Promoción del conocimiento de los postulados del DIH por las partes del conflicto

An earthquake is a sudden process in the Earth’s crust or mantle caused by tectonic stress. The earthquake location specifies the spatial position and time of occurrence for an earthquake. The location may refer to the earthquake hypocentre and corresponding origin time, a mean or centroid of some spatial or temporal characteristic of the earthquake, or another property of the earthquake that can be spatially and temporally localized (Lomax et al., 2009). Thus, the hypocentre or absolute location of an earthquake is defined by:

h = (x,y,z,t) (2.1)

where x and y are the coordinates in the horizontal plane (epicentre), z is the hypocentral depth and t is the origin time.

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Given that earthquakes occur deep in the Earth, earthquake location can be considered as an indirect process involving remote observations. Usually, determining a hypocentral location requires both the identification of seismic phases at several seismometers or seismic stations and knowing the velocity structure of the medium, in this case the Earth, between the hypocentre and the observing stations.

Earthquake location is a classical inverse problem in seismology: given the observations (phases travel times) we must find the model (source location) that “fits” them. This is commonly solved by the match (or misfit) of the observed travel times and the prediction of those arrivals for a source location given a known velocity structure. Predicted arrival times are usually calculated from tracing of seismic waves’ ray paths through a medium with known velocity structure, but also from full-waveform inversions and other techniques.

Considering a homogeneous and isotropic medium with seismic wave velocity v, the arrival time at an ith station ti, is given by

!! =  !+ !!!!

!  _{!  !}

!!!!!  !!!!!

!   (2.2)

with ti, xi, yi and zi being our observations and t, x, y and z our model or source location

that must fit them. Consequently, the earthquake location is ordinarily (see section 2.1.1 below) obtained by reducing the misfit between the predicted and the observed arrival times.

2.1.1. Linearized location method

Although earthquake location is a non-linear problem (see below), direct and iterative linearized inversion methods are commonly used to rapidly obtain hypocentral solutions and error estimates. Most of the computer programs used to obtain earthquake locations are based on linearized methods (e.g. HYPOELLIPSE, HYPO71) developed after the technique proposed by Geiger (1912). Geiger’s technique is based on the least-square

Chapter  2  –  Theory  and  Methods

method, in which travel-time residuals are iteratively minimized in order to converge to a minimum hypocentral solution.

A travel-time residual corresponds to the difference between the observed travel-time and a predicted or calculated one. Following Equation 2.2, for an observation at the ith seismic station, with observed arrival time ti, the residual travel-time between the

predicted and observed time is given by

ri = ti - ci (2.3)

where ri is the residual travel-time and ciis the calculated or predicted travel-time.

Non-zero residual times can be due to (i) incorrect reading of the phase arrivals (noisy signals, experience-related analyst errors, etc), (ii) incorrect assumed velocity model for the prediction of the travel time (including attenuation effects, simplistic assumptions, etc) or (iii) an incorrect choice of earthquake source parameters.

Dismissing reading errors and inaccuracies of the velocity model, the only source of time residuals corresponds to incorrect source parameters. Therefore, the travel-time of a phase would be determined exclusively by the hypocentral coordinates (x,y,z,t). We then find the “correct” source parameters that minimize the residual times such as ri = 0.

In order to do this, we firstly define a trial origin time and hypocentre (x0, y0, z0, t0) and

calculate its residual time. The selection of an initial trial hypocentre is made based on, for example, the location of the closest station (smallest travel time), approximate depth range of the seismicity, etc. We then calculate the residual time for a new hypocentral location (x0+ΔX , y0+ΔY, z0+ΔZ, t0+ΔT), with Δ being small perturbations on each one

of the spatio-temporal coordinates (X,Y,Z,T). The residual time for this new location can be calculated from the partial derivatives of the travel-time evaluated at the new hypocentral parameters. This is solved through a Taylor series expansion:

!!    =     !!_! !ℎ_!" ! !!!  !ℎ!"      (2.4)

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where hj are the estimated (not true) hypocentral parameters and Δh represents the variations or perturbations on the hypocentral parameters with respect to the initial trial hypocentre. The partial derivatives of travel time with respect to the hypocentral parameters (!t/!h) express the relative influence of each one of the spatio-temporal coordinates on a given travel time datum (Thurber, 1993). The usage of partial derivatives linearizes the location problem as seen in Equation 2.4.

Geiger’s least-squares method implies that the perturbations on the hypocentral parameters are selected such as the root-mean-square (RMS) of the residual is minimized. We then repeat this process iteratively until the perturbations Δ on the hypocentral parameters (smaller after every iteration) do not reduce the RMS of the residuals anymore, hopefully reaching a minimum solution.

2.1.2. Simultaneous inversion of travel time data for hypocentral and velocity structure parameters – The minimum 1-D velocity model

As seen above, misfits between calculated and observed travel times are produced by timing errors, hypocentre location errors, inaccurate and/or simplistic assumed velocity model or, most likely, a combination of all the above. Accurate location of earthquakes requires the most reliable possible information about the velocity structure of the study region, and the simultaneous solution of the coupled hypocentre velocity model problem (e.g. Kissling, 1988; Kissling et al., 1994).

Though, for simplicity, in Section 2.1.1 the importance of an accurate velocity model was dismissed, the evident reciprocal dependency of hypocentral and velocity model parameters requires the simultaneous and correct determination of them. The determination of the unknown hypocentral parameters and velocity model from a set of arrival times is called the coupled hypocentre velocity model problem (Kissling, 1988). In matrix notation, the coupled hypocentre velocity model parameter relation can be written as (Kissling et al., 1994):

Chapter  2  –  Theory  and  Methods

where

r is the vector of travel time residuals

H is the matrix of partial derivatives of travel time with respect to hypocentral parameters

Δh is the vector with adjustments to hypocentral parameters

M is the matrix of the partial derivatives of travel time with respect to velocity model parameters

Δm is the vector with adjustments to velocity model parameters e is the vector containing travel time errors

The determination of Δm, and therefore, of our 1-D minimum velocity model can be obtained through a least-squares solution for a system of linear equations:

Δm = (MT M)-1 MT r (2.6)

where MT _{is the transposed matrix M.}

Further, following the method introduced by Aki et al. (1977), the least squares solution can be solved using damped parameters, which allow the stabilization of the solution avoiding underestimated parameters:

m = (MT M + L)-1 MT r (2.7)

where L is the diagonal matrix containing the damping parameters.

The damped least squares solution shown in equation (2.7) is implemented in the program VELEST (Kissling et al., 1994; Kissling et al., 1995) in order to find the

minimum1-D velocity model. The concept of minimum 1-D velocity model (Kissling