Need to Combine Data . . . Joint Inversion: An . . . Proposed Solution – . . . Towards a Model Fusion: Case of . . . Numerical Example: . . . Fast, Practical Alternative Model Fusion: Case of . . . Conclusions to Joint Inversion Acknowledgments of Multiple Datasets: Title Page Model Fusion ◭◭ ◮◮ ◭ ◮ Omar Ochoa Page 1 of 32 Department of Computer Science Go Back University of Texas at El Paso El Paso, TX 79968, USA Full Screen omar@miners.utep.edu Close Quit
Need to Combine Data . . . 1. Need to Combine Data from Different Sources Joint Inversion: An . . . • In many areas of science and engineering, we have dif- Proposed Solution – . . . ferent sources of data. Model Fusion: Case of . . . Numerical Example: . . . • For example, in geophysics, there are many sources of Model Fusion: Case of . . . data for Earth models: Conclusions – first-arrival passive seismic data (from the actual Acknowledgments earthquakes); Title Page – first-arrival active seismic data (from the seismic ◭◭ ◮◮ experiments); ◭ ◮ – gravity data; and – surface waves. Page 2 of 32 Go Back Full Screen Close Quit
Need to Combine Data . . . 2. Need to Combine Data (cont-d) Joint Inversion: An . . . • Datasets coming from different sources provide compli- Proposed Solution – . . . mentary information. Model Fusion: Case of . . . Numerical Example: . . . • Example: different geophysical datasets contain differ- Model Fusion: Case of . . . ent information on earth structure. Conclusions • In general: Acknowledgments – some of the datasets provide better accuracy and/or Title Page spatial resolution in some spatial areas; ◭◭ ◮◮ – other datasets provide a better accuracy and/or ◭ ◮ spatial resolution in other areas or depths. Page 3 of 32 • Example: Go Back – gravity measurements have (relatively) low resolu- tion; Full Screen – each seismic data point comes from a narrow tra- Close jectory of a seismic signal – so resolution is higher. Quit
Need to Combine Data . . . 3. Joint Inversion: An Ideal Future Approach Joint Inversion: An . . . • At present: each of the datasets is often processed sep- Proposed Solution – . . . arately. Model Fusion: Case of . . . Numerical Example: . . . • It is desirable: to combine data from different datasets. Model Fusion: Case of . . . • Ideal approach: use all the datasets to produce a single Conclusions model. Acknowledgments • Problem: in many areas, there are no efficient algo- Title Page rithms for simultaneously processing all the datasets. ◭◭ ◮◮ • Challenge: designing joint inversion techniques is an ◭ ◮ important theoretical and practical challenge. Page 4 of 32 Go Back Full Screen Close Quit
Need to Combine Data . . . 4. Proposed Solution – Model Fusion: Main Idea Joint Inversion: An . . . • Reminder: joint inversion methods are still being de- Proposed Solution – . . . veloped. Model Fusion: Case of . . . Numerical Example: . . . • Practical solution: to fuse the models coming from dif- Model Fusion: Case of . . . ferent datasets. Conclusions • Simplest case – data fusion, probabilistic uncertainty: Acknowledgments – we have several measurements (and/or expert esti- Title Page mates) � x 1 , . . . , � x n of the same quantity x . ◭◭ ◮◮ def – each measurement error ∆ x i = � x i − x is normally ◭ ◮ distributed with 0 mean and known st. dev. σ i ; x i − x ) 2 Page 5 of 32 � n ( � – Least Squares: find x that minimizes ; 2 · σ 2 Go Back i =1 i � n x i · σ − 2 Full Screen � i i =1 – solution: x = . Close � n σ − 2 i Quit i =1
Need to Combine Data . . . 5. Data Fusion: Case of Interval Uncertainty Joint Inversion: An . . . • In some practical situations, the value x is known with Proposed Solution – . . . interval uncertainty. Model Fusion: Case of . . . Numerical Example: . . . • This happens, e.g., when we only know the upper bound Model Fusion: Case of . . . ∆ i on each measurement error ∆ x i : | ∆ x i | ≤ ∆ i . Conclusions • In this case, we can conclude that | x − � x i | ≤ ∆ i , i.e., Acknowledgments def that x ∈ x i = [ � x i − ∆ i , � x i + ∆ i ]. Title Page • Based on each measurement result � x i , we know that ◭◭ ◮◮ the actual value x belongs to the interval x i . ◭ ◮ • Thus, we know that the (unknown) actual value x be- Page 6 of 32 longs to the intersection of these intervals: Go Back n � def x = x i = [max( � x i − ∆ i ) , min( � x i + ∆ i )] . Full Screen i =1 Close Quit
