Error

whitson⁺ facilitates regression on parameters to minimize the sum of squares of weighted residuals in the context of observed data and corresponding predictions. The objective function in is defined as:

$$\chi^2 = \sum_{i=1}^{N} (r_i w_i)^2$$

Here, $r_i$ represents the RMS residual error between observed measurement $i$ and its corresponding prediction, while $w_i$ is the user-assigned weighting factor for that residual. Ideally, these weighting factors should be inversely proportional to the standard deviations of the residuals. Minimizing $\chi^2$ in provides the maximum likelihood estimation of the model parameters, assuming independent and normally distributed residuals.

The default weighting factors are 1, while they can also be set to 0 (no importance), and 10 (high importance). Reported is a root-mean-square (RMS) residual error ($E_{\text{RMS}}$) defined as:

$$E_{\text{RMS}} = \sqrt{\frac{\chi^2}{\sum_{i=1}^{N} (w_i)^2}}$$

This metric is related to $\chi^2$ but is more easily interpreted.

The residuals are calculated as relative percentages using the formula:

$$E_i = \frac{100(y_{i,\text{hist}} - y_{i,\text{sim}})}{y_{i,\text{ref}}}$$

where $y_{i,\text{hist}}$ is the historical production value, $y_{i,\text{sim}}$ is the simulated value, and $y_{i,\text{ref}}$ is the reference value for observation $i$. Historical production values that are 0 are excluded from the error calculation, while historical production values that have a higher error than 100% is set to 100% error (to avoid that individual outliers dominate the whole error calculation). A failed numerical run is attributed an error of 101%, ensuring that successful cases are always favored.