Seismic Inverse Problem Drawbacks of the . . . It Is Necessary to Take . . . How We Can Use . . . Explicit Expert . . . Using Expert Knowledge in How We Can Use . . . Implicit Expert . . . How We Can Use . . . Solving the Seismic Inverse Implicit Expert . . . A General Problem Problem General Problem: . . . Result: . . . Reduction Matthew G. Averill, Kate C. Miller, G. Randy Keller, Proof Vladik Kreinovich, Roberto Araiza, and Scott A. Starks Acknowledgments Pan-American Center for Earth and Environmental Studies Title Page University of Texas at El Paso, El Paso, TX 79968, USA averill@geo.utep.edu, miller@geo.utep.edu, keller@utep.edu, ◭◭ ◮◮ vladik@utep.edu, raraiza@cs.utep.edu, sstarks@utep.edu ◭ ◮ Page 1 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 1. Seismic Inverse Problem It Is Necessary to Take . . . How We Can Use . . . • Problem: to determine the geophysical structure of a region. Explicit Expert . . . • Solution: we: How We Can Use . . . Implicit Expert . . . – measure seismic travel times, and How We Can Use . . . – reconstruct velocities at different depths from this data. Implicit Expert . . . A General Problem • Difficulty: the inverse problem is ill-defined: General Problem: . . . – large changes in the original distribution of velocities can lead to Result: . . . Reduction – very small changes in the resulting measured values. Proof • Conclusion: many different velocity distributions are consistent with the same Acknowledgments measurement results. Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 2. Drawbacks of the Existing Approach It Is Necessary to Take . . . How We Can Use . . . • Situation: because of the non-uniqueness, the velocity distribution that is Explicit Expert . . . returned by the existing algorithm is usually not geophysically meaningful. How We Can Use . . . • Example: it predicts velocities outside of the range of reasonable velocities at Implicit Expert . . . this depth. How We Can Use . . . Implicit Expert . . . • Current solution: a geophysicist adjusts the initial approximation so as to A General Problem avoid this discrepancy. General Problem: . . . • Problem: several iterations are needed; it is very time-consuming. Result: . . . Reduction • Problem: adjustment requires special difficult-to-learn skills. Proof • Result: the existing tools for solving the seismic inverse problem are not as Acknowledgments widely used as they could be. Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 3. It Is Necessary to Take Expert Knowledge Into It Is Necessary to Take . . . Consideration How We Can Use . . . Explicit Expert . . . • Objective: make the tools for processing seismic data more accessible. How We Can Use . . . Implicit Expert . . . • Solution: incorporate the expert knowledge into the algorithm for solving the How We Can Use . . . inverse problem. Implicit Expert . . . • Example why expert knowledge is needed: velocity is outside the interval of A General Problem values which are possible at this depth for this particular geological region. General Problem: . . . Result: . . . • Corresponding expert knowledge: the intervals of possible values of data. Reduction • What needs to be done: modify the inverse algorithms in such a way that the Proof velocities are always within these intervals. Acknowledgments Title Page • Question: how can we do it? ◭◭ ◮◮ ◭ ◮ Page 4 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 4. How We Can Use Interval Uncertainty It Is Necessary to Take . . . How We Can Use . . . • How algorithms work now: Explicit Expert . . . – start with a reasonable velocity model; How We Can Use . . . Implicit Expert . . . – predict traveltimes x i between stations; How We Can Use . . . def – use the difference ∆ x i = � x i − x i , where � x i are measured values, to Implicit Expert . . . adjust the velocity model: A General Problem ∗ divide ∆ x i by the length L of the path; General Problem: . . . ∗ add ∆ x i /L to all slownesses along the path. Result: . . . Reduction • How to modify when we know the interval [ s j , s j ] of possible slownesses: Proof – first, we compute the next approximation s ( k ) Acknowledgments to the slownesses, j Title Page – then, we replace s ( k ) with the nearest value within the interval [ s j , s j ]. j ◭◭ ◮◮ ◭ ◮ Page 5 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 5. Explicit Expert Knowledge: Fuzzy Uncertainty It Is Necessary to Take . . . How We Can Use . . . • Experts can usually produce an wider interval of which they are practically Explicit Expert . . . 