How are Volumes Calculated?

Volume calculation is one of the optional features of EnviroInsite's 2D and 3D contouring routines. To enable the volume calculation feature, check the Estimate Volume box on the Options tab of the 3D Contour dialog box. The program will report the volume within each contour interval and the average concentration value within that interval. These values may be converted to mass by taking the product of the porosity, the average concentration, and the volume. Do not forget to account for adsorption to grain surfaces.

EnviroInsite uses a Monte Carlo integration method for volume estimation. The following links provide some information on Monte Carlo integration:

http://mathworld.wolfram.com/MonteCarloIntegration.html

https://en.wikipedia.org/wiki/Monte_Carlo_integration

The following is a quick summary of the implementation of the method in EnviroInsite. First, a specified number of sample points are randomly assigned locations within the contour grid. Then, the value at each point is estimated based on the same interpolation method specified for the contouring procedure. Finally, the volume associated with each point (i.e. the grid volume divided by the number of points) is tallied and added to the total volume. This procedure is repeated over multiple iterations, where the number of sample points is increased by the number of initial points between iterations. The progress towards convergence is documented by reporting the interval volumes and average values after each iteration.

A step-by-step description of the method is as follows:

1.Set the number of points to be sampled.

2.Calculate the volume associated with each point as V_i = (Total Grid Volume) / (Number of Points).

3.Randomly assign the point locations within the 3D grid used to generate the contours using a uniform distribution.

4.Determine the concentration at each point by interpolating from the measure values.

5.Calculate the volume within each interval by taking the product of the number of points within the interval and the volume associated with each point as calculated in Step 2.

6.Calculate the average concentration within each interval.

7.Report the result for the give number of sample points.

8.Exit if the number of intervals has been reached.

9.Double the number of points to be sampled and return to Step 2.