Co-seismic change in ocean bottom topography: Implication to absolute global mean sea level change

Geodesy and Geodynamics 10 (2019) 179e186

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Co-seismic change in ocean bottom topography: Implication to absolute global mean sea level change Jiangcun Zhou a, b, *, Heping Sun a, b, Jianqiao Xu a, b, Xiaodong Chen a, b, Xiaoming Cui a a b

State Key Laboratory of Geodesy and Earth's Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan, 430077, China University of Chinese Academy of Sciences, Beijing, 100049, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 August 2018 Accepted 10 April 2019 Available online 19 April 2019

Earthquakes perturb both the ocean bottom topography due to displacements of sea floor and the geoid due to mass redistribution, which induces the relative sea level (RSL) change. However, the relative global mean sea level (GMSL) change is zero in that sea water mass is conserved. But the absolute GMSL change is not zero because earthquakes displace total ocean mass with respect to the Earth's center of mass (CM) which remains unchanged after an earthquake. This displacement, i.e. the absolute GMSL change, may be detectable by altimetry since the satellites are orbiting around CM. In this paper, we proposed a method to estimate co-seismic absolute GMSL change caused by earthquakes based on the point dislocation theory for a spherically symmetric, non-rotating, elastic and isotropic (SNREI) Earth. This change can be directly connected to the perturbation of ocean bottom topography. We first computed co-seismic displacements as well as the change in geo-potential and solved the sea level equation to validate the insignificance of the oceans' feedback, i.e. the loading effect due to RSL change, to co-seismic displacements. The results imply that the loading effect due to RSL change is negligible on displacements while is considerable on geoid. We then computed the absolute GMSL change caused by co-seismic vertical and horizontal displacements by making use of the integrated Green's function method. The numerical results show that a large earthquake may raise the absolute GMSL by magnitude of sub-millimeter and the recent three large events cause GMSL to rise about one millimeter, in which the contribution from horizontal displacement is non-negligible. © 2019 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Point dislocation theory SNREI earth Sea level equation Absolute GMSL Ocean bottom topography

1. Introduction Sea level change is becoming a more and more important scientific issue involving a wide range of disciplines which have

* Corresponding author. State Key Laboratory of Geodesy and Earth's Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan, 430077, China. E-mail addresses: [email protected] (J. Zhou), [email protected] (H. Sun), [email protected] (J. Xu), [email protected] (X. Chen), [email protected] (X. Cui). Peer review under responsibility of Institute of Seismology, China Earthquake Administration.

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both scientific and social impacts. It has affected our habitats. For example, a village was displaced due to increasing sea level in Torres islands [1]. The results from altimetry concluded that current trend in the absolute global mean sea level (GMSL) change was about 3.4 ± 0.4 mm/a [2]. Specifically, due to the melting of the glaciers, a heavier sea level increase rate occurred in recent years [3]. Previous studies also showed that the glacial isostatic adjustment (GIA) contributed to the rate by about 0.3 mm/a [4]. Besides, earthquakes also cause sea level change because they give rise to deformation of the Earth as GIA does. When an earthquake occurs, it causes variations both in the Earth's geoid and ocean floor topography, hence in RSL. Many investigations have been conducted on this problem [5e9]. Melini et al. [7] gave a gravitationally self-consistent, integral sea level equation to compute earthquake-induced sea level change, which also considered ocean mass conservation of realistic oceans rather than that covering the whole Earth. Broerse et al. [8] applied this

