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 Along a bisecting profile extending from through to at intervals, compute the 5 gravity profiles in mgal by and plot them superimposed on a single graph using different colors or symbols. Compute and plot the total gravity effect of the 5 cylinders by summing their effects at each observation point on the profile. What is the and of the total gravity effect? What is the utility of these statistics for graphing the profile? Suppose you want to estimate the 5 densities ( ) from the total gravity observations in -above [ ]-matrix and least-squares estimates of , and compare the estimated densities with those in the above table. Determine the Choleski factorization of [ ] – i.e., determine a lower triangular matrix such that [ = ]. Find the coefficients of [ ] such that [ = ], and solve the system for the least-squares estimates of . Compare your density estimates with those you obtained in -above.

In this assignment, we will be referring to Fig. 2.1 on page 24/100 of the notes “5642Lectures_2_4.pdf”. We are tasked with considering a set of 5 horizontally infinite cylinders and calculating their gravity profiles in mgal (milligals). These profiles will be computed along a bisecting profile extending from point through to point , with intervals in between.

To begin, we will compute the gravity profiles for each of the 5 cylinders at the specified observation points. The gravity effect of each cylinder will be calculated and expressed in milligals. These gravity profiles will be plotted on a single graph, using different colors or symbols to distinguish between them.

Once we have plotted the individual gravity profiles, we will proceed to compute and plot the total gravity effect of the 5 cylinders. This will involve summing up the effects of each cylinder at every observation point along the profile. By doing so, we will obtain a total gravity effect curve.

Next, we need to determine the maximum and minimum values of the total gravity effect curve. These values represent the highest and lowest points on the graph, respectively. They provide important statistical information about the profile, giving an indication of the range of gravity effects caused by the cylinders.

The utility of these statistics for graphing the profile lies in the ability to identify the maximum and minimum points on the graph. This can help in understanding the overall trend and magnitude of the gravity effects and can aid in interpreting the geophysical significance of the profile.

Moving on, our task is to estimate the densities of the 5 cylinders from the total gravity observations. We will use a least-squares estimation method, utilizing a given matrix of observations and the unknown values of the densities. The matrix will be denoted as [ ] and the estimated densities will be represented as .

By performing the least-squares estimation, we will obtain the estimated density values for each of the cylinders. We can then compare these estimated densities with the tabulated values provided in the above table. This comparison will help evaluate the accuracy of our estimation method.

Furthermore, we need to determine the Choleski factorization of the matrix [ ]. This refers to decomposing the matrix into a lower triangular matrix, denoted as [ ], such that [ = ]. The lower triangular matrix [ ] will contain the coefficients needed for the decomposition.

Additionally, we are required to find the coefficients of the matrix [ ] such that [ = ]. This process will involve modifying the given matrix [ ] by substituting the estimated density values into appropriate positions in the matrix equation.

Once we have obtained the modified matrix equation, we can solve the system to find the least-squares estimates of the densities. These estimates can then be compared with the density values obtained previously.

In summary, this assignment involves various computations and analyses related to gravity profiles of cylinders. These calculations provide important insights into the geophysical properties of the cylinders and can be used to estimate their densities. Additionally, we will perform Choleski factorization and modify matrix equations to obtain least-squares estimates of the densities.

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