SINGULAR VALUE DECOMPOSITION: APPLICATION

SVD (Dongarra, et al., 1979) was performed on the matrix $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$$after parametrizing the model as described by equations (3) , (4) and (5). Pratt and Chapman (1990) suggest a clear way of representing the resultant singular values and singular vectors in model space (${\bf V}$) of the matrices involved in tomographic problems. I will represent my results in a similar fashion including also in the representation the corresponding singular vectors in data space (${\bf U}$). This will help to understand how the data and the model are resolved by using iterative techniques such as conjugate gradients.

The data space (spanned by the columns of $$\mathop{\mbox{\bf U}}_{\mbox{$\sim$}}$$) is shown in Figure . It is a 2-D space of traveltimes where the horizontal axis contains the receiver depths and the vertical axis contains the source depths. If the origin of both axes is the same, the closer a given point is to the main diagonal (near offset), the closer is to zero the corresponding ray angle.

 

data-space
Figure 1 Space used to represent the data: for each source-receiver position the corresponding traveltime is plotted. This space is referred as ${\bf U}$ in subsequent figures.

The geometry used to do the SVD for the different parametrizations is shown in Figure  (Pratt and Chapman, 1990): five sources and five receivers in a constant slowness medium. When the model is isotropic, the matrix $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$$ only depends on the ray geometry and when the model is anisotropic, $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$$ depends on both the ray geometry and the slowness model (constant in this case).

 

SVD-geometry
Figure 2 Recording geometry used to do the SVD for the different parametrizations. The slowness is constant.

Figure  shows the SVD when the model is discretized using 6 horizontal isotropic layers (equation 3). Notice that the differences among the singular values are small. The largest singular values correspond to singular vectors in data and model space whose components are roughly of the same magnitude. We observe that with this parametrization only some ``big structures'' (averages) in data space can be explained whereas in model space all the parameters can be resolved well. By representing the data space as described by Figure 1, it is possible to identify which traveltimes belong to the null space and therefore, cannot be resolved. Errors in these particular traveltimes (noise) will also have little or no effect in the solution.

 

SVD-iso-1d
Figure 3 Singular value decomposition when the model is described as a superposition of 6 horizontal isotropic layers (1-D isotropic). ${\bf U}$ represents the singular vectors in data space and ${\bf V}$ represents the singular vectors in model space. The origin in data space in the upper left corner. The axes in model space are depth (perpendicular to the stratification) and horizontal distance. The gray scale goes from black (negative) to white (positive).

When lateral variations are allowed in the previous parameterization results a matrix $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$$ whose SVD is shown in Figure . The largest singular value corresponds roughly to horizontal layers (in model space) and non-horizontal rays (in data space). As the singular values decrease, the eigenvectors in model space tend to contain more horizontal variations and the eigenvectors in data space tend to expand near and far offsets alternatively (diagonal and non-diagonal structures). The null space of the problem corresponds to ``pure'' horizontal variations in model space and high frequency variations data space (rapid changes among nearby traveltimes). These vectors in the null space of the data are probably not a problem in real applications because if the real velocities do not change rapidly, high frequency variations are unlikely to be found in the measured traveltimes. The presence of noise introduces high frequency variations in the data. Unfortunately, these type of variations are not confined to the null space of the data but are present in singular vectors corresponding to larger singular values. Therefore, in some applications it might be necessary to damp the effect of singular values larger than those contained in the null space in order to attenuate the effect of certain components of the noise.

 

SVD-iso-2d
Figure 4 Singular value decomposition when the model is described as a superposition of 6x4 homogeneous isotropic squared regions (2-D isotropic). ${\bf U}$ represents the singular vectors in data space and ${\bf V}$ represents the singular vectors in model space. The origin in data space is in the upper left corner. The gray scale goes from black (negative) to white (positive).

Figure  shows the SVD of the matrix that result after discretizing the model and computing the data using natural pixels (equation 4). We notice immediately that the matrix for the natural pixels is much better conditioned than the one obtained with square pixels (Figure ). The singular vectors in data space that correspond to the largest singular value are almost identical for both discretizations. However, the ways those data are distributed in the model are different for the different discretizations. For all the eigenvectors in Figure  there is a clear correspondency between the structures represented in data and model space, unlike the eigenvectors in Figure . With the natural pixels all the parameters and data can be resolved because the discretization does not introduce another null space in the problem (different to the null space of the measurements), whereas the discretization of the model in square pixels does.

 

SVD-fatrays
Figure 5 Singular value decomposition when the model is 2-D isotropic and is described as a superposition of 25 natural pixels. ${\bf U}$ represents the singular vectors in data space and ${\bf V}$ represents the sum of the components of each singular vector in model space along the corresponding natural pixels. The origin in data space is in the upper left corner. The gray scale goes from black (negative) to white (positive).

The SVD for a 1-D anisotropic parametrization (equation 5) is shown in Figure . The upper half of each eigenvector in model space corresponds to S_x and the lower half corresponds to S_z. We observe that the largest singular values correspond to S_x in model space and in data space the behavior is similar to the isotropic 1-D case. The null space of the model is spanned by vectors that only contain information about S_z.

 

SVD-ani-1d
Figure 6 Singular value decomposition when the model is horizontally layered and anisotropic (6x2 model parameters). ${\bf U}$ represents the singular vectors in data space and ${\bf V}$ represents the singular vectors in model space. The upper half of each eigenvector in model space corresponds to S_x and the lower half corresponds to S_z. The origin in data space is in the upper left corner. The gray scale goes from black (negative) to white (positive).

 

SVD-ani-2d
Figure 7 Singular value decomposition when the model is 2-D anisotropic. (24x2 model parameters). ${\bf U}$ represents the singular vectors in data space and ${\bf V}$ represents the singular vectors in model space. The upper half of each eigenvector in model space corresponds to S_x and the lower half corresponds to S_z. The origin in data space is in the upper left corner. The gray scale goes from black (negative) to white (positive).

 

model-null
Figure 8 Vectors that span the null space of the model for the SVD shown in Figure 7. Most vectors are contained in the S_z space and therefore, S_z cannot be estimated at the same resolution of S_x from cross-well traveltimes alone.

Figure  shows the SVD when the model is 2-D anisotropic. The result is nearly a combination of the previous ones in model space, namely, vertical variations in S_x correspond to the largest singular values and horizontal variations in S_z to the smallest ones. The null space of the model (Figure ) is formed almost entirely by ``unit'' vectors that describe S_z. For this reason, in the synthetic and real data applications shown in Michelena and Muir (1991), the resolution of the vertical component of the slowness is poor when compared with the resolution of the horizontal component. As expected, S_z is poorly defined by the cross-well recording geometry. Note that by increasing the number of model parameters the condition number of the matrix diminishes when compared with the one of the matrix for the isotropic case (Figure 4).