What do High-Water Marks in Continued Fractions mean? While reading through several articles concerned with mathematical constants, I kept on finding things like this:

The continued fraction for $\mu$(Soldner's Constant) is given by $\left[1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...\right]$. 
The high-water marks are 1, 2, 4, 47, 99, 294, 527, 616, 1152, ... ,
  which occur at positions 1, 2, 3, 12, 70, 126, 202, 585, 1592, ... . 

(copied from here)
I didn't find a definition of high-water marks in the web, so I assume that it's a listing of increasing largest integers, while going through the continued fraction expansion.
Is this correct and is there special meaing behind them?
 A: Your definition seems correct to me -- at least, it agrees with the data you've provided and with my intuition, as a native speaker of English, of how this phrase is used.  I don't know of a specific significance of these high-water marks, but it's well-known that cutting off a continued fraction expansion just before a particularly large coefficient gives a good rational approximation to the number being expanded. For example the sequence of convergents of π begins 3, 7, 15, 1, 292; the continued fraction [3, 7, 15, 1] is the well-known, surprisingly good approximation 355/113.
A: Truncating a continued fraction right before a "high-water" (large) partial quotient $\rm\: a_{i+1}\:$ yields a particularly good rational approximation since 
$$\rm r\ =\ [a_0;\ a_1;\ a_2;\ \cdots\ ]\ \ \Rightarrow\ \ \left|\ r\ -\ \frac{p_i}{q_i}\:\right|\  \le\: \frac{1}{a_{i+1}\:q_i^2}$$
For example
$$\rm \pi\ =\ [3;\ 7;\ 15;\ 1;\ 292;\ \cdots\ ]\ \ \Rightarrow \ \ \left|\ \pi - \frac{355}{113}\:\right|\ \le\: \frac{1}{292\cdot 113^2}\: =\ 2.68\cdot 10^{-7}$$  
In fact we have $\rm\quad \pi\ -\ \dfrac{355}{113}\ =\ {-}2.67\cdot 10^{-7}\:.$
A: The term comes from the common practice of marking bridges or walls with the water level and date during big floods.  So, you can look at these high-water marks to see that in 1985, the river flooded to 5 feet above its current level.  If you come back a decade later, you may find that a new high-water mark has surpassed it.

I haven't seen the term "high-water marks" for continued fractions before (but I like the name).  
I'm sure your interpretation is correct: the $n$th coefficient $a_n$ is a high-water mark if $a_n>a_i$ for $i<n$.  
(Maybe it also qualifies if $a_n \geq a_i$ for $i<n$?  You would have to find a definition or example to settle that question for sure.)
One significance of a high-water mark at position $n$ is that the $(n-1)$st and $n$th convergents will be big improvements in accuracy.  For example, $\pi$ is given by [3;7,15,1,292,...]. In the step before including 292, you go from convergent 333/106=3.14150... to 355/113=3.1415929...  When you include 292, you get 103993/33102=3.1415926530..., gaining 2 or 3 more accurate digits each time.
