Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Towards Combining Interval Algorithm for the . . . and Probabilistic Successes, . . . Case of Interval Prior . . . Uncertainty in Finite A New Linear-Time . . . A New Linear-Time . . . Element Methods Case of Probabilistic . . . Vladik Kreinovich and Roberto Araiza Combination of . . . Department of Computer Science Combination of . . . Effects of . . . Matthew G. Averill Department of Geological Sciences Effects of . . . Acknowledgments University of Texas at El Paso Title Page El Paso, TX 79968, USA emails vladik@utep.edu ◭◭ ◮◮ raraiza@utep.edu, averill@geo.utep.edu ◭ ◮ Page 1 of 23
Data that We Can Use . . . Seismic Inverse . . . 1. Uncertainty in FEM: Probabilistic Approach Seismic Inverse . . . • Traditional approach to FEM: we know the exact equa- Existing Algorithm for . . . tions and the exact values of the parameters x i of these Algorithm for the . . . equations. Successes, . . . • In practice: we only know the approximate values of Case of Interval Prior . . . the corresponding parameters. A New Linear-Time . . . A New Linear-Time . . . • Question: estimate how the uncertainty in the param- Case of Probabilistic . . . eters of the system can affect the result y of applying Combination of . . . the FEM techniques. Combination of . . . • Probabilistic approach – assumption: we know the ex- Effects of . . . act probability distributions of all x i . Effects of . . . • Probabilistic approach – algorithm: e.g., use Monte- Acknowledgments Carlo simulations. Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 23
Data that We Can Use . . . Seismic Inverse . . . 2. Uncertainty in FEM: Interval Approach and Re- maining Problem Seismic Inverse . . . Existing Algorithm for . . . • Interval approach – assumption: sometimes, we only Algorithm for the . . . know the lower bounds x i and the upper bounds x i on Successes, . . . x i . Case of Interval Prior . . . • Example: the probabilities of different values of Young A New Linear-Time . . . modulus may depend on the manufacturing process. A New Linear-Time . . . Case of Probabilistic . . . • Interval approach – objective: find the interval Combination of . . . [ � y − ∆ , � y + ∆] Combination of . . . Effects of . . . that contains y . Effects of . . . • Interval approach – algorithms: presented in talks by Acknowledgments Muhanna, Nakao, Neumaier, Pownuk. Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 23
Data that We Can Use . . . Seismic Inverse . . . 3. Uncertainty in FEM: Remaining Problem Seismic Inverse . . . • In practice: we sometimes have both interval and prob- Existing Algorithm for . . . abilistic uncertainty. Algorithm for the . . . Successes, . . . • Example: Case of Interval Prior . . . – for manufacturing-related parameters, we may only A New Linear-Time . . . know intervals of possible values; A New Linear-Time . . . – for weather-related parameters, we also know the Case of Probabilistic . . . probabilities of different values (e.g., from the weather Combination of . . . records). Combination of . . . • Objective: for different p ∈ [0 , 1], find the value ∆( p ) Effects of . . . that bound ∆ y with probability p . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 23
Data that We Can Use . . . Seismic Inverse . . . 4. General Algorithm Seismic Inverse . . . • Problem – reminder: Existing Algorithm for . . . Algorithm for the . . . – for some x i , we know intervals [ x i , x i ]; Successes, . . . – for some x j , we know the probability distribution; Case of Interval Prior . . . – we want to find the value ∆( p ) that bound ∆ y with A New Linear-Time . . . probability p . A New Linear-Time . . . • Algorithm: Case of Probabilistic . . . – use Monte-Carlo techniques to simulate parameters Combination of . . . x j with known probability distributions; Combination of . . . Effects of . . . – for each such simulation, use interval FEM tech- niques to get an upper bound ∆ for | y − � y | ; Effects of . . . Acknowledgments – after several simulations, we get the resulting bounds Title Page distribution; ◭◭ ◮◮ – from this distribution, we can find the desired bound ∆( p ). ◭ ◮ Page 5 of 23
