Consider a ray tomography experiment, where the goal is to estimate the wave velocity of 9 square cells plotted in each of the cases A and B. The ray paths for source-receiver pairs for both these cases are shown in the figure. Select the correct statement.
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The determinacy of a tomography problem depends on the balance between the number of unknowns (e.g., cell velocities) and the number of independent measurements (e.g., travel times along ray paths), as well as the geometry and coverage of these paths. Insufficient or poorly distributed ray paths can lead to underdetermined or mixed determined systems.
Case A has a unique solution, and B is underdetermined
Both A and B are mixed determined problems
Case A is mixed determined, and B is underdetermined
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The Correct Option isD
Solution and Explanation
Step 1: Understanding the concept of well-determined, underdetermined, and overdetermined problems in ray tomography.
In ray tomography, we aim to determine the velocity distribution within a medium by measuring the travel times of waves along various ray paths passing through it. If we have N unknown parameters (velocities in N cells), we need a sufficient number of independent measurements (travel times along different rays) to solve for these parameters.
• Well-determined: The number of independent measurements equals the number of unknowns. This often leads to a unique solution, provided the ray paths adequately cover the medium.
• Underdetermined: Fewer independent measurements than unknowns. There are infinitely many solutions that satisfy the given data.
• Overdetermined: More independent measurements than unknowns. A unique solution may not exist, but a best-fit (e.g., least squares) solution can be computed.
• Mixed determined: Some parameters are well-constrained while others are poorly constrained, depending on the ray coverage and geometry.
Step 2: Analyzing Case A.
In Case A, there are 9 unknown velocities (one per cell) and 8 ray path measurements. While this is close to the well-determined case numerically, the effectiveness depends on ray path geometry. If some regions (e.g., central cells) are traversed by multiple rays, those values may be better constrained. However, since the number of equations is still less than the number of unknowns and full spatial resolution isn't guaranteed, Case A is classified as a mixed determined problem.
Step 3: Analyzing Case B.
In Case B, there are again 9 unknown velocities but only 6 ray paths—specifically, 3 horizontal and 3 vertical. This number is significantly smaller than the number of unknowns, and the lack of diagonal or oblique rays leads to very limited constraint on cell combinations. Hence, Case B is clearly an underdetermined problem, with infinite possible solutions that satisfy the observed data.
Step 4: Conclusion.
Based on this reasoning:
• Case A: Mixed determined
• Case B: Underdetermined
Final Statement:Case A is mixed determined, and Case B is underdetermined.
