the website http://icgem.gfz-potsdam.de/ICGEM/. The maximum d/o was 96.

The TSH coefficients that corresponded to the SH degree range 30–95 were chosen to maintain consistency with the processing of the GOCE models, and a Gaussian filter with the smoothing radius of 210 km was also applied to the TSH coefficients. The differences in the coefficients between the averages of the monthly models for one year before and after the earthquake were used to calculate the coseismic gravity changes and gravity gradient changes on a sphere with a height of 260 km.

2.5. Forward Modeling Coseismic Gravity and Gravity Gradient Changes

The PSGRN/PSCMP code for the modeling co- and post-seismic response of the Earth’s crust to earthquakes from Wang et al. [41] is used to compute the coseismic gravity changes of the 2011 Tohoku-Oki earthquake based on a five-layer half-space Earth model and the fault slip model from Caltech provided by Wei and Sladen [42]. The layer depths, densities, and seismic velocities are extracted from the CRUST2.0 global tomography model [43], which are derived from the epicenter (38.1°N, 142.8°E) grid cell layer parameters, and are shown in

Table 1. The 5-layer half-space Earth model used in the prediction of co- and post-seismic gravity changes.

Depth (km)	Density (ρ) (103 kg/m3)	VP(km/s)	VS(km/s)	Material Type
0–1	2.10	2.10	1.00	Elastic
1–8	2.70	6.00	3.40	Elastic
8–15	2.90	6.60	3.70	Elastic
15–22	3.05	7.20	4.00	Elastic
22–∞	3.40	8.20	4.70	Biviscous (Burgers body)

. The upper four crustal layers in the model are treated as elastic materials. The bottom half-space mantle layer is treated as biviscous materials, i.e., a Burgers body with a transient (Kelvin) viscosity and a steady state (Maxwell) viscosity (Details about the Burgers viscosities will be described in Section 2.6). The free-air corrections, as calculated by the vertical surface displacements, are added to the calculated coseismic gravity changes from the PSGRN/PSCMP program [9]. A Bouguer layer seawater compensation effect is also corrected according to the vertical displacements with the consideration of the land-ocean differentiation [44]. Since Broerse et al. [44] pointed out that simply considering a global ocean for the seawater correction would lead to non-negligible errors (up to a few 10% for the 2011 Tohoku-Oki earthquake), using a practical land-ocean mask ensures a more reliable seawater effect correction for the model prediction in our study.

The computation region is from 28°N to 48°N latitude and from 132°E to 152°E longitude. The epicenter (38.1°N, 142.8°E) is located at the center of this region. The cell size of the grid is set to 0.1° × 0.1°. The global gridded coseismic gravity changes on a sphere are formed by the forward-modeled coseismic gravity changes, as well as filling in zero values outside the computational region. Based on the global gridded coseismic gravity changes, we estimate the gravitational potential spherical harmonic (SH) coefficients up to d/o 250, by using the classical spherical harmonic analysis approach [45]. Then, based on the estimated gravitational potential SH coefficients, we calculate changes of both the coseismic gravity and radial gravity gradient on the Earth’s surface and on a sphere with a height of 260 km above the WGS84 reference ellipsoid, which are shown in Figure 3 and Figure 4. According to the maximum d/o 250 of the SH coefficients, we use a Gaussian filter with a radius of 110 km here. The epicenter is also shown as a black star in these figures. The units of the gravity and the gravity gradient are mGal (1 mGal = 10⁻⁵ ms⁻²) and mE, respectively. According to Figure 3 and Figure 4, the spatial patterns of the coseismic gravity changes and gravity gradient changes show classic dipole characteristics in the nearly west-to-east direction. Additionally, the attenuation effect of the coseismic gravity change signal at the satellite height is very clear.

The amplitudes of the coseismic gravity changes and coseismic radial gravity gradient changes are reduced by more than a factor of 10 and 20, respectively.