https://doi.org/10.1016/j.geog.2019.04.001 1674-9847/© 2019 Institute of Seismology, China Earthquake Administration, etc. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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equation to investigate the relative sea level (RSL) and the geoid changes due to the 2004 Sumatra earthquake. Later they took the horizontal displacements' effect on sea water column into account, which showed considerable effect of the horizontal displacements on the geoid change [9]. Since sea level change is not uniform in the whole oceans, GMSL is of greater importance for sea level change estimation of the whole globe. However, according to the (relative) sea level equation [7], the relative GMSL change is zero because the ocean mass is conserved. However, this doesn't mean that earthquake will not cause change in the absolute GMSL. Because earthquake will alter the Earth's surface, the topography of the ocean bottom will change consequently. As a result, uplift of the ocean bottom will increase the sea level and vice versa because of conservation of sea water mass. This change can be detected by satellite geodesy such as altimetry because the altimetry satellite is orbiting around the Earth's Center of Mass (CM) which is unchanged after an earthquake. Consequently, the absolute GMSL is used to evaluate global sea level change caused by earthquake. To understand how and to what extent earthquakes affect ocean bottom topography and the absolute GMSL directly promoted the present study. As Broerse et al. [8] pointed out, the loading effect due to RSL change on vertical displacement is small for the 2004 Sumatra earthquake. Is it true for other earthquakes or other finite fault models of the same earthquake? As we know, the near-field displacements are highly dependent on the fault model. However, as will be shown in section 2, the absolute GMSL change is a global integral of displacements. Is it also highly dependent on fault model? These questions also motivated this study. Furthermore, the above-mentioned studies applied the normal mode method and an Earth model which is composed of several homogeneous layers. In this study we used the Preliminary Reference Earth Model (PREM, [10]) which is more realistic. Xu and Sun [11] investigated the Earth's volume change due to earthquake to estimate the Earth's expansion, in which only vertical displacements of degree zero over entire Earth's surface were involved. They focused on the mean value over the whole Earth. However our task considered only the ocean area, i.e. mean value over oceans, in which the orthogonality of spherical harmonics is not applicable any more. The absolute GMSL change is connected to the surface integral of vertical displacements and equivalent vertical displacements caused by horizontal displacements over oceans' area. Hence both spheroidal and toroidal displacements of all degree and order terms will be involved in our task.

2. Theory Earthquake perturbs geoid and induces vertical displacement as well as horizontal displacements of ocean bottom. Therefore position of sea level will be changed if an earthquake occurs. Upward vertical displacement of ocean bottom will lift the above sea water and the lifted sea water will not maintain its shape and redistribute to form a new equilibrium status due to gravitation from both the Earth and sea water. This procedure can be modeled by the sea level equation [7]. On the other hand, the horizontal displacements of ocean bottom also contribute to the column of the lifted sea water. According to Tanioka and Satake [12], the water column height change due to

both the vertical and horizontal ocean bottom displacements can be represented by

hðq; lÞ ¼ ur þ

vD vD u þ u Rvq q R sin qvl l

(1)

in which ur ; uq and ul are the vertical, southward and eastward components of the co-seismic displacement vector u, respectively, in a spherical coordinate system and D is the bathymetric depth of ocean bottom relative to sea level, R is the radius of the Earth. Both the co-seismic displacements and the depth D in Eq. (1) are dependent on colatitudes q and longitude l of the point of interest. Eq. (1) is always valid except that the derivative of D is infinite. This infinite case is for a vertical cliff. However, such an extreme case is unrealistic. The geoid change due to earthquake is the potential change divided by gravity, i.e.

Nðq; lÞ ¼

T

(2)

g

where T is potential change induced by earthquake (see e.g. reference [13]) and g is the normal gravity on the sea level. For the PREM Earth model, the value of g is about 9.816 m/s2. The relative sea level change is [7].

HRSL ¼ N  h  N  h

(3)

in which and hereafter the bar denotes mean over the oceans and an ocean function (which is 1 over oceans and 0 elsewhere) is omitted without confusion because sea level change only occurs over oceans. The last term in the right hand side in Eq. (3) guarantees the conservation of sea water mass [14]. The reason of including this term is as follows: if the earthquake causes gravity potential to increase, it will raise the geoid. However if there is no sufficient sea water to rise to coincide with the geoid, the sea level will form a new equi-potential surface beneath the geoid, and vice versa. So strictly speaking, the sea level in a new equilibrium status is not the geoid any more. Therefore the actual absolute sea level change is

HASL ¼ N  N  h

(4)

As the relative GMSL change is zero, the absolute GMSL change is consequently identical to the global mean of h, i.e.

