Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . Spatial Resolution for How This Idea Is . . . Processing Seismic Data: Limitations of Ray . . . Angular Diversity: A . . . Type-2 Methods for Finding Limitations of the . . . Gauging Uncertainty: . . . the Relevant Granular Gauging Spatial . . . Structure Title Page ◭◭ ◮◮ Vladik Kreinovich, Jaime Nava, Rodrigo Romero, Julio Olaya, Aaron Velasco ◭ ◮ Cyber-ShARE Center, University of Texas at El Paso Page 1 of 18 500 W. University, El Paso, Texas 79968, USA contact vladik@utep.edu Go Back Kate C. Miller Full Screen Department of Geology and Geophysics Close Texas A&M University, College Station, Texas 77843, USA Quit
Need for Seismic Data . . . How Seismic Inverse . . . 1. Need for Seismic Data Processing Uncertainty vs. . . . • It is very important to determine Earth structure: How Traditional . . . How This Idea Is . . . – to find fossil fuels (oil, coal, natural gas), minerals, water; Limitations of Ray . . . Angular Diversity: A . . . – to assess earthquake risk. Limitations of the . . . • Data that we can use to determine the Earth structure: Gauging Uncertainty: . . . – data from drilling boreholes, Gauging Spatial . . . – gravity and magnetic measurements, Title Page – analyzing the travel-times and paths of seismic ways ◭◭ ◮◮ as they propagate through the earth. ◭ ◮ • Active and passive (= from earthquakes) seismic mea- Page 2 of 18 surements are usually the most informative: Go Back – they come from areas of different depth; Full Screen – each sound wave travels along a narrow path, so it provides a detailed structure of the Earth. Close Quit
Need for Seismic Data . . . How Seismic Inverse . . . 2. How Seismic Inverse Problem Is Solved Uncertainty vs. . . . • First, we discretize: divide region into cells, with con- How Traditional . . . stant velocity v j in each cell j . How This Idea Is . . . Limitations of Ray . . . • Once we know v j , we can determine the (fastest) paths Angular Diversity: A . . . of the seismic waves – which leads to Snell’s law: Limitations of the . . . sin( α 1 ) = sin( α 2 ) . Gauging Uncertainty: . . . v 1 v 2 Gauging Spatial . . . Title Page ❅ � � ❅ ✂ ❇ ✂ ✻ ◭◭ ◮◮ ❇ ✂ α 1 v 1 d 1 ❇ ✂ ❇ ✂ ◭ ◮ ❇ ✂ ❄ ❇ ❆ ✁ ✻ Page 3 of 18 ❆ ✁ ❆ ✁ v 2 d 2 α 2 ❆ ✁ Go Back ❆✁ ❄ Full Screen Close Quit
Need for Seismic Data . . . How Seismic Inverse . . . 3. How Seismic Inverse Problem Is Solved (cont-d) Uncertainty vs. . . . • Main idea: the measured travel-time t i along the i -th How Traditional . . . path is t i = � = 1 def How This Idea Is . . . ℓ ij · s j , where s j . v j j Limitations of Ray . . . • Algorithm: repeat the following two steps until the pro- Angular Diversity: A . . . cess converges: Limitations of the . . . Gauging Uncertainty: . . . – based on the current values of s j , find the shortest Gauging Spatial . . . paths and thus, the values ℓ ij ; Title Page – based on the current values ℓ ij , we solve the above ◭◭ ◮◮ system of linear equations, and get the updated s j . ◭ ◮ • Need to find spatial resolution (granularity): it is im- possible to distinguish between s j at two nearby points. Page 4 of 18 • Fact: some reconstructed value s j relate not to an in- Go Back dividual cell but to a cluster of cells (spatial granule). Full Screen Close Quit
Need for Seismic Data . . . How Seismic Inverse . . . 4. Uncertainty vs. Spatial Resolution (Granularity) Uncertainty vs. . . . • There are two types of uncertainty: How Traditional . . . How This Idea Is . . . – the “traditional” uncertainty – due to measurement Limitations of Ray . . . inaccuracy and incomplete coverage. Angular Diversity: A . . . – the spatial resolution – each measured value repre- Limitations of the . . . sents the “average” value over the region ( granule ). Gauging Uncertainty: . . . • Fact: methods of determining traditional uncertainty Gauging Spatial . . . have been traditionally more developed. Title Page • Corollary: the main ideas for determining spatial res- ◭◭ ◮◮ olution comes from these more traditional methods. ◭ ◮ • In view of this, Page 5 of 18 – before we describe the existing methods for deter- Go Back mining spatial resolution, Full Screen – let us describe the corresponding methods for de- termining more traditional uncertainty. Close Quit
