General considerations
The objective is to limit the range of values that are permitted for each 25m x 25m x 25m cell in a model based upon values measured along drill-core. Core diameters are typically 5cm or 6cm, so such small scale information obviously cannot directly represented the expected value of a 15625 m³ cell. In fact, the drill-hole samples only about three parts-per-million of the model cell. There are two ways of proceding:

1. we could reduce the size of the cells, or
2. we could try to scale up the drill hole information.

Wherever possible the first option is preferable because then there would be less inconsistency between information and model. Unfortunately, the size of the model being used at San Nicolas is already large (close to half a million cells) so the second option was used.

Another complication arises because we are only sampling the cell at one position in x and y. Vertical variations within the cell at one horizontal location are well sample, but lateral variations within the cell are unknown. We might expect horizontal inhomogeneities to exist that could increase the range of values that the cell might have to represent.

Choosing upper and lower bounds
To constrain a model we require upper and lower bounds for the values of each model cell. One simple approach would be to assign the minimum and maximum values observed from the drill hole that passed through the cell. However the cell requires a single value throughout so it is unlikely to exhibit either of these extreme values. A cell value closer to the mean is more probable.

The approach taken was to find a mean value of those core measurements within one cell. Upper and lower bounds were chosen as some variation either side of this mean. The choice of variation could be based on the standard deviation of the measurements, however, in that case the uncertainty of the result would depend upon the number samples used. For independent random variables, the uncertainly associated with the mean would be the standard deviation divided by the square-root of the number of samples.

Bounds on density
Density was sampled in drill-core every 1.5m. These core measurements are absolute density values, while the model values are density contrasts relative to an unknown background value. Considering the volcanic host rocks, a background value of between 2.35 g/cm³ and 2.5 g/cm³ is reasonable, and trial inversions were run using both values.

At San Nicolas, inversions were run with bounds on the values of cells that were sampled by a drill hole. Bounds ranged from +-0.05 g/cm³ to +-0.2 g/cm³ on either side of the measured mean density. If the core-measurements only populated half of a cell's height, the variation from the mean was doubled. For cells with no core-measurements, upper and lower density-contrast bounds of 5 g/cm³ and 2 g/cm³ respectively were assigned. In fact it is hard to accurately determine suitable bounds, and this is one area that would benefit from further research.

Bounds on susceptibility
Susceptibility measurements on core were used to set upper and lower bounds on model cell values for inversion of magnetic data in much the same way as density was bounded for gravity inversions. Many susceptibility readings on core were used to estimate one mean value for each model cell. Unlike the density measurements no background level needed to removed. Also, the fact that samples were not evenly collected along the core had to be taken into account when calculating the composite susceptibility value to use for each cell (see the discussion on physical properties).

Upper and lower bounds for cells sampled with drill holes were centered around the composite susceptibility value. Most of the core susceptibility values were around 0.1 x 10^-3 S.I. with some values as high as 100 x 10^-3. The value that was chosen by which to vary the bounds was 5x10^-4.

Including bounds as constraints in the inversion
A brief discussion of how imposing bounds is incorporated into the model objective function is provided on a separate page, as well as in section 4.3 of the "Complete Tutorial" section of the Inversion for Applied Geophysics CD-ROM (citation).

For inversions carried out in this work, the same noise levels, reference model, and choice of regularization parameter were used as for inversions that did not include bounding constraints.