6.2 Homogeneous Probability Distributions

This appendix is reproduced from Mosegaard and Tarantola, 2002.

In some parameter spaces, there is an obvious definition of distance between points, and therefore of volume. For instance, in the 3D Euclidean space, the distance between two points is just the Euclidean distance (which is invariant under translations and rotations). Should we choose to parameterize the position of a point by its Cartesian coordinates {x, y, z}, the volume element in the space would be dV(x, y, z)=dx dy dz, while if we chose to use geographical coordinates, the volume element would be dV(r, ?, ?)= r ² sin ?dr d d ?.

Definition. The homogeneous probability distribution is the probability distribution that assigns to each region of the space a probability proportional to the volume of the region.

Then, which probability density represents such a homogeneous probability distribution? Let us give the answer in three steps.

• If we use Cartesian coordinates {x, y, z} , as we have dV(x, y, z)=dx dy dz , the probability density representing the homogeneous probability distribution is constant: f(x, y, z)=k.

• If we use geographical coordinates {r, ?, ?}, as we have dV(r, ?, ?)=r ² sin ? dr d ? d ? , the probability density representing the homogeneous probability distribution is g(r, ?, ?)=kr ² sin ? .

• Finally, if we...