Formulation of the . . . It Is Important to . . . Detecting . . . Need for Soft Computing Soft Computing Approach to What Is Continuity? Detecting Discontinuities: So How Can We . . . Application to Seismic . . . Seismic Analysis and Beyond Results Discussion Solymar Ayala Cortez 1 , Aaron A. Velasco 1 , and Home Page Vladik Kreinovich 2 Title Page Departments of 1 Geological Sciences and ◭◭ ◮◮ 2 Computer Science ◭ ◮ University of Texas at El Paso, El Paso, TX 79968, USA, sayalacortez@miners.utep.edu, aavelasco@utep.edu, Page 1 of 19 vladik@utep.edu Go Back Full Screen Close Quit
Formulation of the . . . It Is Important to . . . 1. Formulation of the Problem Detecting . . . • Starting from Newton, the main equations of physics Need for Soft Computing are differential equations. What Is Continuity? So How Can We . . . • This fact implicitly implies that all the corresponding Application to Seismic . . . processes are differentiable – and thus, continuous. Results • In practice, we often encounter processes or objects Discussion that change abruptly in time or in space. For example: Home Page – In physics, we have phase transitions when the Title Page properties change abruptly. ◭◭ ◮◮ – In geosciences, we have faults and we have sharp ◭ ◮ boundaries between different layers. Page 2 of 19 Go Back Full Screen Close Quit
Formulation of the . . . It Is Important to . . . 2. It Is Important to Detect Discontinuities Detecting . . . • In civil engineering , discontinuities may indicate crack Need for Soft Computing or faults. What Is Continuity? So How Can We . . . • Finding them can help us check structural integrity of Application to Seismic . . . the structure – e.g., of an airplane or a spaceship. Results • In geosciences , faults are places where most earth- Discussion quakes originate. Home Page • So finding the exact locations of faults may help predict Title Page where earthquakes can happen in the future. ◭◭ ◮◮ • In fracking, it is important to detect possible cracks. ◭ ◮ • Through these cracks, chemicals in the pumped liquid Page 3 of 19 can penetrate into the environment. Go Back Full Screen Close Quit
Formulation of the . . . It Is Important to . . . 3. Detecting Discontinuities Is Often Not Easy Detecting . . . • In some cases, we know the corresponding equations. Need for Soft Computing What Is Continuity? • In such situations, we can use these equations to de- So How Can We . . . velop techniques for detecting discontinuities. Application to Seismic . . . • In many other situations, however, we do not know the Results exact equations describing the process. Discussion • For example: Home Page – while we may know that there is a fault, Title Page – we do not know the exact shape of this fault, and ◭◭ ◮◮ – we do not have a good understanding of how this ◭ ◮ fault interacts, e.g., with seismic waves. Page 4 of 19 • In such situations, all we know is the corresponding Go Back processes are discontinuous. Full Screen • How can we use this information to detect the discon- tinuities? Close Quit
Formulation of the . . . It Is Important to . . . 4. Need for Soft Computing Detecting . . . • At first glance, it looks like the word “discontinuity” Need for Soft Computing has a precise mathematical meaning. What Is Continuity? So How Can We . . . • However, this mathematical definition is not what the Application to Seismic . . . geophysicists have in mind. Results • What they mean is rather a commonsense, informal Discussion meaning of this word. Home Page • Our objective is thus to translate this imprecise mean- Title Page ing into a precise algorithm. ◭◭ ◮◮ • In this translation, it is reasonable to use the technique ◭ ◮ of fuzzy logic . Page 5 of 19 • Indeed, this technique was designed to transform im- Go Back precise (“fuzzy”) expert knowledge into precise terms. Full Screen Close Quit