2.6. Modeling of Post-seismic Gravity Changes

For the GOCE and GRACE observations, the coseismic gravity field changes are estimated by using the differences in coefficients between the pre-earthquake model and the post-earthquake model. Here we note that taking the one-year mean field differences as the coseismic signals will inevitably bring in the impact of the post-seismic effects, which are likely to affect the peak-to-peak range, as well as the spatial pattern of the extracted signals [46]. Han et al. [46] revealed that a dominant post-seismic positive gravity change signal is visible surrounding the epicenter within a couple of years after the 2011 Tohoku-Oki earthquake, with the amplitude up to around +6 μGal under a 500 km spatial scale comparable as GRACE observations. Moreover, they gave the results from aviscoelastic relaxation model and an afterslip model, respectively, and showed that the observed vertical deformation at the coast and offshore agrees better with the viscoelastic relaxation model than the afterslip model. Therefore, we choose the viscoelastic relaxation model to estimate the post-seismic gravity change effect here. Based on our 5-layer half-space model in Table 1, we use Burgers viscosities including a transient viscosity of 10¹⁷ Pa s, and a steady state viscosity of 10¹⁸ Pa s, for the biviscous mantle (with a ratio value of 1 between Kelvin and Maxwell rigidity). Monthly gravity field changes after the Tohoku-Oki earthquake were then calculated according to our biviscous post-seismic model. The model predicted post-seismic gravity changes are comparable with those reported in Han et al. [46] in both amplitude and spatial pattern under a 500 km spatial scale, even though we use different viscosities for the biviscous mantle. Our Burgers viscosities used in the post-seismic model are smaller than those from Han et al. [46] (a transient viscosity of 10¹⁸ Pa s and a steady state viscosity of 10¹⁹ Pa), which might be due to the distinct differences in layer depths between our model and the one used in Han et al. [46].

We converted the gridded gravity changes into geopotential coefficients up to d/o 96, and derived the 1-year mean field after the earthquake to estimate the impact from post-seismic changes.

2.7. Computation of Coseismic Gravity Changes from the Hydrological and Oceanic Mass Redistributions

According to [40,47], the temporal corrections (direct tide, solid Earth tide, ocean tide, pole tide and non-tidal correction) have already been removed from the released GGT observations at each epoch. But only SH coefficients up to d/o 20, estimated from atmospheric and oceanic mass variations, and from seasonal variations of hydrology, are used to model the non-tidal signals [47]. Since the coefficients with the d/o lower than 30 are not included in TSH coefficients, the non-tidal GGT corrections certainly have no contribution to the derived gravity gradients in the manuscript. In order to evaluate the influences of the hydrological and oceanic mass redistributions on coseismic gravity changes, we calculated the pre-earthquake models (data from November 2009 to February 2011) and post-earthquake models (data from March 2011 to May 2012) up to d/o 96 from the ECCO-OBP (Ocean Bottom Pressure from Estimating the Circulation and Climate of the Ocean) [48,49] and GLDAS (Global Land Data Assimilation Systems) [50] models, respectively.

3. Results

3.1. Coseismic Gravity and Gravity Gradient Changes from the forward-modeled TSH Coefficients

Based on the forward-modeled TSH coefficients, the coseismic gravity changes Δg and gravity gradient changes of the components (ΔV_xx, ΔV_yy, ΔV_zz, and ΔV_xz) in the LNOF on a sphere with a height of 260 km above the WGS84 reference ellipsoid were computed, which are shown in Figure 5 and Figure 6. Figure 5 shows that the magnitude of the forward-modeled coseismic gravity changes is approximately 1.2 μGal. The maximum magnitude of the components (ΔV_xx, ΔV_yy, ΔV_zz, and ΔV_xz) is approximately 0.1 mE, which corresponds to the radial component. Compared to Figure 3 and Figure 4, the spatial patterns of Figure 5 and Figure 6 are no longer dipole patterns, and instead show multiple extrema. Additionally, the gravity changes Δg and the changes in the gravity gradient components ΔV_yy and ΔV_zz show multiple rings. The spatial patterns of Δg and ΔV_zz are very similar and nearly isotropic. According to Figure 6, the components (ΔV_xx, ΔV_yy, ΔV_zz, and ΔV_xz) have different spatial patterns. The components ΔV_xx and ΔV_xz have similar spatial patterns, namely, nearly west-to-east stripes and multiple poles in the North-South direction. The ΔV_yy component has a spatial pattern with nearly North-South stripes and multiple poles in the west-to-east direction. Moreover, we also use two other fault slip models, provided by the GSI (Geospatial Information Authority of Japan) and the USGS (United States Geological Survey), to compute the coseismic gravity gradient changes from the TSH coefficients(see the supplementary materials). Compared with the Caltech fault slip model (as mentioned in Section 2.5), the GSI and USGS models have different depth extends. The slips extend to 70 km and 58 km in depth for the GSI and USGS models, respectively, while for the Caltech model, the maximum depth of slip is 47 km. The dip angles of the three slip models are very close, all around 10°. Although calculated with different fault slip models, the spatial patterns of the gravity and gravity gradient changes are similar to the signals plotted in Figure 5 and Figure 6.

