Topography effect on the seismogenic deformation of the earth's surface

TL;DR

This paper compares analytical and numerical models to assess how Earth's topography influences surface displacements after earthquakes, revealing that flat Earth approximations have notable errors especially laterally.

Contribution

It introduces a comparative analysis of flat slab approximation versus topography-inclusive models for earthquake-induced surface displacements.

Findings

Flat Earth approximation has higher error in lateral displacement.

Error in displacement magnitude is within 10% of maximum displacement.

Topography significantly affects surface deformation calculations.

Abstract

A comparison of the displacements of the earth's surface after an earthquake was made, calculating with the analytical expressions coming from an infinite flat slab approximation and compared with these numerically considering the topography of the Earth. One conclusion of this work is that the flat Earth approximation, has a greater error in the lateral displacement than in the vertical one. It can also be noted that the error in the magnitude of the displacement is less or of the order of ten percent of the maximum displacement of the earth's surface.

Tables

Table 1. Table 1: Information about earthquakes.

\brYear	2008	2010	2011	2014
\mrLocation	Wenchuan	Maule	Tohoku-Oki	Iquique
	China	Chile	Japan	Chile
\mrMoment magnitude	7.9 Mw	8.8 Mw	9.1 Mw	8.1 Mw
\mrSlip model	Fielding	Delouis	Hayes	Wei
	2013 [6]	2010 [7]	2011 [8]	2014 [9]
\mr${\Vert \Delta 𝐮\Vert }_{\text{max}}/{\Vert 𝐮\Vert }_{\text{max}}$	$0.58/5.07$	$0.72/6.69$	$1.68/21.6$	$0.13/1.57$
\mr${\left|\Delta u_z\right|}_{\text{max}}/{\left|u_z\right|}_{\text{max}}$	$0.22/4.73$	$0.17/5.34$	$1.15/9.73$	$0.097/0.87$
\mr$|\Delta u_{\mathrm{\{̣}}{}_L$ —_max / —u__L —_max 0.58/3.05 0.71/4.90 1.67/21.10.12/1.49$\mathrm{UNKNOWN}$

Equations

$$\mathbf{u}(\mathbf{x})=$$

$$\displaystyle\!\!-\!{\cal P}\int_{S}\frac{a}{r^{2}}\Big{\{}{\bf\hat{r}}\big{[}-3\left({\bf\hat{r}}\cdot{\bf u}\right)\left({\bf\hat{r}}\cdot d{\bf S}\right)+b\,{\bf u}\cdot d{\bf S}\big{]}-b\big{[}{\bf u}\left({\bf\hat{r}}\cdot d{\bf S}\right)+\left({\bf\hat{r}}\cdot{\bf u}\right)d{\bf S}\big{]}\Big{\}}$$

Full text

1 Introduction

In some fields of Geophysics it is necessary to know the deformation of the surface of the Earth due to the slip in geological faults. It’s known the analytical solutions [1, 2], but where the Earth is considered an infinite flat slab. Currently the effect of introducing real topography approximations has been considered in the literature. General solutions to this problem are numerical. There are examples of these estudies [3, 4, 5]. In this contribution, we develop a numerical method based on the boundary element method, which is applied to a homogenous, isotropic Earth and with the real topography of its surface.

2 Model for the Earth

The Earth is considered to be a linear, homogeneous, isotropic elastic solid of ellipsoidal geometry described by WGS 84. The geological fault is represented by a cutting surface, $\Sigma$, where the abrupt dislocation of one face with respect to the other, tangent to the fault plane, gives rise to the earthquake and the deformation of the external terrestrial surface, $S$.

From the equations that describe this type of solid we can deduce the displacement, ${\bf u}$, of any point ${\bf x}$ on the external surface S, only knowing the displacements in the points ${\bf y}$ of the entire surface

[TABLE]

where ${\bf r}={\bf y}-{\bf x}=r\hat{\bf r}$, $a=1/\left[4\pi(1-\nu)\right]$, $b=1-2\nu$ and $\nu$ is the Poisson’s coefficient.

This integral equation is solved self-consistently using an irregular triangular network (TIN) to represent the elements of the earth’s surface. For the creation of this TIN, we used the real topography of the Earth provided by Gebco [10].

The direction and magnitude of the dislocations in the fault, ${\bf U}$, are considered as input parameters and are obtained from slip models for the fault. There is usually more than one slip model for each earthquake, however, here we use a model for each of the cases with the propose of revealing the effect of the topography and without questioning the accuracy of the slip model.

For each of the earthquakes above considered we choose a model of slip among those published in the literature. We do not take into account the level of accuracy of those models, since at present, we just want to compare the effect of the real topography of the Earth in comparison of the flatness of the Earth assumption.

3 Results

The infinite flat slab model and the real topography of the Earth are compared, for four earthquakes of great magnitude. The information that characterizes these earthquakes is shown in Table 1, where the maximum values of the difference between the two methods are also shown, compared to the absolute value. These comparative values are made for the full displacement vector ${\bf u}$, to its vertical component, $u_{z}$, and to its lateral component $u_{{}_{L}}$.