Case Study Computational . . . What We Plan to Do Available Data: What . . . Fusing Continuous and Probabilistic Approach Discrete Data, on the Resulting Location The Results of the . . . Example of Merging Seismic It May Be Useful to . . . How Accurate Is This . . . and Gravity Models in Home Page Geophysics Title Page ◭◭ ◮◮ Omar Ochoa 1 , 2 , Aaron Velasco 1 , 3 , Vladik Kreinovich 1 , 2 ◭ ◮ 1 Cyber-ShARE Center of Excellence: A Center for Sharing Page 1 of 17 Cyberresources to Advance Science and Education Departments of 2 Computer Science and 3 Geological Sciences Go Back University of Texas at El Paso, El Paso, TX 79968, USA Full Screen omar@miners.utep.edu, velasco@geo.utep.edu, vladik@utep.edu Close Quit
Case Study Computational . . . 1. Case Study What We Plan to Do • To find the density ρ ( v ) at different locations and dif- Available Data: What . . . ferent depths, we can use two types of data: Probabilistic Approach Resulting Location – the seismic data , i.e., the arrival times of signals The Results of the . . . from earthquake and test explosions; It May Be Useful to . . . – the gravity data. How Accurate Is This . . . • Both data provide complementary information: Home Page – seismic data provides information about a narrow Title Page zone around a path; ◭◭ ◮◮ – gravity data provides information about the larger ◭ ◮ area – but with much smaller spatial resolution. Page 2 of 17 • At present, there are no efficient algorithms for pro- cessing both types of data. Go Back Full Screen • So, we must fuse the results of processing these two types of data: a seismic model and a gravity model. Close Quit
Case Study Computational . . . 2. Computational Problem: Need to Fuse Dis- What We Plan to Do crete and Continuous Models Available Data: What . . . • Traditionally, seismic models are continuous : the ve- Probabilistic Approach locity smoothly changes with depth. Resulting Location The Results of the . . . • In contrast, the gravity models are discrete : we have It May Be Useful to . . . layers, in each of which the velocity is constant. How Accurate Is This . . . • The abrupt transition corresponds to a steep change in Home Page the continuous model. Title Page • Both models locate the transition only approximately. ◭◭ ◮◮ • So, if we simply combine the corresponding values value- ◭ ◮ by-value, we will have two transitions instead of one: Page 3 of 17 – one transition where the continuous model has it, Go Back and Full Screen – another transition nearby where the discrete model has it. Close Quit
Case Study Computational . . . 3. What We Plan to Do What We Plan to Do • We want to avoid the misleading double-transition mod- Available Data: What . . . els. Probabilistic Approach Resulting Location • Idea: first fuse the corresponding transition locations. The Results of the . . . • In this paper, we provide an algorithm for such location It May Be Useful to . . . fusion. How Accurate Is This . . . Home Page • Specifically , first, we formulate the problem in the prob- abilistic terms. Title Page • Then , we provide an algorithm that produces the most ◭◭ ◮◮ probable transition location. ◭ ◮ • We show that the result of the probabilistic location Page 4 of 17 algorithm is in good accordance with common sense. Go Back • We also show how the commonsense intuition can be Full Screen reformulated in fuzzy terms. Close Quit
Case Study Computational . . . 4. Available Data: What is Known and What Needs What We Plan to Do to Be Determined Available Data: What . . . • For each location, in the discrete model, we have the Probabilistic Approach exact depth z d of the transition. Resulting Location The Results of the . . . • In contrast, for the continuous model, we do not have It May Be Useful to . . . the abrupt transition. How Accurate Is This . . . • Instead, we have velocity values v ( z ) at different depths. Home Page • We must therefore extract the corresponding transition Title Page value z c from the velocity values. ◭◭ ◮◮ • To be more precise, we have values v 1 , v 2 , . . . , v i , . . . , v n ◭ ◮ corresponding to different depths. Page 5 of 17 • We need to find i for which the transition occurs be- Go Back tween the depths i and i + 1. Full Screen Close Quit