Need to Combine Data . . . 6. Additional Problem: We Also Have Different Spa- Joint Inversion: An . . . tial Resolution Proposed Solution – . . . • In many situations, different measurements have not Model Fusion: Case of . . . only different accuracy, but also different resolution. Numerical Example: . . . Model Fusion: Case of . . . • Example: Conclusions – seismic data leads to higher-resolution estimates of Acknowledgments the density at different locations and depths, while Title Page – gravity data leads to lower-estimates of the same ◭◭ ◮◮ densities. ◭ ◮ • Towards precise formulation of the problem: Page 7 of 32 – High-resolution measurements mean that we mea- Go Back sure the values corresponding to small spatial cells. – A low-resolution measurement means that its re- Full Screen sults are affected by several neighboring spatial cells. Close Quit
Need to Combine Data . . . 7. Towards Formulation of a Problem Joint Inversion: An . . . • What is given: Proposed Solution – . . . Model Fusion: Case of . . . – we have high-resolution estimates � x 1 , . . . , � x n of the Numerical Example: . . . values x 1 , . . . , x n within several small spatial cells; Model Fusion: Case of . . . – we also have low-resolution estimates � X j for the Conclusions weighted averages Acknowledgments n � Title Page X j = w j,i · x i . i =1 ◭◭ ◮◮ • Objective: based on the estimates � x i and � x , we must ◭ ◮ provide more accurate estimates for x i . Page 8 of 32 • Geophysical example: we are interested in the densi- Go Back ties x i . Full Screen Close Quit
Need to Combine Data . . . 8. Model Fusion: Case of Probabilistic Uncertainty Joint Inversion: An . . . We take into account several different types of approximate Proposed Solution – . . . equalities: Model Fusion: Case of . . . • Each high-resolution value � x i is approximately equal Numerical Example: . . . to the actual value x i , with the known accuracy σ h,i : Model Fusion: Case of . . . Conclusions x i ≈ x i . � Acknowledgments • Each lower-resolution value � X j is approximately equal Title Page to the weighted average, with a known accuracy σ l,j : � ◭◭ ◮◮ � X j ≈ w j,i · x i . ◭ ◮ i • We usually have a prior knowledge x pr,i of the values Page 9 of 32 x i , with accuracy σ pr,i : x i ≈ x pr,i . Go Back • Also, each lower-resolution value � X j is approximately Full Screen equal to the value within each of the smaller cells: Close � X j ≈ x i ( l,j ) . Quit
Need to Combine Data . . . 9. Case of Probabilistic Uncertainty: Details Joint Inversion: An . . . • Each lower-resolution value � X j is approximately equal Proposed Solution – . . . to the value within each of the smaller cells: Model Fusion: Case of . . . Numerical Example: . . . � X j ≈ x i ( l,j ) . Model Fusion: Case of . . . • The accuracy of � Conclusions X j ≈ x i ( l,j ) corresponds to the (em- Acknowledgments pirical) standard deviation: Title Page k j � � � 2 , = 1 def σ 2 · x i ( l,j ) − E j � ◭◭ ◮◮ e,j k j l =1 ◭ ◮ where Page 10 of 32 k j � = 1 def E j · x i ( l,j ) . � Go Back k j l =1 Full Screen Close Quit
Need to Combine Data . . . 10. Model Fusion: Least Squares Approach Joint Inversion: An . . . • Main idea: use the Least Squares technique to combine Proposed Solution – . . . the approximate equalities. Model Fusion: Case of . . . Numerical Example: . . . • We find the desired combined values x i by minimizing Model Fusion: Case of . . . the corresponding sum of weighted squared differences: Conclusions � � 2 n m n � � � x i ) 2 ( x i − � 1 Acknowledgments � + · X j − w j,i · x i + σ 2 σ 2 Title Page h,i l,j i =1 j =1 i =1 ◭◭ ◮◮ k j n m ( � � � � X j − x i ( l,j ) ) 2 ( x i − x pr,i ) 2 + . ◭ ◮ σ 2 σ 2 pr,i e,j i =1 j =1 l =1 Page 11 of 32 Go Back Full Screen Close Quit
Need to Combine Data . . . 11. Model Fusion: Solution Joint Inversion: An . . . • To find a minimum of an expression, we: Proposed Solution – . . . Model Fusion: Case of . . . – differentiate it with respect to the unknowns, and Numerical Example: . . . – equate derivatives to 0. Model Fusion: Case of . . . • Differentiation with respect to x i leads to the following Conclusions system of linear equations: Acknowledgments � n � � � Title Page 1 1 w j,i ′ · x i ′ − � · ( x i − � x i ) + · w j,i · X j + σ 2 σ 2 ◭◭ ◮◮ h,i l,j j : j ∋ i i ′ =1 ◭ ◮ � 1 1 · ( x i − � · ( x i − x pr,i ) + X j ) = 0 , σ 2 σ 2 Page 12 of 32 pr,i e,j j : j ∋ i Go Back where j ∋ i means that the j -th low-resolution mea- surement covers i -th cell. Full Screen Close Quit
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