I need to determine the 12VDC Ah capacity of a battery pack, to provide current to a 30A 120VAC circuit thru an inverter with 88% efficiency, where the voltage must not drop below 10vdc, over a period of 3hrs. This is the worst possible scenario. I would expect that the current of 30A would begin to ramp down to 20A after the first 15 min., then hold steady at 20 amps for the remainder of the three hours. As a bit of a buffer, perhaps 80% efficiency should be used for the performance of the inverter. The battery pack will be in a well ventilated position to control fumes and temperature, as will the inverter. I would expect to use deep cycle batteries, but am unsure of size this battery pack will need to be.
$$P = V \cdot I = 120 V \cdot 30 A = 3,600 W = 3.6 kW$$
You're looking for a 4 kW UPS with 3-hour uptime. (I've ignored the reduction to 20 A to give some margin for deterioration of batteries over time.)
Have a look at readymade UPS solutions before you try to roll your own. You'll find all sorts of benefits from reliability, self monitoring, alarms, automatic mains switchover (if mains is available), stable frequency and voltage, etc. As the UPS manufacturers purchase batteries in bulk you may find a readymade solution at lower cost than purchasing components and building your own.
30 Amps at 120 V will be about 340 amps at 12 volts (allowing for 88% inverter efficiency), or about 1000 Ah over 3 hours, but for good battery life, it is generally recommended that you not discharge the battery more than 50%, which would mean you'd need about 2000 Ah of battery capacity.
20 Amps at 120 volts is about 227 A at 12 volts, or 680 Ah over 3 hours.
If you're willing to torture the batteries a bit, you can probably get away with 1200 - 1400 Ah.