## Optimizing model parameterization in seismic traveltime tomography## Contents## IntroductionIn seismic traveltime tomography, a set of linearized equations is solved for the unknown slowness perturbations. The matrix of this set of equations (the tomographic matrix) is usually ill-conditioned, because the model space contains more details than can be resolved using the available data. The condition of the tomographic matrix can be characterized by its singular value spectrum: the larger the normalized singular values and the greater the rank of the matrix, the smaller the null space and the better the condition of the inversion problem. The structure of the tomographic matrix depends on the source-receiver configuration and the model parameterization. Since the source-receiver geometry is often fixed (e.g. in earthquake tomography), conditioning can only be influenced through the model parameterization, i.e. the structure of the model space. Most often in traveltime inversion, the Earth is modelled by regular grid of
cells, where the local slowness is assumed to be constant. The element
G is then
the length of the ith ray in the jth cell. Because of the
regular parameterization not taking into account the raypath geometry, cells not
hit by any rays can often be found and it is also often the case that some rows
of G are linearly dependent. This means that G
may be rank-deficient and has a null space. As a consequence, an infinite number
of solutions can be obtained by adding any linear combination of vectors in the
null space to the solution of the tomographic equations.In order to minimize the null space of the tomographic system of equations, I have proposed a method for finding an optimal, irregular triangular cell parameterization that best suits the given raypath geometry (Wéber 2001). ## Optimization methodThe greater the rank and the larger the singular values of e can be calculated quickly using the power
method, while the sum of the eigenvalues is equal to the trace of
_{1}G. Unfortunately, determining the rank ^{T}Gp
would require the whole singular value decomposition (SVD) of G,
which is a computationally expensive task, given the size of the matrix. However,
p can be estimated by the number of model cells in which ray coverage
is greater than a predefined threshold. This threshold should be chosen according
to the average ray coverage. Although high ray coverage is not a sufficient criterion
to eliminate the null space, it is a necessary criterion for a well-determined inverse
problem.Now, let's define the cost function
G) + ^{T}Gq
where The best parameterization that minimizes the above cost function is expected to consist of irregularly sized cells, so I use Delaunay triangular cells to describe the model. Delaunay triangulation is a procedure to generate a unique set of triangles from arbitrarily distributed nodes in two dimensions. Since the parameterization depends on the nodal configuration, I seek the best model by searching for the optimal nodal distribution. ## Numerical experimentsNow I illustrate the proposed procedure by finding the optimal triangular cell
parameterization for a cross-borehole tomographic inversion problem.
The ray coverage threshold used to estimate The cross-borehole raypath geometry with rectangular grid parameterization and
with the optimized triangular cell parameterization is illustrated, respectively,
in the
The distribution of the triangular cells in Fig.
The figure above compares the normalized eigenvalue spectra of
Further numerical experiments and their detailed discussion can be found in Wéber (2001). ## AcknowledgementsThe reported investigation was financially supported by the Hungarian Scientific Research Fund (No. F019277). |