  We study the finite size fluctuations at the depinning transition for a
one-dimensional elastic interface of size $L$ displacing in a disordered medium
of transverse size $M=k L^\zeta$ with periodic boundary conditions, where
$\zeta$ is the depinning roughness exponent and $k$ is a finite aspect ratio
parameter. We focus on the crossover from the infinitely narrow ($k\to 0$) to
the infinitely wide ($k\to \infty$) medium. We find that at the thermodynamic
limit both the value of the critical force and the precise behavior of the
velocity-force characteristics are {\it unique} and $k$-independent. We also
show that the finite size fluctuations of the critical force (bias and
variance) as well as the global width of the interface cross over from a
power-law to a logarithm as a function of $k$. Our results are relevant for
understanding anisotropic size-effects in force-driven and velocity-driven
interfaces.
