GG413 Geological Data Analysis
Homework #8: Least Square Regression II
Reading: Ch. 3.12-3.13
Due Thu Oct 28, 2014
Please be complete yet concise about writing out your steps and explaining your solutions.
Please attach your Matlab scripts for this problem set as an appendix only
1) On the web you will find a table (hawaii.txt) containing distance, age, and age uncertainty of dated lavas
along the Hawaiian volcano chain. Distance is the great-circle distance from Kilauea (km), age is
radiometric volcano date (in My), and uncertainty is the 1measurement uncertainty (Myr).
(a) Plot the data points with error bars (hint: see “help errorbar”)
(b) Find the non-weighted least squares regression line. Give your solutions for the slope and intercept,
state the rms error (or misfit), and overlay the predictions on your plot of the data.
(c) Find the weighted least squares regression line. Then give your solutions for the slope and intercept,
and state the rms weighted error (or misfit). Create a new plot with the data, error bars, and the new
model line. (hint: for the W matrix, the Matlab function “diag(d)” creates a square matrix with the
values of the vector d on the diagonal).
(d) Briefly comment on the differences between the non-weighted and weighted least-squares solutions,
and their likely cause.
2) For the non-weighted least squares regression in problem 1(b) above...
(a) What is the 95% confidence interval for the true slope (2) and intercept (1)?
(b) Compute and plot the 95% confidence band about the best fit line.
3) The file “faultstep.txt” contains distance (m) and relief (m) for a topographic profile across a small normal
fault. We want to estimate the total vertical offset across the fault. There is also a linear trend in the data
because the fault sits on a gently dipping monocline.
(a) First plot relief versus distance as a scatter plot. Then, make a model function that includes the trend,
as well as the offset of the fault simulated as a heaviside step function (heaviside.m), defined as
 0, (x  x0 )  0
1

H ( x  x0 )   , (x  x0 )  0
2
 1, (x  x0 )  0
(b) First use an estimate of the fault location of x0 = 145 m, and use matrix algebra to solve for the model
coefficients (slope & intercept of the line as well as the amplitude of the fault offset). Plot your model
profile with the data. What is the rms misfit?
(c) What are the standard deviations of the three model coefficients?
(d) You might notice that the fault location, x0, may not be optimal. Now determine the best choice for
x0 that minimizes the total sum of square misfitsE by trying a range of values for x0. That is, define a
dense array of x0 covering a range of possible values, perform the inversion for each value of x0,
empirically evaluate E(x0), plot E versus x0, and find the value of x0 at the minimum E. This is the “grid
search” method for solving this non-linear inverse problem (i.e., H is a non-linear function of x0). Plot
the final model on top of the data. What are the slope, intercept, fault offset, and fault location?
Compare the rms misfit of this solution to the original rms misfit in (b).