Earthquake location problem is old, however, it is still quite relevant. The problem can be stated as to determine the hypocenter (x0,y0,z0) and origin time (t0) of the rupture of fault on the basis of arrival time data of P and S waves. Here, we have considered hypocenter in the cartesian coordinate system. It can be readily converted to geographical coordinates using appropriate conversion factor. The origin time is the start time of the fault rupture.
Here, our data are the P and S arrival times at the stations located at the surface (xi,yi,0). We have also assumed a point source and a constant velocity medium for the simplicity sake.
The arrival time of the P or S wave at the station is equal to the sum of the origin time of the earthquake and the travel time of the wave to the station.
The arrival time, ti=Ti + t0, where Ti is the travel time of the particular phase and t0 is the origin time of the earthquake.
Ti = sqrt((xi-x0)^2 + (yi-y0)^2 + z0^2)/v; where v is the P or S wave velocity.
Now, we have to determine the hypocenter location and the origin time of the Earthquake. This problem is invariably ill-posed as the data are contaminated by noise. So, we can pose this problem as an over-determined case. One way to solve this inconsistent problem for the hypocenter and origin time is to minimize the least square error of the observed and predicted arrival times at the stations.
Before that, we need to formulate the problem. As evident from the above equation, the expression for arrival time and eventually the least square error function is nonlinear. This nonlinear inverse problem can be solved using the Newton’s method. We will utilize the derivative of the error function in the vicinity of the trial solution (initializing the problem with the initial wise guess) to devise a better solution. The next generation of the trial solution is obtained by expanding the error function in a Taylor series about the trial solution. This process continues until the error function becomes considerably small. Then we accept that trial solution as the estimated model parameters.
d = G m, where d is the matrix containing the arrival time data, m is the matrix containing the model parameters and G is the data kernel which maps the data to the model.
There are some instances where the matrix can become underdetermined and the least square would fail such as when the data contains only the P wave arrival times. In such cases the solution can become nonunique. We solve this problem as a damped least square case. Though it can also be using the singular value decomposition which allows the easy identification of the underdetermined case and then partitioning the overdetermined and underdetermined in upper and lower part of the kernel matrix and dealing with both part separately.
m = [G’G + epsilon]G’d
The parameter epsilon is chosen by trial and error to yield a solution that has reasonably small prediction error.
In this problem, we deduce the gradient of the travel time by examining the geometry of the ray as it leaves the source. If the earthquake is moved a small distance s parallel to the ray in the direction of the receiver, then the travel time is simply decreased by s/v, where v is the velocity of the phase. If the earthquake is moved a small distance perpendicular to the ray, then the change in travel time is negligible since the new path will have nearly same length as the old one.
T = r/v
If the earthquake is moved parallel to the ray path,
T” = (r-s)/v = r/v – s/v
delta T = T” – T = r/v – s/v – r/v = – s/v
If the earthquake has moved perpendicular to the raypath,
delta T = 0
So, delta T = – s/v, where s is a unit vector tangent to the ray at the source and points toward the receiver.
This is the Geiger’s Method, which is used to formulate the data kernel matrix.
Let us take a hypothetical earthquake location problem and solve it using this method.