ð HASL ¼ h ¼

hðq; lÞdS S

¼

DV S

(5)

in which S is the area of the oceans and DV is the water column change. As a result, the ocean bottom topography change can be used to estimate the absolute GMSL change. Meanwhile, the redistributed sea water caused by earthquake will result in the loading effect on both the ocean bottom topography and the geoid, which can be modeled by loading theory [15] as below

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1 0 ∞ n   X hn X RSL B HcRSL cosm l þ Hs sinm l P nm C nm nm B C B n¼0 2n þ 1 m¼0 C B C 0 B C ∞ n   0 Load 1 X X B ln dP nm C RSL RSL B C ur Hc cosm l þ Hs sinm l nm nm B 2n þ 1 m¼0 dq C B uLoad C C n¼0 3r B B q C B C B Load C＝ B 0 @ ul A 4pre B X ∞ n   mP C X ln nm C RSL RSL B C Hsnm cosml  Hcnm sinml B N Load sinq C B n¼0 2n þ 1 m¼0 C B C B ∞ C 0 n   B X 1þk X C RSL RSL @ n Hcnm cosml þ Hsnm sinml P nm A 2n þ 1 n¼0 m¼0

181

0

where the left part consists of the three displacement components in spherical coordinate system and the geoid change, from top to bottom respectively, due to loading, r is the density of sea water, re 0 0 0 is the mean density of the Earth, hn ; ln ; kn are Love numbers of degree n for vertical, horizontal displacements and potential, respectively, Hc's and Hs's are the coefficients of the spherical harmonic expansion of RSL, P nm is the fully normalized Legendre function of degree n and order m. This loading effect in turn causes sea water to redistribute. Hence the procedure of solving the sea level change is nonlinear. The solution is obtained by iterative method [7,8]. The uplift of the ocean bottom will decrease the load mass and cause rebound of ocean bottom, and vice versa. This means that loading effect has positive feedback to the absolute GMSL change. So ignoring this loading effect will result in a little bit underestimation of the absolute GMSL change. It is noted that oceans' area also changed slightly because of the horizontal displacements of oceans' boundaries after an earthquake. But it's contribution to h is of one order smaller magnitude, hence negligible. The derivatives of D can be computed by, for example, difference approach in terms of bathymetric data while the co-seismic displacements, including spheroidal and toroidal ones, can be derived by Green's function method [13,16,17]. As will be illustrated in section 3, the loading effect is negligible. Consequently the task of estimating the absolute GMSL change is to compute a surface integral of co-seismic displacements (see Eq. (5)). To precisely carry out the integration, we applied the integrated Green's Function method which was proposed by Goad [18] to compute ocean tide loading. This method can improve the computation precision. Furthermore it can overcome the first order singularity of Green's functions. For the vertical displacement induced by a vertical strike slip fault as an example, the Green's function is

Gðj; AÞ ¼ gðjÞsin 2 A

(7)

where g is an infinite sum of dislocation Love numbers and Legendre polynomials [17], j is the angular distance from the epicenter and A is the azimuth counting counterclockwise from fault line. The integrated Green's function is then defined as

G I ð jÞ ¼

ð jþDj=2 jDj=2

gðjÞsinjdj

Therefore, the water column change in Eq. (5) is then

(8)

(6)

DV ¼

2ðp ð p

½gðjÞsin2AsinjdjdA

0 0

¼

N X M X

GI ðji Þsin 2Aj sinðDAÞOij

(9)

i¼1 j¼1

in which DA is the step in azimuth and Oij is the value of ocean function in a grid indexed by i and j. In a general numerical method, the integrated Green's function is replaced by a product of Dj and the value of the Green's function at a representative point, say the central point of a grid. However, Eq. (9) is a rigorous integral provided the ocean function is accurate enough, hence gives rise to a more precise result. 3. Results and discussions 3.1. Loading effect due to RSL change We first computed the co-seismic displacements, including spheroidal and toroidal ones, as well as the geoid change, based on the point dislocation theory for PREM Earth model. It is noted that this Earth model was slightly modified by averaging the top two layers to omit the ocean layer. The procedures for solving sea level equation are as follows: (a) The vertical and horizontal displacements and geoid changes at central points in global grids with resolution of 0.25  0.25 were computed by Green's function method [13,17]. The Green's functions are computed from dislocation Love numbers up to 32000 . Then the water column height change in each grid was computed via Eq. (1) and the initial RSL change was obtained by subtracting the water column height change from the geoid change. (b) The initial RSL change was mapped onto the oceans and then expanded into spherical harmonics up to degree 720 and order 720. (c) The loading effect due to RSL change on the displacements and geoid change was obtained via Eq. (6). (d) The new RSL change was computed by adding the initial RSL change and the loading effect together and compared with the initial (last) RSL to judge whether the convergence was reached. If not, repeat (b), (c) and (d) till to convergence. In fact, (a) can also be conducted by dislocation Love numbers, which is the base of Green's functions, in the spectral domain. However, the results were not convergent if we truncated the