Data that We Can Use . . . Seismic Inverse . . . 5. Case Study: Determining Earth Structure Seismic Inverse . . . • Importance: civilization greatly depends on the things Existing Algorithm for . . . we extract from the Earth: oil, gas, water. Algorithm for the . . . • Need is growing, so we must find new resources. Successes, . . . Case of Interval Prior . . . • Problem: most easy-to-access mineral resources have A New Linear-Time . . . been discovered. A New Linear-Time . . . • Example: new oil fields are at large depths, under wa- Case of Probabilistic . . . ter, in remote areas – so drilling is very expensive. Combination of . . . • Objective: predict resources before we invest in drilling. Combination of . . . • How: we know what structures are promising. Effects of . . . Effects of . . . • Example: oil and gas concentrate near the top of (nat- Acknowledgments ural) underground domal structures. Title Page • Conclusion: to find mineral resources, we must deter- ◭◭ ◮◮ mine the structure at different depths z at different locations ( x, y ). ◭ ◮ Page 6 of 23
Data that We Can Use . . . Seismic Inverse . . . 6. Data that We Can Use to Determine the Earth Structure Seismic Inverse . . . Existing Algorithm for . . . • Available measurement results: those obtained without Algorithm for the . . . drilling boreholes. Successes, . . . • Examples: Case of Interval Prior . . . A New Linear-Time . . . – gravity and magnetic measurements; A New Linear-Time . . . – travel-times t i of seismic ways through the earth. Case of Probabilistic . . . • Need for active seismic data: Combination of . . . Combination of . . . – passive data from earthquakes are rare; Effects of . . . – to get more information, we make explosions, and Effects of . . . measure how the resulting seismic waves propagate. Acknowledgments • Resulting seismic inverse problem: Title Page ◭◭ ◮◮ – we know the travel times t i ; – we want to reconstruct velocities at different depths. ◭ ◮ Page 7 of 23
Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Hole Tomography Smashed Masked Velocity Models Successes, . . . 9.0 Case of Interval Prior . . . V 0 8.5 e 8.0 A New Linear-Time . . . l 7.5 o -10 7.0 c A New Linear-Time . . . 6.5 Depth (km) i 6.0 t Case of Probabilistic . . . -20 y 5.5 5.0 Combination of . . . ( 4.5 -30 k 4.0 m Combination of . . . 3.5 / 3.0 -40 s Effects of . . . 2.5 ) xri_zpv7_do_sm.vel 2.0 Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 23
Data that We Can Use . . . Seismic Inverse . . . 7. Seismic Inverse Problem: Towards Mathematical Formulation Seismic Inverse . . . Existing Algorithm for . . . • General description: wave equation with unknown v ( x ). Algorithm for the . . . • Difficulty: due to noise, we only know t i . Successes, . . . Case of Interval Prior . . . • Ray approximation: a seismic wave follows the shortest � dℓ A New Linear-Time . . . v → min; Eikonal equation |∇ t | = 1 path t = v . A New Linear-Time . . . Case of Probabilistic . . . • Discontinuity: v is only piece-wise continuous; Snell’s = sin( ϕ ′ ) law describes the transition sin( ϕ ) Combination of . . . . v ′ v Combination of . . . • Ill-posed problem: a change in v outside paths does not Effects of . . . affect observed travel-times t i ; hence, many drastically Effects of . . . different v ( x ) are consistent with observations. Acknowledgments Title Page • Current solution: ◭◭ ◮◮ – start with a meaningful first approximation; ◭ ◮ – use physically motivated iterations. Page 9 of 23
Data that We Can Use . . . Seismic Inverse . . . 8. Seismic Inverse Problem: FEM Approach Seismic Inverse . . . • v ( x ) = v j is constant within each element j . Existing Algorithm for . . . Algorithm for the . . . • We know: travel-times t i between known points A i , B i . Successes, . . . def • We want to find: velocities v j for which t i = t i ( v ) = Case of Interval Prior . . . min � ℓ ij , where: A New Linear-Time . . . v j j A New Linear-Time . . . • min is taken over all paths between A i and B i , and Case of Probabilistic . . . • ℓ ij is the length of the part of i -th path within ele- Combination of . . . ment j . Combination of . . . Effects of . . . • Shortest path: straight inside each element, Snell’s law Effects of . . . on each border. Acknowledgments ; t i = � = 1 def • Simplification: use slownesses s j ℓ ij · s j . Title Page v j j ◭◭ ◮◮ • Problem: system is under-determined. ◭ ◮ Page 10 of 23
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