Need for Seismic Data . . . How Seismic Inverse . . . 5. How Traditional Uncertainty Is Determined: Main Idea Uncertainty vs. . . . How Traditional . . . • Main idea: How This Idea Is . . . Limitations of Ray . . . – First, we add simulated noise to the measured val- Angular Diversity: A . . . ues of traveltimes. Limitations of the . . . – Then, we reconstruct the new values of the veloci- Gauging Uncertainty: . . . ties based on these modified traveltimes. Gauging Spatial . . . – Finally, we compare the resulting velocities with Title Page the originally reconstructed ones. ◭◭ ◮◮ • Main assumption: we know the accuracy of different ◭ ◮ measurements. Page 6 of 18 • Fact: geophysical analysis usually also involves expert knowledge. Go Back Full Screen • Conclusion: it is also necessary to take into account the uncertainty of the expert statements. Close Quit
Need for Seismic Data . . . How Seismic Inverse . . . 6. How This Idea Is Applied to Determine Spatial Resolution Uncertainty vs. . . . How Traditional . . . • First, we add a perturbation of spatial size δ 0 (e.g., How This Idea Is . . . sinusoidal) to the reconstructed field � v ( x ). Limitations of Ray . . . Angular Diversity: A . . . • Then, we simulate the new traveltimes based on the perturbed values of the velocities. Limitations of the . . . Gauging Uncertainty: . . . • Finally, we apply the same seismic data processing al- Gauging Spatial . . . gorithm to the simulated traveltimes and get � v new ( x ). Title Page • If the perturbations are visible in � v new ( x ) − � v ( x ), then ◭◭ ◮◮ details of spatial size δ 0 can be reconstructed. ◭ ◮ • If the perturbations are not visible in � v new ( x ) − � v ( x ), Page 7 of 18 then details of spatial size δ 0 cannot be reconstructed. Go Back • In the geosciences, this method is known as a checker- Full Screen board method. Close Quit
Need for Seismic Data . . . How Seismic Inverse . . . 7. Checkerboard Method: Main Limitation Uncertainty vs. . . . • Main limitation: Its running time is several times higher How Traditional . . . than the original seismic data processing. Indeed, How This Idea Is . . . Limitations of Ray . . . – in addition to applying the seismic data processing Angular Diversity: A . . . algorithm to the original data, Limitations of the . . . – we also need to apply the same algorithm to the Gauging Uncertainty: . . . simulated data – and apply it several times. Gauging Spatial . . . • The computation time drastically increases – and the Title Page whole process slows down. ◭◭ ◮◮ • It is therefore desirable to develop faster techniques for ◭ ◮ estimating spatial resolution, Page 8 of 18 – techniques that will not require new processing of Go Back simulated seismic data Full Screen – and will only use the results of the processing the original seismic data. Close Quit
Need for Seismic Data . . . How Seismic Inverse . . . 8. A Similar Problem Arises For Estimating Tradi- tional Uncertainty Uncertainty vs. . . . How Traditional . . . • As mentioned, the existing methods for determining How This Idea Is . . . traditional uncertainty are also based on Limitations of Ray . . . Angular Diversity: A . . . – simulating errors and Limitations of the . . . – applying the (time-consuming) seismic data pro- Gauging Uncertainty: . . . cessing algorithms to the simulated traveltimes. Gauging Spatial . . . • As a result, the existing methods for determining the Title Page traditional uncertainty are also too time-consuming, ◭◭ ◮◮ • There is a similar need to developing faster uncertainty ◭ ◮ estimation techniques. Page 9 of 18 Go Back Full Screen Close Quit
Need for Seismic Data . . . How Seismic Inverse . . . 9. First Heuristic Idea For Estimating Uncertainty: Ray Coverage Uncertainty vs. . . . How Traditional . . . • The more measurements we perform, the more accu- How This Idea Is . . . rately we can determine the desired quantity. Limitations of Ray . . . Angular Diversity: A . . . • In particular, for each cell j , the value v j affects those traveltime measurements t i for which for which ℓ ij > 0. Limitations of the . . . Gauging Uncertainty: . . . • Thus, the more rays pass through the cell, the more Gauging Spatial . . . accurate the corresponding measurement. Title Page • The number of such rays – called a ray coverage – is ◭◭ ◮◮ indeed reasonably well correlated with uncertainty. ◭ ◮ • Thus it can serve as an estimate for this uncertainty: Page 10 of 18 – the smaller the ray coverage, Go Back – the larger the uncertainty. Full Screen Close Quit
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