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harmonic degree to 720. Sun et al. [17] showed that the truncated harmonic degree should be larger than 10R/d within which d is the hypocenter depth. For an earthquake with hypocenter depth of 30 km for example, the truncation degree is at least 2000. Nevertheless, the truncated degree of 720 is sufficient for loading computation. This entails us to apply the above procedures to solve the sea level equation. Similarly to Broerse et al. [9], we took the horizontal displacement into account when solving the sea level equation. The ETOPO1 model (available at http://www.ngdc.noaa.gov/mgg/ global/global.html) was used. In each cell, the derivatives of ocean depth with respect to longitude and colatitude were approximated by the mean value of the slopes computed by cell points at two edges, respectively. The resolution of this topography model is 1 arc-minute by 1 arc-minute. Therefore, 225 cell values were averaged to obtain the final value in a grid with resolution of 0.25  0.25 . Finally, the derivatives of bathymetric depth with respect to colatitude and longitude were inserted in Eq. (1) to obtain the water column height change due to the horizontal displacements. We focused on the recent three large earthquakes, i.e. the 2004 Sumatra, 2010 Maule and 2011 Tohoku-Oki earthquakes. Fig. 1 shows the results for 2004 Sumatra earthquake. Three finite fault models of this earthquake were used to compute the loading effect. Fig. 1aee show the initial co-seismic vertical displacement, the contribution of the horizontal displacement to the water column height change, the loading effect due to RSL change on the water column height change, the geoid change and the loading effect on geoid change, from top to bottom respectively. Fig. 1aee are of USGS-Ji model [19]. Fig. 1fej show the corresponding items as the same as Fig. 1aee and are of Chlieh model [20]. Similarly, Fig.1k to o are of USGS model [21]. The results are highly dependent on the finite fault models used. For example, the initial co-seismic vertical displacements are about 12 m, 4 m and 3.8 m for the three fault models from left to right, respectively. These three values are not consistent even though scaled by the scalar moments. Consequently, the co-seismic changes are dependent on the detail fault parameters. It is found that the horizontal displacement contributes to the water column height change by about 10% (USGS-Ji), 9% (Chlieh) and 14% (USGS), respectively, and the loading effects due to RSL change are about 0.4%, 0.7% and 0.6% correspondingly, according to the extreme values. This implies the significance of the former effect while insignificance of the latter effect. However, the loading effect on geoid is considerable. For the three fault models, the percentages are 60%, 40% and 20%. Likewise, they are highly dependent on the fault models. As a result, loading effect must be considered in investigation of gravity change such as detecting co-seismic gravity change by GRACE gravimetry. Some studies used a simple method, e.g. zero-order approximation which took only mass term into account, to correct this effect [22,23]. However the exact consideration can be conducted by solving sea level equation. It is also noted that the geoid change is much smaller than the water column height change. Therefore the RSL change is nearly opposite to the water column height change. Comparing Fig. 1f and j with Fig. 2e and f in reference [8], we found a good agreement between them in spatial pattern. However, our results have a bit smaller magnitudes although the same fault model was used. This is because an accumulated scalar moment of 8.22  1022 Nm was used in reference [8] while 6.59  1022 Nm was used in the present study where the shear modulus of PREM model was used. This indicates the importance of using more realistic Earth model.

