Probabilistic and Model Fusion: . . . Model Fusion: . . . Interval - PowerPoint PPT Presentation

Data Fusion under . . . Data Fusion under . . . New Problem: . . . Why This Is Important New Idea: Model Fusion Probabilistic and Model Fusion: . . . Model Fusion: . . . Interval Uncertainty Model Fusion: Interval . . . Model Fusion: Interval . . . of the Results Model Fusion: Interval . . . Preliminary Experiments Acknowledgments of Data Fusion, Home Page with Application Title Page to Geosciences ◭◭ ◮◮ ◭ ◮ Christian Servin 1 , Omar Ochoa 1 , Page 1 of 13 and Aaron A. Velasco 2 1 Department of Computer Science Go Back 2 Department of Geological Sciences Full Screen University of Texas at El Paso El Paso, TX 79968, USA Close contact christians@miners.utep.edu Quit

New Problem: . . . 1. Data Fusion under Interval Uncertainty: Re- Why This Is Important minder New Idea: Model Fusion Model Fusion: . . . • Frequent practical situation: Model Fusion: . . . – we are interested in a quantity u ; Model Fusion: Interval . . . – we have several measurements and/or expert esti- Model Fusion: Interval . . . mates u 1 , . . . , u n of u . Model Fusion: Interval . . . Preliminary Experiments • Objective: fuse these estimates into a single more ac- Acknowledgments curate estimate. Home Page • Interval case: each u i is known with interval uncer- Title Page tainty. ◭◭ ◮◮ • Formal description: for each i , we know the interval ◭ ◮ u i = [ u i − ∆ i , u i + ∆ i ] containing u . � n Page 2 of 13 def • Solution: u belongs to the intersection u = u i of Go Back i =1 these intervals. Full Screen Close

New Problem: . . . 2. Data Fusion under Probabilistic Uncertainty: Why This Is Important Reminder New Idea: Model Fusion Model Fusion: . . . def • Probabilistic uncertainty: each measurement error ∆ u i = Model Fusion: . . . u i − u is normally distributed w/0 mean and known σ i . Model Fusion: Interval . . . • Technique: the Least Squares Method (LSM) Model Fusion: Interval . . . n � ( u − u i ) 2 Model Fusion: Interval . . . → min . 2 σ 2 Preliminary Experiments i i =1 Acknowledgments • Resulting estimate: is Home Page � n u i · σ − 2 Title Page i i =1 u = . � n ◭◭ ◮◮ σ − 2 i i =1 ◭ ◮ • Standard deviation: Page 3 of 13 1 σ 2 = with σ 2 ≪ σ 2 , i . � Go Back n σ − 2 i Full Screen i =1 Close

New Problem: . . . 3. New Problem: Different Resolution Why This Is Important New Idea: Model Fusion • Traditional data fusion: fusing measurement results Model Fusion: . . . with different accuracy. Model Fusion: . . . • Additional problem: different measurements also have Model Fusion: Interval . . . different resolution. Model Fusion: Interval . . . Model Fusion: Interval . . . • Case study – geosciences: estimating density u 1 , . . . , u n at different locations and depths. Preliminary Experiments Acknowledgments • Examples of different geophysical estimates: Home Page – Seismic data leads to higher-resolution estimates Title Page u n of the density values. � u 1 , . . . , � ◭◭ ◮◮ – Gravity data leads to lower-resolution estimates, ◭ ◮ i.e., estimates � u for the weighted average Page 4 of 13 � n u = w i · u i . Go Back i =1 Full Screen Close

New Problem: . . . 4. Why This Is Important Why This Is Important New Idea: Model Fusion • Reminder: there are many sources of data for Earth Model Fusion: . . . models: Model Fusion: . . . – first-arrival passive seismic data Model Fusion: Interval . . . (from the actual earthquakes), Model Fusion: Interval . . . – first-arrival active seismic data Model Fusion: Interval . . . (from the seismic experiments), Preliminary Experiments – gravity data, Acknowledgments Home Page – surface waves, etc. Title Page • At present: each of these datasets is processed sepa- rately, resulting in several different Earth models. ◭◭ ◮◮ ◭ ◮ • Fact: these models often provide complimentary geo- physical information. Page 5 of 13 • Idea: all these models describe the properties of the Go Back same Earth, so it is desirable to combine them. Full Screen Close

New Problem: . . . 5. New Idea: Model Fusion Why This Is Important New Idea: Model Fusion • Objective: to combine the information contained in Model Fusion: . . . multiple complementary datasets. Model Fusion: . . . • Ideal approach: it is desirable to come up with tech- Model Fusion: Interval . . . niques for joint inversion of these datasets. Model Fusion: Interval . . . Model Fusion: Interval . . . • Problem: designing such joint inversion techniques is an important theoretical and practical challenge. Preliminary Experiments Acknowledgments • Status: such joint inversion methods are being devel- Home Page oped. Title Page • Practical question: what to do while these methods are ◭◭ ◮◮ being developed? ◭ ◮ • Our practical solution: fuse the Earth models coming from different datasets. Page 6 of 13 Go Back Full Screen Close

