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How can I systematically search for caverns in Dwarf Fortress? Digging straight down can sometimes miss caverns, so on what levels should I start branching out horizontally to find the first, second, or third cavern layers?

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The advanced world gen parameters include some information on the number of z-levels between various underground features. Assuming that you haven't changed any of them (if you have use your values), the default is 5 z-levels above "Layer 1" which (according to the wiki) is the first cavern layer. This doesn't necessarily mean that the cavern is located 5 z-levels below the ground everywhere. As I understand it this is the distance between the top of the possible cavern and the lowest possible part of the ground in that area. I don't believe that there is a guarantee that the lowest actual ground level will be the lowest possible one, and caverns have funky organic shapes so this is merely the minimum distance between the surface and the cavern. Also this "ground level" can change over various parts of a fortress map.

I myself usually set the z-levels above the first cavern layer to a larger number like 12 (I like lots of room), and will often have to dig down 14 or 15 levels before I find the cavern. Since the caverns are often reasonably tall I would, with the stock 5 z-levels above layer 1, dig down at least 7 or 8 z-levels before expanding sideways. After all it's not the end of the world if you breach a cavern layer below its highest point. Also don't try too hard on those first branching levels, since you might still be above the cavern layer.

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It is highly variable where the caverns may be found. On lower grounds the magma sea can already be on 30Z levels down. On high mountainous sites it can be 200Z levels. Somewhere in between are the caverns generated. In the advanced world gen parameters you can set how much levels between each cavern, the top and the magma sea should be. As well as how much open space inside the caverns.

Without using dfhack reveal, to reveal the map you could try penetrate each 32 tiles with a stairway. Dig down 10 levels for each point and repeat that until you breach something. Once you are down 30 to 40 Z levels you could divide it by 2 and dig down each 16 tiles. You really should have found a cavern by now or you are very "unlucky". For the latter, keep doing this until you breach a cavern.

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