Need to Combine Data . . . Proposed Solution – . . . Numerical Example: . . . Model Fusion: Additional Problem: . . . Auxiliary Problem: . . . A New Approach To Conclusions Acknowledgments Processing Heterogenous Home Page Data Title Page ◭◭ ◮◮ ◭ ◮ Omar Ochoa Department of Computer Science Page 1 of 51 University of Texas at El Paso Go Back El Paso, TX 79968, USA Full Screen Close Quit
Need to Combine Data . . . 1. Need to Combine Data from Different Sources Proposed Solution – . . . • In many areas of science and engineering, we have dif- Numerical Example: . . . ferent sources of data. Additional Problem: . . . Auxiliary Problem: . . . • For example, in geophysics, there are many sources of Conclusions data for Earth models: Acknowledgments – first-arrival passive seismic data (from the actual Home Page earthquakes); Title Page – first-arrival active seismic data (from the seismic ◭◭ ◮◮ experiments); – gravity data; and ◭ ◮ – surface waves. Page 2 of 51 Go Back Full Screen Close Quit
Need to Combine Data . . . 2. Need to Combine Data (cont-d) Proposed Solution – . . . Numerical Example: . . . • Datasets coming from different sources provide compli- mentary information. Additional Problem: . . . Auxiliary Problem: . . . • Example: different geophysical datasets contain differ- Conclusions ent information on earth structure. Acknowledgments • In general: Home Page – 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 51 • Example: Go Back – gravity measurements have (relatively) low spatial resolution; Full Screen – a seismic data point comes from a narrow trajectory Close of a seismic signal – so spatial resolution is higher. Quit
Need to Combine Data . . . 3. Joint Inversion: An Ideal Future Approach Proposed Solution – . . . • At present: each of the datasets is often processed sep- Numerical Example: . . . arately. Additional Problem: . . . Auxiliary Problem: . . . • It is desirable: to data from different datasets. Conclusions • Ideal approach: use all the datasets to produce a single Acknowledgments model. Home Page • 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 51 Go Back Full Screen Close Quit
Need to Combine Data . . . 4. Data Fusion: Case of Interval Uncertainty Proposed Solution – . . . • In some practical situations, the value x is known with Numerical Example: . . . interval uncertainty. Additional Problem: . . . Auxiliary Problem: . . . • This happens, e.g., when we only know the upper bound ∆ ( i ) on each estimation error ∆ x ( i ) : | ∆ x ( i ) | ≤ ∆ i . Conclusions Acknowledgments x ( i ) | ≤ ∆ ( i ) , i.e., • In this case, we can conclude that | x − � Home Page x ( i ) − ∆ ( i ) , � x ( i ) + ∆ ( i ) ]. that x ∈ x ( i ) def = [ � Title Page x ( i ) , we know that the actual • Based on each estimate � ◭◭ ◮◮ value x belongs to the interval x ( i ) . ◭ ◮ • Thus, we know that the (unknown) actual value x be- Page 5 of 51 longs to the intersection of these intervals: Go Back n � x ( i ) = [max( � x ( i ) − ∆ ( i ) ) , min( � x ( i ) + ∆ ( i ) )] . def x = Full Screen i =1 Close Quit
Need to Combine Data . . . 5. Proposed Solution – Model Fusion: Main Idea Proposed Solution – . . . • Reminder: joint inversion methods are still being de- Numerical Example: . . . veloped. Additional Problem: . . . Auxiliary Problem: . . . • Practical solution: to fuse the models coming from dif- Conclusions ferent datasets. Acknowledgments • Simplest case – data fusion, probabilistic uncertainty: Home Page x ( n ) of the same x (1) , . . . , � – we have several estimates � Title Page quantity x . ◭◭ ◮◮ – each estimation error ∆ x ( i ) def x ( i ) − x is normally = � ◭ ◮ distributed with 0 mean and known st. dev. σ ( i ) ; x ( i ) − x ) 2 � n Page 6 of 51 ( � – Least Squares: find x that minimizes 2 · ( σ ( i ) ) 2 ; Go Back i =1 � n x ( i ) · ( σ ( i ) ) − 2 Full Screen � i =1 – solution: x = . Close � n ( σ ( i ) ) − 2 Quit i =1
