Capturing uncertainty in magnetospheric ultra-low frequency wave models

We develop and test an empirical model predicting ground-based observations of ultra-low frequency (ULF, 1-20 mHz) wave power across a range of frequencies, latitudes and magnetic local time sectors. This is parameterized by instantaneous solar wind speed $v_{sw}$, variance in proton number density $var(Np)$ and interplanetary southward magnetic field $B_z$. A probabilistic model of ULF wave power will allow us to address uncertainty in radial diffusion coefficients and therefore improve diffusion modeling of radial transport in Earth's outer radiation belt. Our model can be used in two ways to reproduce wave power; by sampling from conditional probability distribution functions or by using the mean (expectation) values. We derive a method for testing the quality of the parameterization and test the ability of the model to reproduce ULF wave power time series. Sampling is a better method for reproducing power over an extended time period as it retains the same overall distribution while mean values are better for predicting the power in a time series. The model predicts each hour in a time series better than the assumption that power persists from the preceding hour. Finally, we review other sources of diffusion coefficient uncertainty. Although this wave model is designed principally for the goal of improved radial diffusion coefficients to include in outer radiation belt diffusion based modeling, we anticipate that our model can also be used to investigate the occurrence of ULF waves throughout the magnetosphere and hence the physics of ULF wave generation and propagation