# 12 F-h${}^{2}$ analysis

The analysis of $F$ vs $h^{2}$ follows [10, 9, 8].

## 12.1 Window

The window consists of several blocks:

• Parameters allows the user to set the width of the moving average window. The value 1 corresponds to no smoothing. This value is saved in settings.

• Run perform calculation and display curve, see section 12.2.

• Save save parameters and results to given file.

• Graph

• Bottom: display the $F/h^{2}$ and the $\mathrm{d}F/\mathrm{d}h^{2}$ curves.

Stepwise zooming/unzooming can be performed by selecting a range with the mouse and pressing the Zoom/ Unzoom buttons. The graph is restored to its original size by the Restore button. Zooming in the two graphs is independent.

## 12.2 Procedure

1. 1.

We use a moving average with a fixed width and constant weight. This means we substitute a value with its average with $s$ values to the left and to the right, $w=2s+1$

 $\hat{x}_{i}=\frac{1}{w}\sum_{j=-s}^{s}x_{i+j}.$

The value $w=1$ corresponds to the original data. There is only one moving average type defined for both depth and load. Increasing the value of $w$ noise becomes less influential but important small effects can get lost as well. Therefore, the value should not be too large, below 11 is recommended.

2. 2.

The ratio $F/h^{2}$ is calculated for for each data pair $(\hat{h},\hat{F})$ of the (smoothed) loading curve and plotted as a function of the (smoothed) depth $\hat{h}$.

3. 3.

The derivative $\mathrm{d}F/\mathrm{d}h^{2}$ is calculated for for each data pair $(\hat{h},\hat{F})$ of the (smoothed) loading curve and plotted as a function of the (smoothed) depth $\hat{h}$. The derivative is done numerically as the ratio of the derivatives of $F$ and $h$ with respect to the (time)step or index $i$

 $\frac{\mathrm{d}F}{\mathrm{d}h^{2}}=\frac{\mathrm{d}F}{\mathrm{d}i}\left(\frac% {\mathrm{d}h^{2}}{\mathrm{d}i}\right)^{-1}.$

The numerical derivatives are calculated using the three-point formula for equally spaced data.