Formulation of the . . . It Is Important to . . . 5. What Is Continuity? Detecting . . . = t ′ − t is small, def • Continuity of a ( t ) means that if ∆ t Need for Soft Computing def = a ( t ′ ) − a ( t ) should also be small. What Is Continuity? then ∆ a So How Can We . . . • Let µ (∆ t ) be the degree to which ∆ t is small. Application to Seismic . . . • The quantity ∆ may use different units, so a corre- Results sponds to λ · t . Discussion Home Page • So, ∆ t is equivalent to ∆ a/λ , and smallness of ∆ a may be described as µ (∆ a/λ ). Title Page • “If A then B ” means that if A is true, then B is true. ◭◭ ◮◮ • Thus, the degree to which B is true should be greater ◭ ◮ than or equal to the degree to which A is true. Page 6 of 19 • In our case, this means µ (∆ a/λ ) ≥ µ (∆ t ). Go Back • The value µ (∆ t ) decreases with ∆ t , so ∆ a/λ ≤ ∆ t and Full Screen � ∆ a � � � � ≤ λ. Close � � ∆ t � Quit
Formulation of the . . . It Is Important to . . . 6. So How Can We Detect Discontinuity? Detecting . . . • So, we can detect discontinuities by comparing a Need for Soft Computing threshold λ with the ratio What Is Continuity? a ( t ′ ) − a ( t ) � ∆ a � � � So How Can We . . . � � � � � = � . t ′ − t � � � � ∆ t Application to Seismic . . . � � Results • As long as the ratio is below the threshold, we are Discussion continuous. Home Page • Once the ratio is above the threshold, this is an indi- Title Page cation of discontinuity. ◭◭ ◮◮ • In some cases, the values t are equally spaced: ◭ ◮ t 1 , t 2 = t 1 + δt, . . . , t k = t k − 1 + δt, . . . Page 7 of 19 • In such case, the desired ratio is simply proportional Go Back to | a ( t k ) − a ( t k − 1 ) | . Full Screen • In this case, we get an even simpler criterion for de- tecting discontinuity. Close Quit
Formulation of the . . . It Is Important to . . . 7. How Can We Detect Discontinuity (cont-d) Detecting . . . • In general, a process is continuous if | ∆ a/ ∆ t | ≤ λ . Need for Soft Computing What Is Continuity? • We consider the case when the values t are equally So How Can We . . . spaced: t 1 , t 2 = t 1 + δt, . . . , t k = t k − 1 + δt, . . . Application to Seismic . . . • Then, the ratio | ∆ a/ ∆ t | is proportional to Results Discussion | a ( t k ) − a ( t k − 1 ) | . Home Page • So, if the difference | a ( t k ) − a ( t k − 1 ) | does not exceed Title Page λ · δt , then at the location t k , the process is continuous. ◭◭ ◮◮ • If the difference | a ( t k ) − a ( t k − 1 ) | exceeds this threshold, ◭ ◮ then at this location, there is a discontinuity. Page 8 of 19 • This conclusion is consistent with common sense – Go Back which is one more reason to trust it. Full Screen Close Quit
Formulation of the . . . It Is Important to . . . 8. Application to Seismic Analysis Detecting . . . • In this paper, we used the experimental results from Need for Soft Computing the 2014 Southern California study; in this study: What Is Continuity? So How Can We . . . – more than 1000 seismic sensors were placed on a Application to Seismic . . . dense 600 m × 600 m grid Results – on top of one of the known faults – San Jacinto Discussion fault. Home Page • These sensors were in place for a 5-week period. Title Page • As a result, we have the values v ( s, t ) measured by ◭◭ ◮◮ different sensors s at different moments of time t . ◭ ◮ • During this period, the sensors recorded many earth- quakes, both: Page 9 of 19 – weak earthquakes originating in the vicinity of the Go Back fault and Full Screen – stronger earthquake that occurred outside the Close fault. Quit
Formulation of the . . . It Is Important to . . . Detecting . . . Need for Soft Computing What Is Continuity? So How Can We . . . Application to Seismic . . . Results Discussion Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 19 Go Back Figure 1: Sensors placed in the vicinity of San Jacinto fault Full Screen Close Quit
Formulation of the . . . It Is Important to . . . 9. What We Did Detecting . . . • Let t r ( s, E ) be moment when the singal from the earth- Need for Soft Computing quake E reached sensor s . What Is Continuity? So How Can We . . . • For each sensor s , we record the signal v ( s, t ) for 10 Application to Seismic . . . seconds: t r ( s, E ) ≤ t ≤ t r ( s, E ) + 10. Results • The signal interacts with the fault (and with other in- Discussion homogeneities). Home Page • Thus, the shape of the signal at different sensors was Title Page somewhat different. ◭◭ ◮◮ • As a measure of how the earthquake E influenced the ◭ ◮ sensor s , we took: Page 11 of 19 m ( s, E ) = t r ( s,E ) ≤ t ≤ t r ( s,E )+10 | v ( s, t ) | . max Go Back Full Screen Close Quit
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