For the other two large earthquakes, i.e. the 2010 Maule and 2011 Tohoku-Oki earthquakes, we also used three fault models to solve sea level equation, respectively. They are USGS model (available at https://earthquake.usgs.gov/archive/product/finitefault/usp000h7rf/us/1486510671641/web/p000h7rf.fsp), Delouis model [24]and UCSB model (available at http://www.geol.ucsb. edu/faculty/ji/big_earthquakes/2010/02/27/chile_2_27.html) for the former and USGS model (available at http://equake-rc.info/ SRCMOD/searchmodels/listevents/), ARIA model [25] and UCSB model [26]for the latter. Similarly to Fig. 1, the corresponding results are shown in Figs. 2 and 3, respectively. For the 2010 Maule earthquake from Fig. 3, the contributions from the horizontal displacements to the water column height change are about 18%, 21% and 12% for the three fault models, respectively. Correspondingly, the loading effects on the water column height change are about 0.5%, 0.3% and 0.3% while on the geoid are about 40%, 38% and 54%. For the 2011 Tohoku-Oki earthquake from Fig. 2, the contributions from the horizontal displacements to the water column height change are about 13%, 17% and 13% for the three fault models, respectively. Correspondingly, the loading effects on the water column height change are about 0.4%, 0.3% and 0.4% while on the geoid are about 67%, 54% and 59%. As a whole, from above results, we can infer that the loading effect on the water column height change is very small. However this effect is very large on the geoid change. Furthermore, the horizontal displacement effect is non-negligible. Consequently, we can only use initial co-seismic displacements, including horizontal and vertical ones, to perform Eq. (5). 3.2. The absolute GMSL changes To perform an integral over the oceans, an ocean function or a land-sea boundary is needed. We used the land-sea boundary in SPOTL package [27], which has a resolution of 1/64 by 1/64 and is comparable with that of ETOPO1. The steps of the integrated Green's functions were set to be 0.01, 0.1, 0.5 and 1.0 in the spherical distance bands of 0.0 e1.0 , 1.0 e10.0 , 10.0 e90.0 and 90.0 e180.0 , respectively. The step of azimuth was set to be 1.0 . Because the values of Green's functions vary sharply in the nearfield, the step must be smaller for small spherical distance for good computation precision, which is similar to the ocean tide loading computation [27]. Table 1 shows the absolute GMSL changes caused by the above mentioned three large earthquakes. For comparison, results from point sources of these three earthquakes were also listed in the table. The parameters of the point sources are from GCMT (www. gcmt.com). These three large earthquakes all caused the absolute GMSL to rise by magnitude of sub-millimeter. They together raised the absolute GMSL by about 1 mm. In general, the magnitudes are highly dependent on the scalar moments. Actually, it is intuitive because the absolute GMSL is a global integral of displacements. Therefore, the value of the integration ought to reflect the energy of the earthquake. However the existence of the land alters the dependence on the scalar moment unless the co-seismic deformation dominates in the oceans. For example for the 2004 Sumatra earthquake whose epicenter is far away from land, the large values correspond to large scalar moments while values of the other two earthquakes do not. The other factor altering the dependence is the slope of the ocean floor. Since the slope is not regular in the ocean floor, the contribution from the horizontal displacements to the absolute GMSL is highly dependent on the location of the epicenter. Generally speaking, the effect of

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183

Fig. 1. Water column height changes and geoid changes due to the 2004 Sumatra earthquake. (left, middle and right columns are for three different fault models, respectively. The five pictures from top to bottom are for the water column height changes due to co-seismic vertical displacement, horizontal displacement and loading, the co-seismic geoid change and the loading effect on geoid change, respectively. Unit in mm).

horizontal displacement reaches about 10%e20% and the vertical displacement dominates the absolute GMSL change. Nevertheless, although there are large discrepancies between the numerical values of the absolute GMSL change derived from different fault models and point source, one will find that the differences become smaller after scaled by the scalar moment. As a consequence, one can simply use the point source to estimate the absolute GMSL.

4. Conclusion We solved the sea level equation to obtain the co-seismic vertical displacements, geoid and RSL changes with the consideration of ocean's feedback. The loading effect due to RSL changes on displacements is much smaller compared to the co-seismic ones. RSL changes are nearly opposite to the water column height change. However, the loading effect on geoid change is pronounced. The

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Fig. 2. The same as Fig. 1 but for the 2011 Tohoku-Oki earthquake.

horizontal displacements may contribute significantly to geoid change depending on the topography of ocean bottom. We proposed a simple but effective method to estimate absolute GMSL change caused by earthquake based on the point dislocation theory for a SNREI Earth. Absolute GMSL change was estimated in terms of volume change due to earthquake-induced topography

perturbation divided by oceans’ area. The recent three large earthquakes all caused the absolute GMSL to rise by magnitude of sub-millimeter. They together raised the absolute GMSL by about 1 mm. The three large earthquakes occurred in about 7 years. So roughly speaking, the GMSL change rate is about 0.16mm/a which is comparable to that of GIA effect. However, compared with product

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185

Fig. 3. The same as Fig. 1 but for the 2010 Maule earthquake.

