Solve the system for the least-squares estimates

Question

• Refferring to Fig. on page 24/100 of the notes=> 5642Lectures_2_4.pdf, consider a set of 5 horizontally infinite cylinders with the following parameters è

Cylinder #	1	2	3	4	5
d (km)	-34	-20	0	20	35
z (km)	2	3	5	10	10
R (km)	2	1	3	4	3.5
Δρ (gm/cm3)	3.0	5.0	1.0	2.0	1.5

A) Along a bisecting profile extending from d = -64 km through 0 km to +64 km at      1-km intervals, compute the 5 gravity profiles in mgal by

                               g_z = [41.93 Δρ(R²/z)]/[(d²/z²) + 1],

and plot them superimposed on a single graph using different colors or symbols.  Computer software (e.g., IMSL, LINPACK, Matlab, Mathematica, MathCad, Maple, etc.) may be helpful here.

B) Compute and 

1) plot the total gravity effect of the 5 cylinders by summing their effects at each observation point on the profile. 

 2) What is the mean value and standard deviation of the total gravity effect? 

 3) What is the utility of these statistics for graphing the profile?

C) Suppose you want to estimate the 5 densities (Δρ_i) from the total gravity observations in 

B-above.  Determine 1) the [A^TA]-matrix and 2) least-squares estimates of Δρ_i, and 3) compare the estimated densities with those in the above table.

 D)  Determine 

1) the Choleski factorization of [A^TA] - i.e., determine a lower triangular matrix L such that [LL^T = A^TA]. 

2) Find the coefficients of [P] such that [LP = A^TB], and 

3) solve the system for the least-squares estimates of Δρ_i. 

 4) Compare your density estimates with those you obtained in C-above.