Case Study Computational . . . 5. Probabilistic Approach What We Plan to Do def • The difference ∆ v j = v j − v j +1 ( j � = i ) is caused by Available Data: What . . . many independent factors. Probabilistic Approach Resulting Location • Due to the Central Limit Theorem, we thus assume The Results of the . . . that it is normally distributed, with probability density It May Be Useful to . . . � � 1 1 def 2 · σ 2 · (∆ v j ) 2 How Accurate Is This . . . √ p j = 2 · π · σ · exp − . Home Page • The value ∆ v i at the transition depth i is not described Title Page by the normal distribution. ◭◭ ◮◮ • We assume that differences corresponding to different ◭ ◮ depths j are independent, so: Page 6 of 17 � � � � 1 1 2 · σ 2 · (∆ v j ) 2 Go Back L i = p j = √ 2 · π · σ · exp − . j � = i j � = i Full Screen Close Quit
Case Study Computational . . . 6. How to Find the Location: The General Idea What We Plan to Do of the Maximum Likelihood Approach Available Data: What . . . • Reminder: the likelihood of each model is: Probabilistic Approach � � � � 1 1 Resulting Location 2 · σ 2 · (∆ v j ) 2 L i = p j = √ 2 · π · σ · exp − . The Results of the . . . j � = i j � = i It May Be Useful to . . . • Natural idea: select the parameters for which the like- How Accurate Is This . . . lihood of the observed data is the largest. Home Page • The value L i is the largest if and only if − ln( L i ) is the Title Page � 1 (∆ v j ) 2 → min smallest: − ln( L i ) = const + 2 · σ 2 · . ◭◭ ◮◮ i j � = i ◭ ◮ • This sum is equal to � n − 1 � (∆ v j ) 2 = (∆ v j ) 2 − (∆ v i ) 2 . Page 7 of 17 j � = i j =1 Go Back • The first term in this expression does not depend on i . Full Screen • Thus, the difference is the smallest ⇔ the value (∆ v i ) 2 is the largest ⇔ | ∆ v i | is the largest. Close Quit
Case Study Computational . . . 7. Resulting Location What We Plan to Do • We want: to select the most probable location of the Available Data: What . . . transition point. Probabilistic Approach Resulting Location • We select: the depth i 0 for which the absolute value The Results of the . . . | ∆ v i | of the difference ∆ v i = v i +1 − v i is the largest. It May Be Useful to . . . • This conclusion seems to be very reasonable: How Accurate Is This . . . Home Page – the most probable location of the actual abrupt transition between the layers Title Page – is the depth at which the measured difference is the ◭◭ ◮◮ largest. ◭ ◮ Page 8 of 17 Go Back Full Screen Close Quit
Case Study Computational . . . 8. The Results of the Probabilistic Approach are What We Plan to Do in Good Accordance with Common Sense Available Data: What . . . • Intuitively, for each depth i , our confidence that i a Probabilistic Approach transition point depends on the difference | ∆ v i | : Resulting Location The Results of the . . . – the smaller the difference, the less confident we are It May Be Useful to . . . that this is the actual transition depth, and How Accurate Is This . . . – the larger the difference, the more confident we are Home Page that this is the actual transition depth. Title Page • In our probabilistic model, we select a location with ◭◭ ◮◮ the largest possible value | ∆ v i | . ◭ ◮ • This shows that the probabilistic model is in good ac- cordance with common sense. Page 9 of 17 Go Back • This coincidence increases our confidence in this result. Full Screen Close Quit
Case Study Computational . . . 9. It May Be Useful to Formulate the Common What We Plan to Do Sense Description in Fuzzy Terms Available Data: What . . . • Fuzzy logic is known to be a useful way to formalize Probabilistic Approach imprecise commonsense reasoning. Resulting Location The Results of the . . . • Common sense: the degree of confidence d i that i is a It May Be Useful to . . . transition point is f ( | ∆ v i | ), for some monotonic f ( z ). How Accurate Is This . . . • It is reasonable to select a value i for which our degree Home Page of confidence is the largest d i = f ( | ∆ v i | ) → max . Title Page • Since f ( z ) is increasing, this is equivalent to ◭◭ ◮◮ | ∆ v i | → max . ◭ ◮ Page 10 of 17 • Of course, to come up with this conclusion, we do not need to use the fuzzy logic techniques. Go Back • However, this description may be useful if we also have Full Screen other expert information. Close Quit
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