Techniques to show data spanning multiple decades

I have a scatter plot where each point is at integer coordinates that may include 0 for both X and Y. The range of each coordinate is large, but most of the data is clustered around 0.

Ordinarily, I would do something like a log-log plot to show the decades of data. But since there is 0, it's not ideal (I could add a shift, but that makes interpretation of the data more difficult). Additionally, since the data are integers, it looks very banded in log-log plots. Again, relatively unattactive.

An example of the data:

An example of the log-log data where each axis has a shift by 1 before taking the log:

So, is there another type of transformation that would display the data more reasonably? It's important to see all scales of the data.

• Who is the audience / what is the goal for the picture? – MattBagg Dec 10 '12 at 1:22
• @MattBagg This is actually data from physics.SE so let's assume it is people who understand data and can make sense of visualizations of data. The goal is to show interesting trends of the data across all decades. – tpg2114 Dec 10 '12 at 4:10

You could try one of the transformations to approximately constant variance for Poisson data such as $2\sqrt(Y+\frac{3}{8})$.