I have seen most of papers comparing modis aod with sun photometers aod at ground, instead of that is there any other alternative available to compare Modis L3 AOD with PM10, 2.5 mass at ground level.
I would recommend to use MODIS L2 AOD at 10 km spatial resolution to find out relationship with ground PM2.5. You can extract AOD from MODIS using spatial window of 3x3 pixels (average of 9 pixels) over the location of PM2.5 measurement sites, and then you can find out the correlation between AOD and PM2.5.
I came across a few articles, searching for sources of a small but interesting PM event in Flanders, Belgium in July last year: "Detecting source of unexpected peak of PM10 over Flanders (Belgium): a walk through public available information". See my article on researchgate and look for references 20, 21, 22, 23, 24 and 25. You may find different correlations of course, depending on differences in the aerosols, and within the uncertainties of the methods. http://tinyurl.com/pnxlyks
There is no direct relation between PM and AOD. But you can try to find the empirical relations between these two parameter as i have done in my recent publication. I am attached herewith that research article for your reference.
PM10 and PM2.5 are measured close to the surface, near ground level; while AOD is a columnar parameter (integrated through the vertical atmospheric column). PM depends on the mass; while AOD is related to optical property (like extinction) and is a strongly surface area dependent parameter. The two could be related in vertically homogeneous aerosol distribution and under some condtions; yet need not necessarily be in a simple way. The relation could be site-specific and would have large temporal changes. When there is vertical homogeneity in aerosol distribution (mass, type, extinction etc) or even humidity distribution, the relation would increasingly become poor. For example, the same mass concentration of sulphate, dust and BC would produce entirely different AODs and with strongly differing spectral dependency. This would change further, under different humidity conditions. A crude analogy would be trying to relate the height of a person with the girth of the wrist .
We should completely agree with Krishma Moorthy. A comparison of "optical concentration" with "gravimetric concentration" is not easy nor immediate since light scattering and absorption depends upon size, refractive index, and so on. When you add to such uncertainty the one caused by a different distribution of particulate matter with height, than you easily get into a hopeless matter. What we can say is that when AOD is high, PM would be high too, but the final correlation will be very poor. Since PM10 or PM2,5 have legal significance, any trial to correlate it to AOD will be hopeless and not useful at all.
There may be a suitable approach to correlate AOD with PM2.5 according to a recent review: Yuanyuan Chu et al., "A Review on Predicting Ground PM2.5 Concentration Using Satellite Aerosol Optical Depth", Atmosphere 2016, 7(10), 129; doi:10.3390/atmos7100129 (http://www.mdpi.com/2073-4433/7/10/129). It's open access. 116 studies are explored, check carefully the different approaches