clear; close all; global G epsilon; epsilon=1.0e-6; % Velocity parameters vpvs = 1.78; vp=6.5; vs=vp/vpvs; % defining the region of the problem which contains both the source and the receiver %We take a 100x100 units^2 area and the depth of 100 units. The surface is 0 units depth. xmin=-100; xmax=100; ymin=-100; ymax=100; zmin=-100; zmax=0; % stations: x, y, z coordinates (xi, yi, 0) sxm = [-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]'*ones(1,9)*10; sym = 10*ones(9,1)*[-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]; sx = sxm(:); sy = sym(:); Ns = length(sx); % num of stations sz = zmax*ones(Ns,1); %zeros % true earthquakes locations Ne = 20; % number of earthquakes M = 4*Ne; % 4 model parameters, x, y, z, and t0, per earthquake extrue = random('uniform',xmin,xmax,Ne,1); %x0 eytrue = random('uniform',ymin,ymax,Ne,1); %y0 eztrue = random('uniform',zmin,zmax,Ne,1); %z0 t0true = random('uniform',0,0.2,Ne,1); %t0 mtrue = [extrue', eytrue', eztrue', t0true']'; %column matrix containing all the true model parameter values Nr = Ne*Ns; % number of rays, that is, earthquake-stations pairs N = 2*Ne*Ns; % total data; Ne*Ns for P and S each % Generating the true data based on the true earthquake and station locations dtrue=zeros(N,1); %allocating space for all data for i = [1:Ns] % loop over stations for j = [1:Ne] % loop over earthquakes dx = mtrue(j)-sx(i); % x component of the displacement obtained by the difference between station and source location x component dy = mtrue(Ne+j)-sy(i); % y component of the source- receiver displacement dz = mtrue(2*Ne+j)-sz(i); % z component of displacement between source and receiver r = sqrt( dx^2 + dy^2 + dz^2 ); % source-receiver distance k=(i-1)*Ne+j; dtrue(k)=r/vp+mtrue(3*Ne+j); %P arrival time for each station-source pair obtained by summing the travel time with the origin time of the earthquake dtrue(Nr+k)=r/vs+mtrue(3*Ne+j); % S arrival time end end % Generating observed data by adding gaussian noise with standard deviation 0.2 sd = 0.2; dobs=dtrue+random('normal',0,sd,N,1); % observed data %% Determining the predicted arrival time using the Geiger's Method % inital guess of earthquake locations mest = [random('uniform',xmin,xmax,1,Ne), random('uniform',ymin,ymax,1,Ne), ... random('uniform',zmin+2,zmax-2,1,Ne), random('uniform',-0.1,0.1,1,Ne) ]'; % Here, we take a random initial guess %Formulating the data kernel matrix and estimating the predicted models for iter=[1:10] %for 10 iterations (termination criteria) % formulating data kernel G = spalloc(N,M,4*N); %N- total num of data,2*Ne*Ns; M is total num of model %parameters, 4*Ne dpre = zeros(N,1); %allocating space for predicted data matrix for i = 1:Ns % loop over stations for j = 1:Ne % loop over earthquakes dx = mest(j)-sx(i); % x- component of displacement obtained using the initial guess dy = mest(Ne+j)-sy(i); % y- component of displacement dz = mest(2*Ne+j)-sz(i); % z- component of displacement r = sqrt( dx^2 + dy^2 + dz^2 ); % source-receiver distance for each iteration k=(i-1)*Ne+j; %index for each ray dpre(k)=r/vp+mest(3*Ne+j); %predicted P wave arrival time dpre(Nr+k)=r/vs+mest(3*Ne+j); %predicted S wave arrival time %First half of data kernel matrix correspoding to P wave G(k,j) = dx/(r*vp); % first column of data kernel matrix G(k,Ne+j) = dy/(r*vp); % second column of data kernel matrix G(k,2*Ne+j) = dz/(r*vp); % third column of data kernel matrix G(k,3*Ne+j) = 1; % fourth column of data kernel matrix % Second half of the data kernel matrix corresponding to S wave G(Nr+k,j) = dx/(r*vs); G(Nr+k,Ne+j) = dy/(r*vs); G(Nr+k,2*Ne+j) = dz/(r*vs); G(Nr+k,3*Ne+j) = 1; end end % solve with dampled least squares dd = dobs-dpre; dm=bicg(@dlsfun,G'*dd,1e-5,3*M); solving using the biconjugate method %solving the damped least square equation G'dd = [ G'G + epsilon* I] dm % We use biconjugate method to reduce the computational cost (see for the dlsfun at the bottom) mest = mest+dm; %updated model parameter end % Generating the final predicted data dpre=zeros(N,1); for i = 1:Ns % loop over stations for j = 1:Ne % loop over earthquakes dx = mest(j)-sx(i); dy = mest(Ne+j)-sy(i); dz = mest(2*Ne+j)-sz(i); r = sqrt( dx^2 + dy^2 + dz^2 ); k=(i-1)*Ne+j; dpre(k)=r/vp+mest(3*Ne+j); % S-wave arrival time dpre(Nr+k)=r/vs+mest(3*Ne+j); % P- wave arriavl time end end % Calculating the data and model misfit expre = mest(1:Ne); % x0 eypre = mest(Ne+1:2*Ne); %y0 ezpre = mest(2*Ne+1:3*Ne); %z0 t0pre = mest(3*Ne+1:4*Ne); %t0 dd = dobs-dpre; %residual of observed and predicted arrival time E = dd'*dd; %error fprintf('RMS traveltime error: %f\n', sqrt(E/N) ); Emx = (extrue-expre)'*(extrue-expre); %misfit for x0 Emy = (eytrue-eypre)'*(eytrue-eypre); %misfit for y0 Emz = (eztrue-ezpre)'*(eztrue-ezpre); %misfit for z0 Emt = (t0true-t0pre)'*(t0true-t0pre); %misfit for t0 fprintf('RMS model misfit: x %f y %f z %f t0 %f\n', sqrt(Emx/Ne), sqrt(Emy/Ne), sqrt(Emz/Ne), sqrt(Emt/Ne) );
function y = dlsfun(v,transp_flag) global G epsilon; temp = G*v; y = epsilon * v + G'*temp; return