Referring to Fig. 2.1 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/cm |
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(R^{2}/z)]/[(d^{2}/z^{2}) + 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**]-matrix and

^{T}A**2)**least-squares estimates of

**Δρ**, and

_{i}**3)**compare the estimated densities with those in the above table.

** **

**D) **Determine **1)** the Choleski factorization of [**A ^{T}A**] – i.e., determine a lower triangular matrix

**L**such that [

**LL**=

^{T}**A**].

^{T}A**2)**Find the coefficients of [

**P**] such that [

**LP**=

**A**], and

^{T}B**3)**solve the system for the least-squares estimates of

**Δρ**.

_{i}**4)**Compare your density estimates with those you obtained in

**C**-above.