Yes, you can charge each battery independently from separate solar panels as long as each is electrically isolated.
However, it would make more sense to combine the panels and charge the whole 24 V battery as one. Lead-acid batteries are fairly forgiving, especially in that you can charge them with a current-limited voltage supply. A 12 V battery can be held indefinitely at about 13.6 V without harm. It will draw some charge current at that level, but this current won't do long term harm even if the battery is already fully charged.
A power supply that takes the solar panel output and makes the roughly 27.2 V output should work fine. The more cells in series, which in this case is 12 instead of the usual 6 of a "12 V" battery, the more you have to worry about cell imballance. You might therefore want to be a bit conservative with the float voltage and/or make the current limit lower than usual. The easiest way to do that is to size the solar panel such that it simply can't produce enough power to hurt the battery in full sunlight. Then the power supply only needs to try to produce about 27 V, but fold back as necessary limited by the power from the solar panel. This scheme will give up fast charge time, but if that is not a issue it's intrisically safe operation makes things easier.
Taking the question at face value, ALL the answers given are wrong [ :-) ] and your question and a subsequent comment do not match.
You say you have 4 x 12V batteries to make a 48V supply.
Your diagram implies (but does not state) that the panel is 24V rated.
In a comment you say
I don't think I know the PV array's knew voltage. I know it produces up to 18V in full sunlight though. – Wurlitzer Jul 10 '12 at 23:00
An 18V O/C panel is 12V rated, not 24V as per diagram.
Cheap and nasty solution:
Use a relay (or relays) to swap the panel between batteries. Either two positions if 24V panel or 4 positions if 12V panel.
A simple sequencer will be required to control the relays.
Very fast swapping rate reduces relay lifetime, and controller will take a time to settle so "dead time" % is higher.
Very slow swapping rate increases risk of imbalance.
If you swapped every minute or even every 5 minutes the long term differences will be minimal.
A good "real" solution is to use a 48V panel - or 2 x 24V or 4 x 12V panels in series.
Also suitable would be to use a 12V to 48V converter. Some MPPT controllers are available which will both up-convert the input voltage and also optimise loading to maximise panel power output.
Best Answer
Yes, that is perfectly reasonable, but also connect the 12V points across the two banks. So you have two banks connected in series, where each bank is a 200Ah and 150Ah in parallel. Also, if there is a chance things may get out of balance, you can put a protection circuit across each 12 bank (like ~16V zeners).
EDITED: Thought I better change the 15V suggestion to 16V. Just make sure a standard 14.4 volt charge (per 12V bank) doesn't cause notable zener current.