Outdoor baths are generally called 露天風呂 (roten-buro, open air bath) as opposed to 温泉 (onsen, hot spring). Many onsen are also roten-buro. Some roten-buro heat their water by burning wood, coal, or natural gas. However, all onsen derive their hot water from geothermal sources.
In the case of the above bath, it is described as an onsen in the accompanying text. The water source on the far right agrees with this fact; the geothermal water flows from it constantly, overflowing the bath. Water heated by a fuel source would not be wasted in this manner. Additionally, a fuel heated outdoor bath would be small (similar to the one pictured) and covered when not in use to limit heat loss.
