Sensor Distribution Design of Travel Time Tomography in Explosion

2.1. Effect of Eigenvalue and Rank on Inversion

In Equation (3), with a given data vector t₀, the model vector s₀ ∈S and then |t₀ − Ds₀|² is minimized. This is accomplished by pre-multiplying Equation (3) by D^T and taking a matrix inverse:

As the N ×N square matrix L=D^TD is often near-singular, it results in instability in the solution. That is, some of its eigenvectors {e_i: i=1,⋯,N} have extremely small eigenvalues {λ_i:i=1,⋯,N}. Measurement errors in data space T propagate into the solution S in parallel to each eigenvector with an amplification 1/λ_i. Hence, when small eigenvalues exist, the solution becomes unstable and the inverse problem is ill-posed [10]. When an eigenvalue is zero, the corresponding eigenvector in the data space can not be mapped into the model space [11]. The greater the rank is and the larger the eigenvalues are, the more stable the inversion problem is and the more independent pieces of information may be gained from the data. Hence, optimal sensor distribution can be obtained by maximizing the magnitude of eigenvalues of L and the rank of D. The evaluating function is:

The evaluating function E₁ has two components, one represents the relative distribution of eigenvalues and the other indicates the relative size of null space. If the value of E₁ is minimal, the inversion results are optimal and stable.

2.2. Effect of Condition Number on Inversion

The Equation (3) is ill-posed if small perturbations of matrix D or T can result in a large change in solutions [12].

Assuming that the observed data T has a minor perturbation of δT, the perturbation of solution is δS. Equation (3) becomes:

Then,

According to the property of subordinate norm, ‖δS‖=‖D⁻¹δT‖≤ ‖D⁻¹‖‖δT‖ and $\Vert S\Vert \ge \frac{\Vert DS\Vert }{\Vert D\Vert }=\frac{\Vert T\Vert }{\Vert D\Vert }$. It concludes that $\frac{\Vert \delta S\Vert }{\Vert S\Vert }\le \frac{\Vert D^{-1}\Vert \Vert \delta T\Vert }{\Vert T\Vert /\Vert D\Vert }$. That is $\frac{\Vert \delta S\Vert }{\Vert S\Vert }\le \mathit{\text{cond}}\left(D\right)\frac{\Vert \delta T\Vert }{\Vert T\Vert }.$

Then,

Supposing that T is accurate and the matrix D has a minor perturbation of δ_D, the corresponding perturbation of solution is δS. Supposing that S _c = S + δS is the solution of perturbation equation,

Similarly the following condition can be obtained:

When the matrix D has a minor perturbation, the error of solution is also determined by the condition number.

Therefore, the condition number is the second judging index of matrix D's quality. The smaller condition number means the more well-posed equation.

The evaluating function expressed by the condition number can be written as:

Optimal sensor distribution can reduce the condition number and the more stable inversion solutions can be obtained.

2.3. Effect of Ray Coverage on Inversion

When designing sensor location, rays coverage should be enlarged as much as possible, i.e., rays density and orthogonality should be maximized. The ray density represents the number of rays passing through each cell. The cells not being hit by any ray are the main factors giving rise to the null space. They make some column vectors of matrix D be zero (d_j = 0) so that the equation can not be resolved. Therefore, tomography with sparse rays must ensure that any column vector is non-zero. Increasing the ray density can avoid zero vectors.

Ray orthogonality is measured by maximal sinusoidal quantity of angle between rays [13]. The small orthogonality makes some rows in D linearly dependent.

The greater the ray density is, the better the orthogonality is and the smaller inversion error can be achieved. The evaluating function expressed by rays density and orthogonality can be written as:

2.4. Evaluation Method

As sensor distribution can be optimized by minimizing E₁ and E₂ and maximizing E₃, an evaluating function is defined as:

There are many factors on the ill-posed and underdetermined equations except for the above analysis. However, the value of _E can be used as a judging index with regard to optimizing sensor distribution and the experiment. The detailed process is as follows and a flow chart is illustrated in Figure 2.

(1)

Dividing cells according to model character.

(2)

Giving a distribution model randomly according to the number of sensors and calculating matrix D and the rank of D.

(3)

If all column vectors of matrix D are non-zero and D is full rank, make this distribution model as an initial model; otherwise, go to Step (2).

(4)

When the number of initial model is equal to the given number K, the optimization algorithm is employed to obtain the optimum value of E and sensor distribution.

3.1. Sub-Region and Multi-Scale Cells Partition

The test region is divided into cells and each cell has the same velocity. The dividing pattern of cells affects matrix D. We propose a sub-region and multi-scale cell partition method. Cells are divided in terms of the solution distribution, i.e., the smaller size and the higher density of cells correspond to the bigger changing gradient of solutions, and the bigger size and the lower density correspond to the smaller changing of solutions.

When the explosive is exploding in air, the shock wave overpressure attenuates quickly in the near region to the explosive and the attenuation becomes slow with the increase in distance. According to the shock wave characteristic, the smaller cells are adopted in the near region to the explosive while the bigger cells are adopted in the far region. In order to avoid a row vector to be linearity dependent on matrix D aroused by symmetry, cells are divided into different size in the symmetrical region with consideration of the resolution of inversion.

3.2. Optimizing Sensor Distribution Based on Adaptive Escaping Particle Swarm Optimization Algorithm

4.1. Comparison of Cell Performance

To verify the sensor distribution design, a test region is 16 m × 16 m, explosive is placed in the lower-left and sensors are distributed on the region boundary. Two cell patterns are used in dividing the test region. The first one is a uniform rectangle with 49 cells as shown in Figure 4a. The second one is sub-region and multi-scale cells as shown in Figure 4b. The velocity of shock wave decreases exponentially with the distance according to the prior information. In the near region to the explosive, the attenuation amplitude of velocity is larger and the cells are smaller and denser. In the far region the attenuation amplitude of velocity is smaller and the cells are bigger and sparser. The region is divided into seven sub-regions according to the proportional distance and different size cells in the symmetry region are placed in order to avoid a row vector to be linearity dependent on matrix D. The total cells number is 58.

The influence of cells on matrix D is compared with the same sensor distribution.

(1)

To ensure each cell being passed through by at least one ray, the lowest number of sensor required by uniform rectangle cells is 9 as shown in Figure 4a. The multi-scale cells require at least five sensors as shown in Figure 4b.

(2)

Each index of matrix D in uniform rectangle cells and multi-scale cells is compared with 13 sensors and the same distribution. Simulation results are given in Table 1. It is clear that the indexes in multi-scale cells are superior to those of the uniform rectangle cells. The matrix D in multi-scale cells is full rank mostly and the condition number is far smaller than that of uniform rectangle cells. The ray density and orthogonality in multi-scale cells are larger than that of uniform rectangle cells generally.

4.2. Optimizing Sensor Distribution Based on Multi-Scale Cells