EARTHSC 5642 Spring 2015 Dr. von Frese EARTHSC 5642 Spring 2015 Dr. von Frese Homework 5.2 A) Compute and plot 17 gravity effects (gz) of the buried horizontal cylinder with radius R = 3 km centered on the cylinder at the station interval of 1 km. B) Compute the Fast Fourier Transform (FFT) for the travel-time signal (gz) using the attached description of the FFT in Summary of Jenkins and Watts (1968) procedure(see the attached Appendix A7.3). Some information about the assignment can be find below in the solution of the exercise 1.1 that I have already done. I have provide two solutions the first one was obtained using matlab and the second excel but they are both the same thing. (Note: Assignment is Homework 5.2 only) 1) Partition the (gz) observations successively into halves and use an appropriate version of eq. (A7.3.5) in APPENDIX A7.3 from Jenkins and Watts (1968) to construct the transform. Show all details of the partitioning and calculations of the transform coefficients. 2) Describe in no more than a single, half-page paragraph how the FFT was taken. 3) List and plot the coefficients of the cosine and sine transforms for (gz). 4) List and plot the coefficients of the amplitude and phase spectra for (gz). C) Inverse transform the FFT to estimate the original (gz) observations. 1) Compute the synthesis of the signal coefficients showing all calculations. 2) Describe in no more than a single, half-page paragraph how the IFFT was taken. 3) Plot up and analyze the differences between the FFT-estimates and original observations. D) Determine the second horizontal derivative ∂2gz/∂d2 from the FFT of (gz). 1) What are the transfer function coefficients that take the second horizontal derivative in the f-frequency domain? 2) Apply the second derivative coefficients to the FFT of (gz) and inverse transform and plot the results. 4) How do the results in D.2 compare with the analytical horizontal second derivative gravity effects of the buried horizontal cylinder? Exercise 1.1 You have taken a job at the Johnson Space Flight Center in Houston (TX). In the desk that you were assigned, you find papers with a list of raw travel-time data for the free falls of a feather and a rock hammer. The intriguing thing about the two lists of numbers is that they are exactly the same i 1 2 3 4 5 6 7 8 ti(s) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 zi(ft) 25.0 25.7 27.7 31.0 35.6 41.6 48.9 57.5 9 10 11 12 13 14 15 16 17 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 67.5 78.8 91.4 105.3 120.6 137.2 155.1 174.4 194.6 Explore the inverse properties of numerical differentiation and integration for the above profile of travel-time data – i.e., A) Plot the travel-time data profile using appropriate units. B) Compute and list the 15 horizontal derivative values that may be defined from the successive 3-point data sequences. C) Find the derivative values for i = 1 and 17 using the 2nd Fundamental Theorem of Calculus (i.e., a ...