When you crop, and display/print the same size , the apparent softness increases (so called COC - a term that I hate). This decreases the DOF.
I see two actions: one is croping, another is resizing to get the same image size. Which of those is responsible for softness change?
Both, together. Just cropping without resizing doesn't change anything, you will just see a smaller portion of the image.
What actually changes the DOF is the
enlargement . Cropping and resizing results in more enlargement, but there are other ways of increasing the enlargement (like simply increasing viewing size, zooming).
When you enlarge an image, all the optical imperfections become more visible. Everything appears less sharp, and a part of what previously appeared as acceptably sharp will now go past the threshold into softness.
It's looks like a loop: method of measuring dof require to do something that changes this parameter.
There are 3 parameters : viewing size, enlargement and visual acuity. All three have been melted into one, the CoC. Of course, to do that some assumptions are made on the viewing distance and on the average visual acuity. When in some situation one of the 3 parameters differs significantly from the assumption (e.g. viewer is blind as a bat) the CoC would need to be changed accordingly (in this example, increased, because a person with poor eyesight will not notice a much larger degree of softness in the picture).
For average situations, the CoC's have already been computed, they are here:
http://www.dofmaster.com/digital_coc.html
You can see the CoC doesn't change with pixel count, but it changes with format, the more a format is cropped, the bigger the enlargement, hence the smaller the Coc.
--
Kind regards,
Bogdan