Geophysics437, Fall 2008:
Assignment #2, due Friday, October 10.
Corrections, detrending and interpolation of 2-D gravity and magnetic data collected in Southern Alberta.

Materials to hand in:
1) Hard copy of report or send the file by email (suggested volume 10–12 pages including figures; combine figures, make subfigures).
2) Send your Matlab program and input files by email.

In this report you will prepare the gravity and magnetic data that you collected so that these data may be analyzed in your next assignment. After explaining how the data were collected, the first goal in this assignment will be to write two main Matlab programs, one for gravity and the other for magnetics. The programs will do the following: 1) input data, 2) perform corrections (see below), 3) output the corrected data into a file. These two main programs will be made up of several parts (see b–f below). The parts referred to above may be either subroutines (i.e. programs called by the main program) or simply sections within the main program. As with all good programming, please document the important steps by using explanatory comments within the program.

a) Write an introduction to the report. State the dates and location of the field school survey. Describe how, and the conditions under which, the gravity and magnetic data were collected, what equipment was used. Do not include anything that you have already described in your previous report, just refer to your report #1. Preferable format of your report would be one and half – double spaced. Try to keep it to not more than 10 – 12 pages with figures (combine figures, make subfigures). Include references and acknowledgment. Appendix with your Matlab codes is additional pages.

b) Use least-squares method to find the density for the gravity profile from the first assignment. Compare result with the assignment #1 and comment on it.

c) Using the raw gravity data, start with connection of the base stations between each other. Please relate everything to your base station #2. After this perform the drift, latitude and free-air corrections to obtain the Free Air Anomaly. Outline your procedure and list and explain important new procedures but do not repeat the explanations from the first assignment, just refer them.

d) Interpolate the gravity and topography data onto a 2-D grid of constant spacing. Note that the main space coordinate should be distance away from the gravity base station. The grid spacing resolution should reflect the gravity data. Using the Matlab "subplot" function, create three figures: 1.The interpolated, but uncorrected gravity; 2. The topography (i.e. the elevation); 3. The free-air anomaly. Include figure captions that explain the figures and give the UTM coordinates of the gravity base station. In the text of the report, discuss the relationship between the topography and the Free-Air Anomaly. Are the results consistent with what you expect?

e) Using the original (not interpolated) Free Air Anomaly data, write an inversion program (in Matlab) to remove the regional trend approximated by plain and determine the near surface density.   Illustrate the regional trend (combine with another figure, use subplot). Use the method of least squares discussed in class. Compare this 2-D density with the 1-D density of just one profile that you obtained in (b) and in the assignment #1.   Using two different near surface densities (one obtained from the 2-D dataset, another – from profile data), obtain two Bouguer gravity datasets. As with the Free-Air Anomaly, interpolate the Bouguer anomalies onto an equally spaced grid and present the result as a figure with two subplots. Provide a brief discussion of methods and results. Describe the least-squares method application to inverse the 2-D gravity and magnetic data sets in more details.

f) Plot the diurnal variations of the magnetic field as a function of time for any chosen day of 2006 magnetic survey (1 figure) as an example (you may want to use "subplot" here).   Use the least squares method to perform a two-dimensional detrending of the diurnally corrected magnetic data for the dataset of 2004-2005-2006-2007 approximating the regional trend by a plain. What would you suggest to improve the regional trend fit? Use Matlab to interpolate the raw magnetic data, as well as the corrected magnetic data, into a uniform grid. Create images of the raw and corrected magnetic field and show the regional trend (could be one or two figures).   Calculate and show the detrended anomaly.  Explain the difference between the corrected vs. raw magnetic field.  Is there any evidence of bias introduced by the grid interpolation?

g) Appendix contains your Matlab code(s) and finding coefficients of matrix a from example given in class. x=[0,2,5], y=[1,3,4], a=[a₀,a₁],
calculate by hand a₀and a₁ with two alternative methods shown in the class:
1) A=X^-1Y;
2) A=(X^TX) ^-1X^TY,
where X^-1 is inverse matrix and X^T is transpose matrix. Show every step of calculation.

See the map of the area with 3 gravity profiles marked on it: Gravity Profiles 2005.pdf.
Download gravity data gravity2005.xls
Download magnetic data to work with: in TXT format magndata200405062007.txt or in XLS format magndata2004050607.xls. These are data collected from 2004, 2005, 2006 and 2007.
Geology of the studied area: ReportWarner.pdf
Manual for the magnetometer GSM-19.
Manual for the gravimeter used to acquire the gravity data for this course CG-5ShortFormManual.pdf.
Here from you may download three examples prepared for you by Dennis:
matrix_example
grid_example
interp_example