  In order to perform probabilistic tsunami hazard assessment (PTHA) based on
subduction zone earthquakes, it is necessary to start with a catalog of
possible future events along with the annual probability of occurance, or a
probability distribution of such events that can be easily sampled. For
nearfield events, the distribution of slip on the fault can have a significant
effect on the resulting tsunami. We present an approach to defining a
probability distribution based on subdividing the fault geometry into many
subfaults and prescribing a desired covariance matrix relating slip on one
subfault to slip on any other subfault. The eigenvalues and eigenvectors of
this matrix are then used to define a Karhunen-Lo\`eve expansion for random
slip patterns. This is similar to a spectral representation of random slip
based on Fourier series but conforms to a general fault geometry. We show that
only a few terms in this series are needed to represent the features of the
slip distribution that are most important in tsunami generation, first with a
simple one-dimensional example where slip varies only in the down-dip direction
and then on a portion of the Cascadia Subduction Zone.