3.2. Post-seismic Gravity Changes from the Viscoelastic Model

We compute the 1-year mean post-seismic radial gravity gradient changes (ΔV_zz) based on the TSH coefficients method from the computed viscoelastic relaxation models in Section 2.6, which are shown in Figure 7. A Gaussian filter with the smoothing radius 210 km is applied to the TSH coefficients. The maximum and minimum values of the radial gravity gradient changes from the viscoelastic relaxation model are 0.0095 mE, and -0.0047 mE, respectively. This 1-year mean field will be used in removing the post-seismic effect, and deriving the coseismic gravity signals both from the GOCE and GRACE data (in Section 3.4 and Section 3.5). Comparison between our viscoelastic relaxation results and those from Han et al. [46] under the SH degree 60 truncation has also been provided in the supplementary of the manuscript.

3.3. Coseismic Gravity Changes from the Hydrological and Oceanic Mass Redistributions

Based on the TSH coefficients method, we compute the coseismic radial gravity gradient changes (ΔV_zz), which are shown in Figure 8. According to Figure 8, the radial gravity gradient changes derived from the hydrological and oceanic mass redistributions are very small. The maximum and minimum value of the radial gravity gradient changes from both of GLDAS and ECCO-OBP are 0.0022 mE, and −0.0027 mE, respectively. Nevertheless, the contribution from the hydrological and oceanic mass redistributions will be removed in deriving the coseismic gravity signals both from GOCE and GRACE data.

3.4. GRACE-derived Coseismic Gravity Changes and Gravity Gradient Changes

The coseismic gravity changes and gravity gradient changes from GRACE are shown in Figure 9 and Figure 10. It should be noted that before computing the coseismic gravity changes, we removed the post-seismic effects from the one-year mean post-seismic model, and the contribution from the hydrological and oceanic mass redistributions. According to Figure 9 and Figure 10, the coseismic gravity and gravity gradient changes from GRACE have nearly the same spatial pattern, showing only clear multi-pole characteristics in the North-South direction alongside west-to-east stripes. Only the gravity gradient changes ΔV_xx and ΔV_xz have similar spatial patterns compared to the forward-modeled coseismic signals plotted in Figure 6. Additionally, the gravity changes and the gravity gradient components ΔV_yy and ΔV_zz have very different spatial patterns compared to those from the forward-modeled coseismic signals. The spatial stripes for the component ΔV_yy are oriented west-to-east, which is completely different from the North-South stripes of the forward-modeled coseismic signal. The reason for this situation should be that GRACE’s satellite is in a polar orbit with an inclination of 89.5°, so the inter-satellite range-rate observations along the orbit are almost in the North-South direction. Thus, these inter-satellite range-rate observations are not sensitive to signals in the west-to-east direction.

Furthermore, when comparing the forward-modeled coseismic results in Figure 5 and Figure 6 with those from the GRACE data in Figure 9 and Figure 10, the magnitudes of the coseismic gravity changes from GRACE are smaller than those from the forward model.

