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 ...

1. 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).

2. 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

3. 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

4. 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 function can be
determined from the integral of its derivative) given by equation
(1.5) in the GeomathBook.pdf (p. 18/153) and equation (4.ii) in

5. the 5642Lectures_1.pdf (p. 16/21).
D) Plot the complete derivative profile using appropriate units.
E) Numerically integrate the derivative profile and compare to
the original data profile.
F) Compute and list the 16 integral values that may be defined
from the successive 2-point data sequences of the travel-time
data.
G) Using the 1st Fundamental Theorem of Calculus (i.e., a
function can be determined from the derivative of its integral),
find the integral value for i = 1.
H) Plot the complete integral profile using appropriate units.
I) Numerically differentiate the integral profile and compare to
the original travel-time data profile.
J) Compute, list, and plot the derivative of the derivative profile
in D.
K) Compute, list, and plot the integral of the profile in J and
compare to profile D.
L) How can you account for the intercepts of the data sets?
M) On which planetary body of the solar system might these
data have been observed? Why?
********

6. Solution
s using below matlab
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Exercise1
%Description: This function completes sections A-M of
Exercise 1.1.
%Initial data is entered first and then a series of calculations
and
%displays are executed in order to understand the falling
motion of a
%feather and a rock hammer.
%Data Input (i is index, t is time (seconds), z is distance fallen
(feet))
ti=[0.0:0.5:8.0];
zi=[25.0 25.7 27.7 31.0 35.6 41.6 48.9 57.5 67.5 78.8 91.4
105.3 120.6 137.2 155.1 174.4 194.6];
%A) Plot the travel-time data profile using appropriate units

7. figure(1);
hold on
plot(ti,zi);
title('A)Travel-Time Data Profile')
xlabel('Time (s)');
ylabel('Distance Fallen (ft)');
hold off
%B) Compute and list the 15 horizontal derivative values that
may be
%defined from the successive 3-point data sequences.
dzdt=zeros(1,17);
for i=2:16
dzdt(i)=(zi(i+1)-zi(i-1))/(ti(i+1)-ti(i-1));
end
fprintf('The 3-point first derivative (velocity) beginning at time
0.0, 0.5,... 7.0 are:n %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f
%-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-
.fn',dzdt(2:16))
%C) Find the derivative values for i = 1 and 17 using the 2nd
Fundamental

8. %Theorem of Calculus (i.e., a function can be determiend from
the integral
%of its derviatve) given by equation (1.5).....
dzdt(1)=dzdt(2)-2*(zi(2)-zi(1))/(ti(2)-ti(1));
dzdt(end)=2*(zi(end)-zi(end-1))/(ti(end)-ti(end-1))-dzdt(end-1);
%D) Plot the complete derivative profile using appropriate units
figure(2);
hold on
plot(ti,dzdt);
title('D)Velocity Data Profile')
xlabel('Time (s)');
ylabel('Velocity (ft/s)');
hold off
%E) Numerically Integrate the derivative profile and compare it
to the
%original data profile.
zcalc=zeros(1,17);
zcalc(1)=25.0; %Input from problem
calculatez;
function calculatez

9. for i=2:16
zcalc(i)=zcalc(i-1)+(dzdt(i-1)+4*dzdt(i)+dzdt(i+1))/6*(ti(i)-
ti(i-1));
end
zcalc(end)=zcalc(16)+dzdt(16)*(ti(end)-ti(end-1));
end
figure(1)
hold on
plot(ti,zcalc,'-r');
legend('Given Position Data','Calculated Position Data');
hold off
%F) Compute and list the 16 integral values taht may be defined
from the
%successive 2-point data sequences of the travel-time data.
fInt=zeros(16,1);
for i=1:16
fInt(i)=(zi(i+1)+zi(i))/2*(ti(i+1)-ti(i)); %Trapezoid rule
end
fprintf('nThe integral values calulated, beginning at time 0.0,
0.5,... 7.0 are:n %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-
.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-.2f %-.2fn',fInt)

10. %G) Find the integral value for i=1;
%?????????
fInt=[0; fInt];
%H)
figure(3);
hold on
plot(ti,fInt);
title('H)Integral Data Profile')
xlabel('Time (s)');
ylabel('Distance*Time (ft*s)');
hold off
%I)
zcalcint=zeros(1,17);
zcalcint(1)=25.0; %Given initial value
for i=2:16
zcalcint(i)=(fInt(i+1)-fInt(i-1))/(ti(i+1)-ti(i-1));
end
zcalcint(end)=(fInt(end)-fInt(end-1))/(ti(end)-ti(end-1));
figure(1)
hold on
plot(ti,zcalcint,'-r');

11. legend('Given Position Data','Calculated Position Data');
hold off
The data sets are very well known information collected to
analyze and double the check the calculation above.
Z is the dependent vertical variable, 25’ since 1=i=25z
In the solar system, these data can be observed on the planet
mercury since the distance given is in accordance with it.

12. ******