Need to Combine Data . . . 6. Towards Formulation of a Problem Proposed Solution – . . . • What is given: Numerical Example: . . . Additional Problem: . . . – we have high spatial resolution estimates � x 1 , . . . , � x n Auxiliary Problem: . . . of the values x 1 , . . . , x n in several small cells; Conclusions – we also have low spatial resolution estimates � X j for Acknowledgments the weighted averages Home Page n � Title Page X j = w j,i · x i . i =1 ◭◭ ◮◮ x i and � • Objective: based on the estimates � X j , we must ◭ ◮ provide more accurate estimates for x i . Page 7 of 51 • Geophysical example: we are interested in the densi- Go Back ties x i . Full Screen Close Quit
Need to Combine Data . . . 7. Model Fusion: Case of Probabilistic Uncertainty Proposed Solution – . . . We take into account several different types of approximate Numerical Example: . . . equalities: Additional Problem: . . . Auxiliary Problem: . . . • Each high spatial resolution value � x i is approximately Conclusions equal to the actual value x i , w/known accuracy σ h,i : Acknowledgments x i ≈ x i . � Home Page • Each lower spatial resolution value � X j is approximately Title Page equal to the weighted average, w/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 8 of 51 x i , with accuracy σ pr,i : x i ≈ x pr,i . Go Back • Also, each lower spatial resolution value � X j is ≈ the Full Screen value within each of the smaller cells: Close � X j ≈ x i ( l,j ) . Quit
Need to Combine Data . . . 8. Case of Probabilistic Uncertainty: Details Proposed Solution – . . . • Each lower spatial resolution value � X j is approximately Numerical Example: . . . equal to the value within each of the smaller cells: Additional Problem: . . . Auxiliary Problem: . . . � X j ≈ x i ( l,j ) . Conclusions Acknowledgments • The accuracy of � X j ≈ x i ( l,j ) corresponds to the (em- Home Page 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 9 of 51 k j � = 1 def E j · x i ( l,j ) . � Go Back k j l =1 Full Screen Close Quit
Need to Combine Data . . . 9. Model Fusion: Least Squares Approach Proposed Solution – . . . • Main idea: use the Least Squares technique to combine Numerical Example: . . . the approximate equalities. Additional Problem: . . . Auxiliary Problem: . . . • We find the desired combined values x i by minimizing Conclusions the corresponding sum of weighted squared differences: Acknowledgments � � 2 n m n � � � x i ) 2 ( x i − � 1 Home Page � + · 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 10 of 51 Go Back Full Screen Close Quit
Need to Combine Data . . . 10. Model Fusion: Solution Proposed Solution – . . . • To find a minimum of an expression, we: Numerical Example: . . . Additional Problem: . . . – differentiate it with respect to the unknowns, and Auxiliary Problem: . . . – equate derivatives to 0. Conclusions • Differentiation with respect to x i leads to the following Acknowledgments system of linear equations: Home Page � 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 11 of 51 pr,i e,j j : j ∋ i Go Back where j ∋ i means that the j -th low spatial resolution estimate covers i -th cell. Full Screen Close Quit
Need to Combine Data . . . 11. Simplification: Fusing High Spatial Resolu- Proposed Solution – . . . tion Estimates and Prior Estimates Numerical Example: . . . • Idea: fuse each high spatial resolution estimate � x i with Additional Problem: . . . a prior estimate x pr,i . Auxiliary Problem: . . . 1 1 Conclusions • Detail: instead of · ( x i − � x i ) + · ( x i − x pr,i ) , we σ 2 σ 2 Acknowledgments pr,i h,i have a single term σ − 2 f,i · ( x i − x f,i ) , where Home Page x i · σ − 2 h,i + x pr,i · σ − 2 Title Page � def def pr,i σ − 2 = σ − 2 h,i + σ − 2 x f,i = , pr,i . σ − 2 h,i + σ − 2 f,i ◭◭ ◮◮ pr,i ◭ ◮ • Resulting simplified equations: � n � Page 12 of 51 � � 1 σ − 2 w j,i ′ · x i ′ − � f,i · ( x i − x f,i ) + · w j,i · X j + Go Back σ 2 l,j j : j ∋ i i ′ =1 Full Screen � 1 · ( x i − � X j ) = 0 . Close σ 2 e,j j : j ∋ i Quit
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