3.5. GOCE-derived Coseismic Gravity Changes and Gravity Gradient Changes

We use the same method of processing the forward-modeled coseismic signals to compute the coseismic gravity and gravity gradient changes from GOCE, which are shown in Figure 11 and Figure 12. The post-seismic effects and the contribution from the hydrological and oceanic mass redistributions are also removed like GRACE-derived coseismic gravity signals. A Gaussian filter with a radius of 210 km was also applied. Figure 11 shows that the magnitude of the forward-modeled coseismic gravity changes is approximately 2.4 μGal. The maximum magnitude of the components (ΔV_xx, ΔV_yy, ΔV_zz, and ΔV_xz) is approximately 0.15 mE, which corresponds to the radial component.

4. Discussion

When compared to the forward-modeled coseismic signals in Figure 7 and Figure 8, the spatial patterns of the coseismic signals in Figure 11 and Figure 12 from the GOCE observations, although noisy, are somewhat similar, exhibiting multiple extrema. The spatial patterns of the gravity changes Δg and gravity gradient changes of the radial component ΔV_zz exhibit multiple rings. According to Figure 12, the components ΔV_xx and ΔV_xz show similar spatial patterns, i.e., the west-to-east stripes and multiple poles in the North-South direction. Note that, there are more gravity gradient change peaks compared to the modeled gravity gradient changes in the area of interest, which is the same situation with Figure 8 in Fuchs et al. [28]. Referring to Fuchs et al. [28], we also use the averaged accuracy of gravity gradients before the 2011 earthquake to represent the accuracy of the derived coseismic gravity gradients. The deviations of the mean (1σ) in 0.5° grid cells for the differences of gravity gradients (ΔV_xx, ΔV_yy, ΔV_zz and ΔV_xz) between the GOCE-derived pre-earthquake model and the GOCO03S [32] are computed, which are 0.018, 0.022, 0.036 and 0.024 mE, respectively. According to Figure 12, we could see that there is a significant change (greater than 3σ) in gravity gradients between pre- and post-earthquake gravity gradients.

We note that the uncertainties (σ) might be underestimated because the two models are not completely independent and the errors are not stationary, but we will not do further assessment here. Additionally, according to Figure 5, Figure 6, Figure 11 and Figure 12, the amplitudes of the coseismic gravity gradient changes are larger than the forward-modeled coseismic signals, and the geographical positions of the multi-poles are different, which is similar to what is seen in Fuchs et al. [28]. The geographic positions of these multi-poles from the GOCE data are located approximately 200 km to the northeast of the modeled results. The reasons for these discrepancies are still not very clear. According to Fuchs et al. [28], Heki et al. [51] and Feng et al. [52], the differences between the results from the forward model and GOCE data might be caused by systematic errors in the SGG observations and the weak sensitivity of SGG observations to the location of the earthquake. In addition, the large uncertainties of fault slip models also contribute to the discrepancies. Dai et al. [16] show that the fault slip model still leads to around 40% relative difference of gravity changes compared to the GRACE observations, even inverted with a multiple data source, which is likely, because the current commonly-used dislocation models are not sophisticated enough. The noticeable differences between model prediction and observation need to be investigated by further studies. According to Figure 7, Figure 10 and Figure 12, the maximum and minimum values from the viscoelastic relaxation model are nearly 15–21% of the ones (max: 0.031 mE, min: −0.046 mE) derived from the GRACE one-year mean field differences, and nearly 3–6% of the ones (max: 0.161 mE, min: −0.167 mE) derived from the GOCE one-year mean field differences. According to Figure 8 and Figure 12, the maximum and minimum value of the radial gravity gradient changes from both of GLDAS and ECCO-OBP are nearly two orders of magnitude smaller than the GOCE-derived coseismic gravity gradient signals.

In order to show the gravity change time series, we processed the GOCE SGG data to obtain the monthly gravity field models with the time period from November 2009 to May 2012. The data processing strategies are the same with that determining the pre-earthquake and post-earthquake models above. According to the GOCE daily reports EGG [53], there are a lot of special events, several data gaps and frequent calibrating operations (about once a month), from November 2009 to May 2012. Therefore, we are unlikely to be able to derive continuous monthly time-variable gravity field models due to the practical data quality. In addition, the solutions after the earthquake have larger oscillations, which agree with the fact that there are more special events in SGG data after the earthquake than those before the earthquake for most months [53]. Figure 13a,b show the radial gravity gradient changes at A (40.6°, 141.2°) and B (40.2°, 147.2°) points on Figure 12 derived from the monthly time-variable gravity field models before and after the earthquake, while Figure 13c,d shows the average radial gravity gradient changes at a negative area [(40°~41°), (140.5°~142.5°)] around A and a positive area [(39°~40.5°), (147~148.5°)] around B.

