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An Example of "Boot-Strapping" In Applied Physics By Patrick Bruskiewich ©PSCB2012 Abstract In mathematical physics a number of specialized techniques are learned with the passage of time, techniques which become part of an applied physicist's box of magic. One technique is colloquially known as "Boot Strapping", by which a minimum of data can be used to yield a maximum of results. Sometimes this involves taking a series of under described equations (say a single equation in two unknowns), some additional good data points and a technique to more fully analyze the problem. In Geophysics there is a nice example of "Boot Strapping" in how one can estimate both the speed of a "shot wave" in seismic imaging, and the depth of an interface or feature within the earth. This practical seismic imaging example is well suited for a first year physics course studying wave motion. Rudiments of Seismic Imaging Seismic imaging is a very well developed technology, which provides geophysicists an opportunity to study the structure of the earth to a dozen or so kilometres deep and to prospect for much needed resources. Seismic imaging relies on ray tracing and model building. At its rudiments is a simple model of a "shot" providing an acoustical wave that passes through an isotropic and homogenous medium until the acoustical wave is reflected off an interface, and back to an array of microphones on the earth's surface which records the time-signature of acoustical wave (refer to Fig. 1: Acoustical Wave Passing through a Medium). Consider a surface SS and an interface RR. A shot or explosion is set off at "shot point" and an acoustical wave travels through the isotropic and homogenous medium at some unknown speed v to an interface at some unknown depth h. shntpnint ^^ Fig. 1: Acoustical Wave Passing through a Medium The microphones Pi and P2 are located in a line at measured distances from the shot point O where microphone Pq is located (refer to Fig. 2: Position of Microphones Pi and P2). R R ^ Microphone Position Fig. 2: Position of Microphones P ^ and P ? The Microphone Pi is a distance Xi from the shot point O, while the microphone P2 is a distance X 2 from O. The microphones Pi and P2 record the reflected acoustical wave at times Ti and T2 (refer to Fig. 3: Time Signature of the Acoustic Wave). Elementary Concept of Seismic Reflection From this arrangement let us use the "Boot Strapping" technique to estimate the velocity of the acoustical wave v, and the depth of the interface h. By analogy, the reflection of an acoustical wave off the interface RR is similar to the optics of viewing a reflection off a mirror. The image that we see in a mirror appears to be behind the mirror, at an Imaginary Point O' a distance similar to that from which the mirror is away from the viewer. From the standpoint of ray tracing, the seismic imaging is similar. We arrive at an arrangement that the Imaginary Point O' is a distance of 2 times the depth h from the shot point (Refer to Fig. 3: The Imaginary Point O' in the Seismic Image). — r O'lmaginary Paint Fig. 3: The Imaginary Point O' in the Seismic Image By inspection, it is evident that the time it takes for the acoustical wave to travel from O to T and back to the Microphone Pi is the same time it takes for a wave to travel from the Imaginary Point O' to Pi. If the distance from the shot point O to Microphone Pi is Xi and the time it takes for the wave to travel from the shot point O to Microphone Pi is Ti then by Pythagoras' Theorem (vT x f=Xf + (2h) 2 which is a single equation with two unknowns. Similarly for Microphone P2 is at X 2 we have ,2 ,^9 /~ r \2 {vT 2 )=Xl + {2h) With two measurements of time we have now two equations in two unknowns, v and h. It always helps to have some data to work with. Let us say then that the seismic array has gathered the following data Microphone Distance X in metres Time T in seconds PO 1.60 PI 1,000 1.649 P2 2,000 1.787 Table 1: Data collected at Microphones P0, PI and P2 We can take any pair of data points and solve first for the velocity of the seismic wave v and then for the depth of the feature or interface h. Algebraic Approach Let us use the data for PO and PI . By inspection (2hf =v% 2 =v 2 T l 2 -X This leads to 2 X, IX, 2 which for our chosen data yields v = 2508m/s and a depth h of h = 2006m Geometric Approach A second technique is to solve the problem geometrically. There is an angle i at the imaginary image point that defines the angle Z PO O' PI. By inspection the sin of this angle is . . x smz vT If we were to follow the wave front at the instant at which it emerged from the ground we would have a similar triangle with an emergence angle i (refer to Fig. 4: The Emergence Angle i). PO dx PI vdt Fig. 4: The Emergence Angle i This triangle is similar to the triangle in Fig. 3. By inspection we see that . . vAT sm/ = AX from which we determine that X vAT , XfAX^ vT AX 2 V = T K ATj which ultimately means that X( AX\ v = T K ATj Choose the midpoint between PI and P2. For this point and t = \{t 1+ t 2 ) We can now also measure a difference in time AT and a difference in distance AX from the shot point O, that is AT = T 2 -T l and AX = X 2 -X 1 If we combined these four simple expressions into the equation for the velocity v we find that ] x]-x] v = ■ ' Ti-T( (which is what we would arrive at using the Algebraic Approach and points PI and P2). Using the data for the microphones at PI and P2 we find that v = 2515m/s and a depth h of h = 2012m Conclusion While this example is known by any geophysicist worth their weight in rock, it is nonetheless a good practical example to introduce the "Boot Strapping" technique to freshman physics students (while we have one equation with two unknowns, a little ingenuity helped you to get around that limitation). This example is also good for showing that algebraic and geometric techniques many times complement each other. ©PSCB2012