100% certain. How We Can Use . . . • In addition, experts can also produce narrower intervals about which their Implicit Expert . . . degree of certainty is smaller. How We Can Use . . . Implicit Expert . . . • As a result, instead of a single interval, we have a nested family of intervals A General Problem corresponding to different levels of uncertainty. General Problem: . . . • In effect, we get a fuzzy interval (of which different intervals are α -cuts). Result: . . . Reduction • Previously: a solution is satisfying or not. Proof • New idea: a satisfaction degree d . Acknowledgments Title Page • Specifics: d is the largest α for which all s i are within the corresponding α -cut intervals. ◭◭ ◮◮ ◭ ◮ Page 6 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 6. How We Can Use Fuzzy Uncertainty It Is Necessary to Take . . . How We Can Use . . . • Objective: find the largest possible value α for which the slownesses belong Explicit Expert . . . to the α -cut intervals. How We Can Use . . . • Possible approach: Implicit Expert . . . How We Can Use . . . – try α = 0, α = 0 . 1, α = 0 . 2, etc., until the process stops converging; Implicit Expert . . . – the solution corresponding to the previous value α is the answer. A General Problem General Problem: . . . • Comment: Result: . . . – this is the basic straightforward way to take fuzzy-valued expert knowl- Reduction edge into consideration; Proof – several researchers successfully used fuzzy expert knowledge in geo- Acknowledgments physics (Nikravesh, Klir, et al.); Title Page – we plan to add their ideas to our algorithms. ◭◭ ◮◮ ◭ ◮ Page 7 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 7. Implicit Expert Knowledge: Interval Uncertainty It Is Necessary to Take . . . How We Can Use . . . • Situation: sometimes, velocities are in the interval, but the geophysical struc- Explicit Expert . . . ture is still not right. How We Can Use . . . • Explanation: Implicit Expert . . . How We Can Use . . . – algorithms assume that the measured errors are independent and nor- Implicit Expert . . . mally distributed; A General Problem � N def x i ) 2 ; General Problem: . . . – so, stopping criterion is MSE E = ( x i − � i =1 Result: . . . – for geophysically meaningless models, E is small, but some differences Reduction x i − � x i are large. Proof Acknowledgments • Solution: require that | x i − � x i | ≤ ∆ for some bound ∆. Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 8. How We Can Use Interval Uncertainty It Is Necessary to Take . . . How We Can Use . . . • Problem: how can we guarantee that we only get solutions which are physical Explicit Expert . . . in the above sense? How We Can Use . . . • Traditional approach: once the mean square error is small, we stop iterations. Implicit Expert . . . How We Can Use . . . • Natural new idea: continue iterations until all the differences | x i − � x i | are Implicit Expert . . . under ∆. A General Problem • Question: what if this does not happen? General Problem: . . . Result: . . . • Similar question: what traditional algorithms do if we do not MSE small? Reduction • Answer to similar question: restart computations with a different starting Proof velocity model. Acknowledgments Title Page • Solution to our problem: restart computations with a different starting ve- locity model. ◭◭ ◮◮ ◭ ◮ Page 9 of 16 Go Back Full Screen Close Quit
Seismic Inverse Problem Drawbacks of the . . . 9. Implicit Expert Knowledge: Fuzzy Uncertainty It Is Necessary to Take . . . How We Can Use . . . • Experts cannot always provide us with exact upper bounds ∆. Explicit Expert . . . • Instead, they give different bounds with different degrees of certainty – i.e., How We Can Use . . . a fuzzy number. Implicit Expert . . . How We Can Use . . . • Natural idea: find the largest α for which all all the differences x i − � x i fit into Implicit Expert . . . the corresponding α -cut intervals. A General Problem • Algorithm: try α = 0, α = 0 . 1, etc. General Problem: . . . Result: . . . • Remaining problem: detect velocity models that are not geophysically rea- Reduction sonable. Proof • Solution: data fusion, i.e., take into consideration other geophysical and ge- Acknowledgments ological data (gravity map, geological map, etc.). Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 16 Go Back Full Screen Close Quit
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