According to Figure 13, although the radial gravity gradient change time series contain some significant oscillations, they are still comparable with the co-seismic gravity change time series from GRACE for some smaller earthquake (Mw ≈ 8.5) given by some previous studies (see as Figure 2 of Han et al. [54], and Figure 6 and Figure 7 of Chao and Liau [55]). In order to show the changes more clearly, we use a 1D median filter (with 5th order) to process those four time series, and the obtained filtered results denoted by the red curves in Figure 13. Those filtered curves show obvious steps before and after (gray areas in Figure 13) the earthquake. We further use step functions to fit the original observations (the obtained fitting curves are also plotted in Figure 13; blue curves), and the step values can be estimated at the same time (see Figure 13). Here we use a bootstrap procedure (a Monte Carlo process, see Efron and Tibshirani [56]) to estimate the uncertainties for those step values. This method has been widely used for the uncertainty estimations in different geophysical research [57,58,59,60,61,62]. Here we give a simple example to explain the error estimation process (more details can be found in Shen and Ding [60]):

(1)

A step function s(t) is used to fit the original time series (the earthquake time is used as a priori information in the way some previous studies have done), then the step value s is obtained at the same time.

(2)

300 different random Gaussian white noise time series n_i(t) are synthesized, which have the same length, and with the same mean power in the frequency domain (i.e., almost the same noise level; here we use mean power P = 1.3 × 10⁻² mE²/cpy; see the supplementary Figure S3). Note that here we ignore the small differences of the oscillation amplitudes in the radial gravity gradient change time series (see Figure 13), because the time points are limited, the 300 different noise time series can almost reproduce all possible different oscillations.

(3)

The fitted step function s(t) is added in the 300 noise time series n_i(t), and 300 new noisy time series f_i(t) = s(t) + n_i(t) are obtained.

(4)

For the 300 noisy time series f_i(t), repeating the process 1) and 2) for all of them, then the 300 new step values s_i can be estimated. The standard deviation of those different S_i is used as the final error estimation for the step value s.

From Figure 13, we can see that the step values are clearly over three times their corresponding uncertainties (the double uncertainties are denoted by the blue areas in Figure 13 as the corresponding two standard deviations). Statistically, we may suggest that those steps represent the coseismic gravity gradient change signals caused by the 2011 Tohoku-Oki earthquake.

To reveal how the spatial patterns of the GOCE and modeled gravity gradient changes agree with each other, we computed the correlation coefficients (see

Table 2. Correlation coefficients, RMS of observed gravity and gravity gradients (ΔV_xx, ΔV_yy, ΔV_zz, ΔV_xz and Δg), RMS of the differences between the observed and forward modeling signals. The observed signals are from GRACE and GOCE. The positions for the computation are located in the region of 28°N–48°N, 132°E–152°E. The unit of ΔV_xx, ΔV_yy, ΔV_zz, and ΔV_xz is mE and the unit of Δg is μGal.

	GOCE					GRACE
Quantities	ΔVxx	ΔVyy	ΔVzz	ΔVxz	Δg	ΔVxx	ΔVyy	ΔVzz	ΔVxz	Δg
Correlation coefficients	0.68	0.56	0.61	0.68	0.63	0.85	0.53	0.68	0.80	0.74
RMS of observed signals	0.044	0.041	0.078	0.056	1.179	0.010	0.005	0.015	0.012	0.179
RMS of differences	0.038	0.036	0.066	0.047	1.043	0.007	0.012	0.015	0.008	0.158

