  Thermodynamic depth is an appealing but flawed structural complexity measure.
It depends on a set of macroscopic states for a system, but neither its
original introduction by Lloyd and Pagels nor any follow-up work has considered
how to select these states. Depth, therefore, is at root arbitrary.
Computational mechanics, an alternative approach to structural complexity,
provides a definition for a system's minimal, necessary causal states and a
procedure for finding them. We show that the rate of increase in thermodynamic
depth, or {\it dive}, is the system's reverse-time Shannon entropy rate, and so
depth only measures degrees of macroscopic randomness, not structure. To fix
this we redefine the depth in terms of the causal state
representation---$\epsilon$-machines---and show that this representation gives
the minimum dive consistent with accurate prediction. Thus, $\epsilon$-machines
are optimally shallow.
