Okay increasing the saved image file 'depth' is not a problem. Just add
-depth 16 just before the fine image save.
This however does not change the image depth or quality, while the image is in memory! That is a compile time setting. So increasing the color value quality is already done for you.
http://www.imagemagick.org/Usage/basics/#depth
ASIDE: be careful with the height map images. These are typically saved without any 'colorspace correction'. that is they are saved using a 'linear RGB' colorspace. in IM v6 this is the default, so you chould have no problems, but IM v7 and other image processing programs will typically automatically think of a saved image as using non-linear sRGB colorspace. Caution and testing is recommended.
For details about why images are typically saved using a non-linear colorspace corrected form see...
http://www.imagemagick.org/Usage/color_ ... perception
To the problem at hand...
The problem is that your hight map as been mangled, so as to produce 'staircased values'.
For an example of this see this example where a linear gradient was compressed to 4 bit depth/quality
http://www.imagemagick.org/Usage/basics ... um_effects
Now assuming that the height map is smooth, a blur should produce the right result. The problem is that if the image is not smooth, that is you have sudden 'jumps' or edges (cliffs in terms of a height map) then you need to try and preserve the sharpness of those edges.
This sounds more like adaptive-blur or possibly LAT image processing. IM has these, but I have not played with them much. Try out these local area operators, and let us know how they with before and after images, and the command.
The other suggestion I have is to find areas which are 'flat', and clear them (edge detect). You can then 'hole-fill' those flat areas to interpolate the values. See one example of hole filling images.
Sparse Color Shepards, a Blur Alternative
http://www.imagemagick.org/Usage/canvas/#sparse_blur
If you have a small example image I can experiment with it to see what I can do.