Table 1 Co-seismic absolute GMSL changes due to recent three large earthquakes. Event

2004 Sumatra

2010 Chile

2011 Tohoku-Oki

Moment (1022 Nm) Vertical (mm) Horizontal (mm) Total (mm)

9.49/6.59/2.57/3.95 0.48/0.30/0.12/0.24 0.08/0.02/0.02/0.00 0.56/0.32/0.13/0.25

2.25/2.19/2.46/1.86 0.21/0.22/0.20/0.21 0.03/0.01/0.02/0.01 0.24/0.23/0.22/0.22

4.37/5.53/4.74/5.31 0.25/0.25/0.31//0.30 0.06/0.01/0.09/0.04 0.31/0.27/0.40/0.34

Notes: “Vertical” and “horizontal” mean contributions from vertical and horizontal displacements, respectively, and “total” means the sum of both. Values separated by the slashes are computed from the three finite fault models as mentioned in the test and the point source, from left to right respectively.

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of other source such as ice melting, the value is much smaller. Nevertheless, from our results, one will understand how much the co-seismic deformation of the Earth affects the absolute GMSL. Although the co-seismic deformation is highly dependent on the used fault model, the absolute GMSL is not because it is a global integral of the displacements. Therefore, to simplify the computation, a point source can be applied. It is noted that we only considered the co-seismic effect on the absolute GMSL. To entirely evaluate how the absolute GMSL changes after an earthquake, other effect such as the post-seismic effect should also be considered. We will investigate this effect in our future work. Conflicts of interest The authors declare that there is no conflict of interest. Acknowledgments We are grateful to Prof. Haoming Yan for providing us with the WFL Fortran functions to carry out the spherical harmonic analysis. This study was financially supported by the “973” project of China (Grant No. 2014CB845902) and the NSFC projects (Grant Nos. 41874026, 41374025 and 41621091). References rie. Ballu, M. Bouin, P. Sime oni, W.C. Crawford, S. Calmant, J.M. Bore , [1] Vale T. Kanas, B. Pelletier, Comparing the role of absolute sea-level rise and vertical tectonic motions in coastal flooding, Torres Islands (Vanuatu), PNAS 108 (2011) 13019e13022. [2] R.S. Nerem, D. Chambers, C. Choe, G.T. Mitchum, Estimating mean sea level change from the TOPEX and Jason altimeter missions, Mar. Geodes. 33 (supp 1) (2010) 435. [3] S. Yi, W. Sun, K. Heki, A. Qian, An increase in the rate of global mean sea level rise since 2010, Geophys. Res. Lett. 42 (2015) 3998e4006. [4] W.R. Peltier, Chapter 4: global glacial isostatic adjustment and modern instrumental records of relative sea level history, Int. Geophys. 75 (2001) 65e95. [5] D. Melini, A. Piersanti, G. Spada, G. Soldati, E. Casarotti, E. Boschi, Earthquakes and relative sea level changes, Geophys. Res. Lett. 31 (2004) L09601. [6] D. Melini, A. Piersanti, Impact of global seismicity on sea level change assessment, J. Geophys. Res. 111 (2006) B03406. [7] D. Melini, G. Spada, A. Piersanti, A sea level equation for seismic perturbations, Geophys. J. Int. 180 (2010) 88e100. [8] D. Broerse, L. Vermeersen, R. Riva, W. van der Wal, Ocean contribution to coseismic crustal deformation and geoid anomalies: application to the 2004 December 26 Sumatra-Andaman earthquake, Earth Planet. Sci. Lett. 305 (2011) 341e349. [9] T. Broerse, R. Riva, B. Vermeersen, Ocean contribution to seismic gravity changes: the sea level equation for seismic perturbations revisited, Geophys. J. Int. 199 (2014) 1094e1109. [10] A. Dziewonski, D. Anderson, Preliminary reference earth model, Phys. Earth Planet. In. 25 (1981) 297e356.

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Jiangcun Zhou. PhD in geophysics, associate research fellow at the Institute of Geodesy and Geophysics, Chinese Academy of Sciences. His research interests focus on numerical simulations of elastic, viscoelastic and poroelastic deformation of the Earth due to internal and external forces such as tide generating force and dislocation.