Seismic inversion constrained by stress value changes

Figures & data

Figure 1. Sample image and its coordinate system in frequency domain. (a) 50*50 sample image. (b) Coordinate system of the sample image in frequency domain.

Figure 2. Workflow of PSO algorithm.

Figure 3. Feasibility experiment results. (a) Actual wave velocities and coordinates. (b) Normalized true wave velocity distribution after stress change. (c) Normalized true wave velocity distribution after stress change when $D\text{ = 1}$. (d) Inversion result before stress change when $D\text{ = 1}$. (e) Inversion result after stress change when $D\text{ = 1}$.

Figure 4. Energy ratio and distance of feasibility experiment results. (a) Global best and mean fitness function values. (b) Energy ratio of inversion results to true velocity distributions. (c) Distance between inversion results and true velocity distributions.

Table 1. Meanings of symbols in feasibility experiment.

Symbol	Meaning
rb	Inversion result before stress change
ra	Inversion result after stress change
mb	True model before stress change as shown in Figure 1(a)
ma	True model after stress change as shown in Figure 3(b)
mbD1	Frequency-domain-truncated true model before stress change when $D\text{ = 1}$
maD1	Frequency-domain-truncated true model after stress change when $D\text{ = 1}$ as shown in Figure 3(c)

Figure 6. Distances and variances of these distances. (a) Distance between inversion results and true wave velocity distributions. (b) Variances of the distances.

Table 2. Meanings of symbols in contrast experiment.

Symbol	Calculation method	Meaning
A	$\Vert mbD1+maD1-mb-ma{\Vert }_2$	Distance between frequency-domain-truncated true models and true models, the meanings of $mbD1$, $maD1$, $mb$ and $ma$ are same with the symbols in Table 1
B1,B2,B3	$\Vert r{b}_i+r{a}_i-mb-ma{\Vert }_2$ $i=1,2,3$	Distance between inversion results and true models, the subscript $i$ indicates the experiment number. Meanings of $rb$ and $ra$ are same with the symbols in Table 1
B4	$\frac{{\sum }_{i=1}^3{\Vert r{b}_i+r{a}_i-mb-ma\Vert }_2}{3}$	Mean distance between inversion results and true models
C1,C2,C3	$\Vert r{b}_i+r{a}_i-mbD1-maD1{\Vert }_2$ $i=1,2,3$	Distance between inversion results and frequency-domain-truncated true models
C4	$\frac{{\sum }_{i=1}^3{\Vert r{b}_i+r{a}_i-mbD1-maD1\Vert }_2}{3}$	Mean distance between inversion results and frequency-domain-truncated true models
P1,P2,P3	$\Vert r{b}_i+r{a}_i-mb-ma{\Vert }_2$ $i=4,5,6$	Distance between inversion results and true models
P4	$\frac{{\sum }_{i=4}^6{\Vert r{b}_i+r{a}_i-mb-ma\Vert }_2}{3}$	Mean distance between inversion results and true models
Q1,Q2,Q3	$\Vert r{b}_i+r{a}_i-mbD1-maD1{\Vert }_2$ $i=4,5,6$	Distance between inversion results and frequency-domain-truncated true models
Q4	$\frac{{\sum }_{i=4}^6{\Vert r{b}_i+r{a}_i-mbD1-maD1\Vert }_2}{3}$	Mean distance between inversion results and frequency-domain-truncated true models
Var-B	$\frac{{(B1-B4)}^2+{(B2-B4)}^2+{(B3-B4)}^2}{2}$	Variance of B1, B2 and B3
Var-C	$\frac{{(C1-C4)}^2+{(C2-C4)}^2+{(C3-C4)}^2}{2}$	Variance of C1, C2 and C3
Var-P	$\frac{{(P1-P4)}^2+{(P2-P4)}^2+{(P3-P4)}^2}{2}$	Variance of P1, P2 and P3
Var-Q	$\frac{{(Q1-Q4)}^2+{(Q2-Q4)}^2+{(Q3-Q4)}^2}{2}$	Variance of Q1, Q2 and Q3

Data availability statement

All the computer codes are available at: https://github.com/feiyucq/Seismic-Tomography-with-Stress-Change-Direction-Constraint.git