We have Philip's equation to evaluate Cumulative Infiltration of water into soil:
$$z=s\sqrt t+kt$$ Where $z$ is Cumulative Infiltration and $t$ is Cumulative time and $s,k$ are coefficients. Here is the method I've taught to find these coefficients:
We equate $Q$ with differences of measured infiltration and infiltration from the equation: $$Q=s\sqrt t+k.t-z$$ So for the measurement of the $i$th step we have the following equation w $$Q_i=s_i\sqrt{t_i}+k_it_i-z_i$$ Then to find $s$ , $k$ we need to solve the system of equations obtained form $\frac{\partial Q}{\partial s}=0$ and $\frac{\partial Q}{\partial k}=0$
So I am trying to find the coefficients of $z=s\sqrt t+kt$ for the following datas:
$$\begin{array}{|c|c|} \hline t& z \\ \hline 5&2 \\ \hline 15& 4.5\\ \hline 45& 9\\ \hline 75& 13 \\ \hline 110& 17.5\\ \hline \end{array}$$
By following the procedure I got
$$Q_1=s_1\sqrt{5}+5k_1-2$$ $$Q_2=s_2\sqrt{15}+15k_2-4.5$$ $$Q_3=s_3\sqrt{45}+45k_3-9$$ $$Q_4=s_4\sqrt{75}+75k_4-13$$ $$Q_5=s_5\sqrt{110}+110k_5-17.5$$ And $\frac{\partial Q}{\partial s}=\sqrt {t_i}$ , $\frac{\partial Q}{\partial k}=t_i$ And I didn't get any system of equations . can you please help me? I am confused.
You have $n$ data points $(t_i,z_i)$ and you need to adjust the model $$z=s\sqrt t+kt$$ This is a multilinear regression with predictors $\sqrt t$ and $t$ without constant term.
If your prefer, let $t_i=u_i^2$ to make $$z= s u +k u^2$$
Just do it with Excel (polynomial trend line).
Edit
I shall use the most basic approach using normal equations, having the (may be wrong) feeling that you are not very comfortable with matrices.
In the least-square sense, you want to minimize the sum of squares of absolute errors that is to say $$\text{SSQ}=\sum_{i=1}^n r_i^2 \qquad \text{where} \qquad r_i=s\sqrt {t_i}+kt_i- z_i$$ Write the partial derivatives $$\frac{\partial \text{SSQ}}{\partial s}=2 \sum_{i=1}^n r_i \frac{\partial r_i}{\partial s}=2 \sum_{i=1}^n \Big[s\sqrt {t_i}+kt_i- z_i \Big]\sqrt {t_i} $$ $$\frac{\partial \text{SSQ}}{\partial k}=2 \sum_{i=1}^n r_i \frac{\partial r_i}{\partial k}=2 \sum_{i=1}^n \Big[s\sqrt {t_i}+kt_i- z_i \Big]{t_i} $$ Set these partial derivatives equal to $0$ since you are looking for the minimum and expand to get $$s\sum_{i=1}^n t_i + k\sum_{i=1}^n t_i^{3/2}-\sum_{i=1}^n t_i^{1/2}z_i=0$$ $$s\sum_{i=1}^n t_i^{3/2} + k\sum_{i=1}^n t_i^{2}-\sum_{i=1}^n t_iz_i=0$$ So, two linear equations is $(s,k)$. Just solve them.