# Picking a probability distribution for observed intensities

I have an experiment that measures "intensity" (in this case, electron density of a molecule) on a grid. The values it gives are non-negative,. I'd like to write a likelihood for this observation given a model calculated on the same grid, assuming the grid measurements are independent.

Since the values are non-negative, and I would like to be somewhat tolerant of large errors, I originally thought I'd choose a lognormal distribution (here for a single data point):

$$L(I_{exp}|I_{mod})=\frac{1}{\sqrt{2\pi}\sigma I_{exp}}\exp{\left[-\frac{\log^2(I_{exp}/I_{mod})}{2\sigma^2}\right]}$$

Here's the problem: the data values $I_{exp}$ are sometimes zero, because the experimental device cannot go below some degree of precision. But the lognormal goes to $0$ as $I_{exp}\rightarrow0$ So at the low values it might be more accurate to pick a truncated normal distribution.

Are there any distributions that 1) have non-zero probability at 0 and 2) are more tolerant of large errors than a normal pdf?