PROBLEM #$2[11$ points$]$

The two$-$term infiltration equation of Philip $(1957)$ computes the cumulative vertical infiltration, $($cm$),$ of water into a deep$-$drained homogeneous soil as function of time, $($hour$),$ as follows

$(1)$

where denotes the soil sorptivity and signifies the saturated soil hydraulic conductivity. These latter two coefficients are so$-$called soil properties.

$2\mathrm{.}1.$ Please use dimensional anlysis to determine the units of the soil sorptivity, $,$ and the saturated soil hydraulic conductivity, $.$ Please write down the units here $[1$ point$]$

We would like to evaluate Equation $(1)$ in its ability to describe measured infiltration data. We collected the following experimental data

Table $1$: Measured cumulative infiltration data

$($hour$)|0\mathrm{.}010|0\mathrm{.}020|0\mathrm{.}050|0\mathrm{.}100|0\mathrm{.}200|0\mathrm{.}500|1\mathrm{.}000$

$($cm$)|0\mathrm{.}051|0\mathrm{.}075|0\mathrm{.}123|0\mathrm{.}176|0\mathrm{.}264|0\mathrm{.}451|0\mathrm{.}682$

We would like to use the tabulated data to evaluate Philip's two$-$term expression in Equation $1.$

$2\mathrm{.}2.$ Setup the resulting system of equations in matrix form, $.$ Each entry of the design matrix, coefficient vector and the data vector should be defined below $-$ and is either a numerical value or a symbol.

$[$Note: by clicking on X$,$ c and$/$or d you can modify the content and size of each array using the Equation editor$]$

Please, also copy the MATLAB code you wrote for this system $[2$ points$]$

$\%$ TYPE YOUR MATLAB CODE HERE

A $=[0\mathrm{.}010$; $0\mathrm{.}020$; $0\mathrm{.}050$; $0\mathrm{.}100$; $0\mathrm{.}200$; $0\mathrm{.}500$; $1\mathrm{.}000]$;

b $=[0\mathrm{.}051$; $0\mathrm{.}075$; $0\mathrm{.}123$; $0\mathrm{.}176$; $0\mathrm{.}264$; $0\mathrm{.}451$; $0\mathrm{.}682]$

$2\mathrm{.}3.$ Please solve for the unknown coefficients using least squares. Enter your solution of the unknown coefficients in the vector below using at least three significant digits $[1$ point$]$

Please, also copy the MATLAB code you wrote for this solution

$\%$ TYPE YOUR MATLAB CODE HERE

$2\mathrm{.}4.$ Plot the observed data, $,$ as function of time, $,$ using a red plus symbol. Then, add to this figure in a solid blue line the simulated infiltration curve of Equation $(1)$ using the least squares solution of $2\mathrm{.}3.$ Make sure that you use a sufficient number of time values when plotting the simulated curve $($it should look smooth$).$ Add labels to both your axes and a legend $[1$ point$]$

$\%$ TYPE YOUR MATLAB CODE HERE

$\%$ WHEN YOU PLOT A FIGURE, END THE MATLAB BOX WITH "hold off"

$2\mathrm{.}5.$ Compute the vector of residuals. Your code should print the values of the residuals $[1$ point$]$

$\%$ TYPE YOUR MATLAB CODE HERE

$2\mathrm{.}6.$ Is the solution unique or not? Please explain your answer in words and confirm using MATLAB code $[1$ point$]$

$\%$ TYPE YOUR MATLAB CODE HERE

$2\mathrm{.}7.$ Now we have determined the optimal coefficients of Equation $(1),$ at what time, $,$ does the function predict that the cumulative infiltration, $,$ is equal to $2$ cm$?[2$ points$]$

$[$HINT: You can treat this as an intersection problem!$]$

$\%$ TYPE YOUR MATLAB CODE HERE

Insert your answer here, $=\_\_\_\_\_\_\_\_\_\_\_\_,($at least $3$ significant digits$)$ in units of $\_\_\_\_\_\_\_\_$

$2\mathrm{.}8.$ To determine whether the two$-$term Equation $(1)$ sufficiently well describes the measured cumulative infiltration curve, we consider the following extension

$(2)$

where is a third unknown coefficient. Which expression, the two$-$term of Equation $(1)$ or the three$-$term of Equation $(2)$ receives most support from the experimental data of Table $1?$ Please explain in detail how you got to your answer and demonstrate your answer with MATLAB code below $[2$ points$]$

$\%$ TYPE YOUR MATLAB CODE