A number of different methods for solving the problem of computing the first arrival travel time of a seismic or electromagnetic wave traveling between a source in one borehole and a receiver in another borehole has been implemented in the m-file 'sippi_forward_traveltime'.
[d,forward,prior,data]=sippi_forward_traveltime(m,forward,prior,data,id,im)
In order to use this m-file to describe the forward problem specify the 'forward_function' field in the forward structure using
forward.forward_function='sippi_forward_traveltime';
In order to use sippi_forward_traveltime, the location of the sources and receivers must be specified in the forward.S and forward.R. The number of columns reflect the number of data, and the number of rows reflect whether data are 2D (2 columns) or 3D (3 columns):
forward.S % [ndata,ndim] forward.R % [ndata,ndim]
Using for example the data from Arrenęs, the forward geometry can be set up using
D=load('AM13_data.mat');
forward.sources=D.S;
forward.receivers=D.R;
In addition the method used to compute the travel times must also be given (see below).
In order to use the geometry from the AM13 reference data, and the Eikonal solution to the wave-equation, the forward structure can be defined using
D=load('AM13_data.mat');
forward.forward_function='sippi_forward_traveltime';
forward.sources=D.S;
forward.receivers=D.R;
forward.type='eikonal';
Ray type models are based on an assumption that the wave propagating between the source and the receiver has infinitely high frequency. Therefore the travel time delay is due to the velocity along a ray connecting the source and receiver.
The linear so-called straight ray approximation, which assumes that the travel time for a wave traveling between a source and a receiver is due to the travel time delay along a straight line connecting the source and receiver, can be chosen using
forward.type='ray'; forward.linear=1;
The corresponding so-called bended-ray approximation, where the travel time delay is due to the travel time delay along the fast ray path connecting a source and a receiver, can be chosen using
forward.type='ray'; forward.linear=0;
When sippi_forward_traveltime has been called once, the associated forward mapping operator is stored in 'forward.G' such the the forward problem can simply be solved by calling e.g. 'd{1}=forward.G*m{1}'
Fat type model assume that the wave propagating between the source and the receiver has finite high frequency. This means that the travel time is sensitive to an area around the raypath, typically defined using the 1st Fresnel zone.
A linear fat ray kernel can be chosen using
forward.type='fat'; forward.linear=1; forward.freq=0.1;
and the corresponding non-linear fat kernel using
bforward.type='fat'; forward.linear=0; forward.freq=0.1;
Note that the center frequency of the propagating wave must also be provided in the 'forward.freq' field. The smaller the frequency, the 'fatter' the ray kernel.
For 'fat' type forward models we rely on the method described by Jensen, J. M., Jacobsen, B. H., and Christensen-Dalsgaard, J. (2000). Sensitivity kernels for time-distance inversion. Solar Physics, 192(1), 231-239
Using the Born approximation, considering only first order scattering, can be chosen using
forward.type='born'; forward.linear=1; forward.freq=0.1;
For a velocity field with small spatial variability one can compute 'born' type kernels (using 'forward.linear=0', but as the spatial variability increases this is not possible.
For the 'born' type forward model we make use if the method described by Buursink, M. L., Johnson, T. C., Routh, P. S., and Knoll, M. D. (2008). Crosshole radar velocity tomography with finite‐frequency Fresnel volume sensitivities. Geophysical Journal International, 172(1), 1-17.
The eikonal solution to the wave-equation is a high frequency approximation, such as the one given above.
However, it is computationally more efficient to solve the eikonal equation directly, that to used the 'forward.type='ray';' type forward model.
To choose the eikonal solver to compute travel times use
forward.type='eikonal';
The Accurate Fast Marching Matlab toolbox : http://www.mathworks.com/matlabcentral/fileexchange/24531-accurate-fast-marching is used to solve the Eikonal equation.
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