New Problem: . . . 6. Model Fusion: Statistical Case Why This Is Important New Idea: Model Fusion • Objective: find the values u 1 , . . . , u n of the desired quan- Model Fusion: . . . tity in different spatial cells. Model Fusion: . . . • Geophysical example: u i is the density at different Model Fusion: Interval . . . Model Fusion: Interval . . . 1 km × 1 km × 1 km cells . Model Fusion: Interval . . . • Input: we have Preliminary Experiments Acknowledgments – high-resolution measurements, i.e., values � u i ≈ u i Home Page with st. dev. σ i ; u ( k ) cor- Title Page – lower-resolution measurements, i.e., values � responding to blocks of neighboring cells: ◭◭ ◮◮ � u ( k ) ≈ w ( k ) · u i , with st. dev. σ ( k ) . ◭ ◮ � i i Page 7 of 13 • Additional information: a lower-resolution measure- Go Back ment result is representative of values within the block. Full Screen Close

New Problem: . . . 7. Model Fusion: Statistical Case (cont-d) Why This Is Important New Idea: Model Fusion • Formal description: when w ( k ) u ( k ) ≈ u i , � = 0, we have � i Model Fusion: . . . with st. dev. δ ( k ) . Model Fusion: . . . • How to estimate δ ( k ) : as the empirical st. dev. within Model Fusion: Interval . . . the block. Model Fusion: Interval . . . Model Fusion: Interval . . . • High-resolution values (reminder): � u i ≈ u i w/st. dev. σ i . Preliminary Experiments • Lower-resolution values (reminder): Acknowledgments � u ( k ) ≈ w ( k ) · u i , with st. dev. σ ( k ) . � Home Page i i Title Page • LSM Solution: minimize the sum ◭◭ ◮◮ u ( k ) − � w ( k ) · u i ) 2 ( � � � � � i u i ) 2 u ( k ) ) 2 ◭ ◮ ( u i − � ( u i − � i + + . σ 2 ( δ ( k ) ) 2 ( σ ( k ) ) 2 Page 8 of 13 i i i k k • How: differentiating w.r.t. u i , we get a system of linear Go Back equations with unknowns u 1 , . . . , u n . Full Screen Close

New Problem: . . . 8. Model Fusion: Interval Case Why This Is Important New Idea: Model Fusion • Quantities of interest: values u 1 , . . . , u n of the desired Model Fusion: . . . quantity in different spatial cells. Model Fusion: . . . • Objective: find the ranges u 1 , . . . , u n of possible values Model Fusion: Interval . . . of u 1 , . . . , u n . Model Fusion: Interval . . . • High-resolution measurements: values � u i ≈ u i with Model Fusion: Interval . . . bound ∆ i : Preliminary Experiments u i − ∆ i ≤ u i ≤ � � u i + ∆ i . Acknowledgments Home Page u ( k ) correspond- • Lower-resolution measurements: values � ing to blocks of neighboring cells: Title Page � u ( k ) ≈ w ( k ) · u i , with bound ∆ ( k ) . ◭◭ ◮◮ � i i ◭ ◮ • Resulting constraint: Page 9 of 13 � u ( k ) + ∆ ( k ) . u ( k ) − ∆ ( k ) ≤ w ( k ) � · u i ≤ � Go Back i i Full Screen Close

New Problem: . . . 9. Model Fusion: Interval Case (cont-d) Why This Is Important New Idea: Model Fusion • Additional information: a priori bounds on u i : Model Fusion: . . . u i ≤ u i ≤ u i . Model Fusion: . . . Model Fusion: Interval . . . • Additional information: a priori bounds on the changes Model Fusion: Interval . . . between neighboring cells: Model Fusion: Interval . . . − δ ij ≤ u i − u j ≤ δ ij . Preliminary Experiments Acknowledgments • High-resolution measurements (reminder): Home Page � u i − ∆ i ≤ u i ≤ � u i + ∆ i . Title Page ◭◭ ◮◮ • Lower-resolution measurements (reminder): � u ( k ) − ∆ ( k ) ≤ u ( k ) + ∆ ( k ) . ◭ ◮ w ( k ) � · u i ≤ � i Page 10 of 13 i • Objective: minimize and maximize each u i under these Go Back constraints. Full Screen Close

New Problem: . . . 10. Model Fusion: Interval Solution Why This Is Important New Idea: Model Fusion • Problem. Minimize (Maximize) u i under the following Model Fusion: . . . constraints: Model Fusion: . . . • u i ≤ u i ≤ u i . Model Fusion: Interval . . . • − δ ij ≤ u i − u j ≤ δ ij . Model Fusion: Interval . . . Model Fusion: Interval . . . • � u i − ∆ i ≤ u i ≤ � u i + ∆ i . u ( k ) − ∆ ( k ) ≤ � u ( k ) + ∆ ( k ) . w ( k ) Preliminary Experiments • � · u i ≤ � i Acknowledgments i Home Page • Current solution method: linear programming. Title Page • Objective: provide more efficient algorithms for specific geophysical cases. ◭◭ ◮◮ • Preliminary results: some such algorithms have been ◭ ◮ developed. Page 11 of 13 Go Back Full Screen Close