Accurately reproducing this effect was an interesting challenge.
The right hand image attached below is about as close as I could come to the original (on the LHS) without explicitly introducing shading inside each circle (as suggested by
lizzader ).
I followed Sam's approach with respect to the order of layers and using the multiply layer blend mode (with each layer near full opacity), but differed in that I did the simulation in the Lab color space, and did some minor tweaking using Topaz Adjust as the last step before conversion back to sRGB for posting. Working in Lab produced a major improvement in simulation accuracy compared to working in sRGB. I have a sneaking suspicion that working in CMYK might even be better, but, to be honest, I don't feel like putting any more effort in this little exercise.
BTW, to avoid the difficulty of visually comparing images of two different sizes and/or with the circles located differently, I did all my work at 180px image size, and tried to place the circles very close to the originals. You'll notice that my attempt doesn't have any JPG compression artifacts, whereas the original does. In such a small image, these may actually visually seem to smear things out a bit and my help with the "smoothness" sought by the OP, but I didn't bother to add in any fake JPG artifacts to mine to test this.
Hopefully, the OP will return to discuss this more -- perhaps giving some context to this little exercise.
Cheers,
Tom M