) of the gravity changes and gravity gradient changes (ΔV_xx, ΔV_yy, ΔV_zz, ΔV_xz and Δg) between the observed models (GOCE and GRACE) and the forward model for the TSH coefficients (see Figure 2). The Root Mean Square (RMS) of observed (ΔV_xx, ΔV_yy, ΔV_zz, ΔV_xz and Δg) of GOCE/GRACE, and the differences between the observed and forward modeling signals, are also computed and shown in Table 2. According to Table 2, both of GOCE and GRACE do not perform well, because all the correlation coefficients are less than or equal to 0.85. Although most correlation coefficients corresponding to GOCE are lower than those of the GRACE time-variable gravity field models, the correlation coefficient corresponding to the ΔV_yy component from GOCE is slightly larger than the one of GRACE. Thus, for the SH degree range 30–95, the GOCE mission could reveal a greater signal of coseismic gravity gradient changes in the west-to-east direction than GRACE, because GRACE has sensitive observations along the North-South direction. The coefficient for ΔV_zz is very close to the value in Fuchs et al. [28]. All RMS of observed signals are reduced after subtracting the forward-modeled coseismic signals from the GOCE-derived results. However, because the amplitude of the results from GRACE is only about half of the one from the forward model, the RMS of GRACE’s ΔV_yy increases when the modeled signal is subtracted from the observed signal and the RMS of ΔV_yy doesn’t change. Of course, compared to GOCE, GRACE performs relatively well in the SH degree range 30–95, especially for the ΔV_xx component, which has the maximum correlation coefficient because only inter-satellite range-rates are observed along its orbit.

5. Conclusions

We employed the least squares method with a bandpass auto regression moving average filter, to recover two global gravity field models up to degree and order 220, from GOCE satellite gravity gradient data from the 15 March 2011 to the 31 May 2012 after the 2011 Japan Tohoku-Oki earthquake event. Then, we used the recovered models to estimate the coseismic gravity and gravity gradient changes that were caused by the earthquake by subtracting the pre-earthquake model from the post-earthquake model. This approach is different from that in Fuchs et al. [28]. They used the diagonal components (V_xx, V_yy, and V_zz) along the orbit to construct the new vertical gravity gradient V_zz, which was used to present the coseismic gravity gradient changes. To extract the coseismic gravity signals from the recovered global gravity field models, we proposed TSH coefficients according to the influences of colored noise in the SGG data and the polar gap problem on the recovered models.

The gravity changes Δg and the gravity gradient changes (ΔV_xx, ΔV_yy, ΔV_zz, and ΔV_xz) were computed by the GOCE-derived tailored spherical harmonic (TSH) coefficients, GRACE-derived TSH coefficients and the forward-modeled TSH coefficients. In data processing, the degree range 30–95 was used to obtain the TSH coefficients. A Gaussian filter with a radius of 210 km was also applied to the TSH coefficients according to the maximum frequency of 95. When comparing the coseismic gravity signals from GOCE, the forward model, and GRACE, the spatial patterns of the coseismic gravity and gravity gradient changes from GOCE were analogous to those from the forward model. There is a significant change (greater than 3σ) in the gravity gradients between the pre- and post-earthquake gravity gradients, which is associated with the earthquake. Moreover, the radial gravity gradient changes from the derived monthly time-variable gravity field models before and after the earthquake show obvious steps before and after the earthquake, whose amplitudes are at a level over three times that of their corresponding uncertainties, and thus significant. Statistically, it is reasonable to infer that these steps represent the coseismic gravity gradient change signals caused by the 2011 Tohoku-Oki earthquake. For the GGT components (ΔV_xx, ΔV_zz, and ΔV_xz), the correlation between the GOCE-derived coseismic gravity signals and the forward-modeled coseismic results was weaker than that from GRACE. However, the correlation coefficient that corresponded to the ΔV_yy component from GOCE is larger than that from GRACE. The component ΔV_yy had spatial patterns that included North-South stripes and multiple poles in the west-to-east direction. This situation means that the GOCE mission might reveal more coseismic gravity signals in the West-East direction than GRACE in the SH degree range 30–95. The estimated time series of radial gravity gradient changes (see Figure 13) suggest that the coseismic gravity changes can be reproduced from GOCE observations through the proposed method.