Either Paint Shop Pro or Photoshop will do the job. Personally, I prefer Paint Shop Pro to Adobe's cumbersome, unconventional interfaces.
1. Edge preseriving smooth. This filter smooths the noise while at the same time, identifying the edges in the image and not blurring them to oblivion. Without this smoothing process, the later contrast enhancements multiply the noise to a point where its too distracting to discern any features. This filter isnt perfect of course, and rain wrapped storm bases don't have razor sharp edges, so you will still lose some fine detail.
2. Contrast Enhancement. This can be done manually with curves or with an automatic filter. Mike H is a master of modifying the curves to really make an image "pop", but I just ran it through the automatic process on this one for reasons of time. You'll want to adjust the settings so that the area of interest is in the center of the greyscale range, and so that the brightest area of the image is almost white, and the darkest area almost black (for analysis purposes, not for show).
3. Paint Shop Pro's clarify effect. What this feature does is locally enhance the contrast as opposed to the contrast of the entire image. This is extremely handy for bringing out features in all areas of the image without oversaturating or wiping out entire sections. If you have yourself a rain wrapped tornado, and the camera is able to barely discern the edges, this filter will bring it right out. It doesn't look pretty at the harsh levels I took these images to, but its great for analysis purposes.