Digging Deeper—2D Gravity Surveying ∗

Figure 7.14. Selected right singular vectors plotted versus the depth t. Each singular vector carries information about the solution in a certain depth interval only, starting from the smallest depths.

discussed in Section 7.6. The next section gives another example of depth resolution in geophysics.

The key to analyzing and understanding depth resolution often lies in a study of the right singular vectors v_i that provide the basis for the regularized solution. Some of these singular vectors are shown in Figure 7.14 for the depth profiling problem; for clarity we plot the elements of the singular vectors versus the abscissas tj = sin(τj) to better illustrate the interplay between the singular vectors and the depth. Clearly, each singular vector carries information about the solution in a certain depth interval only, starting from the smallest depths for small indices i . When we include the first k SVD components in a regularized solution, then we can recover details in those depth ranges for which the singular vectors v1, . . . , vk carry information. The combined inspection of the right singular vectors and the Picard plot (which reveals the SVD components that can be recovered from noisy data) is a convenient way to obtain the desired insight about depth resolution.

In Section 8.3 we return to this basis and demonstrate that there is actually a better basis for solving this problem than the SVD basis that underlies the analysis here.

7.9 Digging Deeper—2D Gravity Surveying

Throughout this book we have used the gravity surveying problem from Section 2.1 as a simple test problem. Real-world gravity surveying problems are, of course, much more complex than this simple one-dimensional problem; in particular, one needs a 3D reconstruction of the mass density distribution below the ground. Such large-scale problems are, however, beyond the scope of this book. As a compromise, in this section we study the inverse potential field problem associated with a 2D gravity surveying problem, which still gives insight into more realistic problems as well as providing another example of the concept of depth resolution.

To set the stage, let f (t, z ) denote the mass density distribution as a function of the horizontal and vertical coordinates t and z . We consider the domain Ω = [0, 1]× [0, d], which represents a vertical “slice” below the ground, from the surface

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158 Chapter 7. Regularization Methods at Work: Solving Real Problems at z = 0 to the depth z = d . We ignore all mass outside⁷ this slice. Following the derivation from Section 2.1 it is easy to see that the vertical component of the gravity field due to f (t, z ) at the surface is given by

g(s) = 1

0

d 0

z

(z²+ (s− t)²)^3/2d z d x , 0≤ s ≤ 1. (7.13) Typically the depth d of the domain Ω is much smaller than the width of the domain;

here we use d = 0.1.

We discretize the domain Ω into a grid of Nt× Nz pixels, and in the center of each pixel we conceptually place a point source whose magnitude we wish to find from samples of the measured field g(s) at m equidistant points on the surface. This corre-sponds to a very simple Galerkin discretization with delta functions as basis functions, leading to a system with n = N_t·Nz unknowns (the source magnitudes) organized into a vector x ∈ Rⁿ(similar to the organization of the unknowns in the image deblurring and tomography problems above). The coefficient matrix A ∈ R^m×n can easily be constructed from submatrices obtained from calls to the function gravity in Regu-larization Tools. We also construct an exact solution which consists of two smoothly varying mass concentrations located at (t, z ) = (¹₃,¹₃d ) and (t, z ) = (¹¹₁₆,²₃d ); see Figure 7.16 for an example with Nt = 40 and Nz = 20.

Figure 7.15 shows the Picard plots for the exact and the noise-corrupted right-hand side, using m = n = 800. We use the noise level η = 10⁻⁴, which is clearly visible as the plateaux where the right-hand side coefficients |ui^Tb| level off for the noisy problem. The solution coefficients do not (overall) increase, showing that the discrete Picard condition is satisfied. From these Picard plots we expect that we can recover about 400 SVD components for the noise-free data (the remaining coefficients are contaminated with rounding errors) and about 240 SVD components for the noisy data. We also see that it makes sense to use TSVD truncation parameters k that are multiples of N_t = 40, the number of discretization points in the t direction.

Figures 7.16 and 7.17 show TSVD solutions for the noise-free and noisy prob-lems, respectively. Both figures demonstrate that as more SVD components are in-cluded in the regularized solutions, we obtain better reconstructions of the larger source in the left part of the domain Ω. For small values of k the reconstructed source is clearly too shallow, and the more SVD components we include the more correct the localization of this source; about 400 components are needed to compute a reasonably good reconstruction. We also see that the smaller and deeper source in the right part of the domain cannot be reconstructed well—even in the noise-free problem—due to the influence of the rounding errors. For the noisy data, the maxi-mum truncation parameter before the noise sets in is k = 280, and the corresponding TSVD solution is not able to produce the correct sources: the bottom part of the larger source is not reconstructed correctly, and the smaller and deeper source is both too shallow and too weak.

When too few SVD components are included in the reconstruction, then the source is artificially placed too close to the surface. This resembles the behavior

7An alternative model, sometimes referred to as “2¹₂-dimensional,” assumes that the function f (t, z ) extends infinitely in the third dimension; this model has a different kernel than the one con-sidered here.

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7.9. Digging Deeper—2D Gravity Surveying^∗ 159

0 50 100 150 200 250 300 350 400

10⁻¹⁰ 10⁻⁵ 10⁰

σ_i

| u_i^Tb |

| u_i^Tb | / σ_i

0 50 100 150 200 250 300 350 400

10⁻¹⁰ 10⁻⁵ 10⁰

σ_i

| u_i^Tb |

| u_i^Tb | / σ_i

Figure 7.15. Picard plots for the noise-free problem (top) and the problem with noisy data (below). The singular values beyond i = 410 level off at the plateaux around 3· 10⁻¹³.

of the formulation ambiguity mentioned in Section 2.4; but it is an artifact of the discretization and the oversmoothed regularization.

Clearly, the depth resolution depends on the number of SVD components in-cluded in the regularized solution. To see how this is reflected in the right singular vectors vi, it is convenient to lump information about Nt = 40 singular vectors at a time by computing the vectors w1, w2, . . . whose elements are the RMS values of the elements of 40 consecutive singular vectors. For example, the first two vectors have elements given in terms of v₁, . . . , v₄₀ and v₄₁, . . . , v₈₀by

(w1)i=

⎛

⎝^Nt

j =1

v_{i j}²

⎞

⎠

1 2

, (w2)i=

⎛

⎝ ^2Nt

j =N_t+1

v_{i j}²

⎞

⎠

1 2

, i = 1, . . . , n.

Figure 7.18 shows the vectors wl in the form of Nz × Nt images similar to the SVD solutions, and we see that the large elements in each vector wl are confined to a few layers in the solution around the l th layer. This implies that each sequence of 40 right singular vectors mainly carries information about the solution at the l th layer (this resembles the localization of the singular vectors in Figure 7.14 for the depth profiling problem). The singular vectors vi beyond i = 400 are heavily influenced by rounding errors, and so are the corresponding vectors w_l.

In conclusion, we see again how a combined inspection of the Picard plot and the right singular vectors—this time in the form of the vectors w_l in Figure 7.18—

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160 Chapter 7. Regularization Methods at Work: Solving Real Problems

Figure 7.16. TSVD solutions for the noise-free 2D gravity surveying problem, and the exact solution. For k = 440 the solution is heavily influenced by rounding errors.

provides valuable insight about the depth resolution available in a given noisy problem.

Extensions of these ideas to 3D potential field problems are introduced and discussed in [15].