Yes it should be under the asphalt shingles.
Here is an example; http://www.fema.gov/pdf/rebuild/mat/fema499/hgcc_fact19.pdf
Here is Owens Cornings installation instructions for their standard 3 tab;
It includes a recommendation for 15 lb felt.
Notice it states that the underlayment (roofing felt) is required to meet the UL fire rating.
About three years ago I inspected a home that I found had no roofing felt, and the 3 tab shingles had been replaced recently. The homeowner was also a contractor, who stated that he never installed felt on any homes that he roofed. :shock:
So we debated back and forth, and finally I just called up the local AHJ, and they said there were absolutely no exceptions to having the roofing felt. It was required by their standards.
Your local building official may see it otherwise.
The felt adds a back up layer of protection if the roof shingles leak. Just like house wrap under siding (when properly installed)
…Hey at least the roof sheathing in your picture is plywood…and not OSB (was wood)