  Temperature-dependent spin-Seebeck effect data on Pt|YIG
(Y$_3$Fe$_5$O$_{12}$)|GGG (Gd$_3$Ga$_5$O$_{12}$) are reported for YIG films of
various thicknesses. The effect is reported as a spin-Seebeck resistivity
(SSR), the inverse spin-Hall field divided by the heat flux, to circumvent
uncertainties about temperature gradients inside the films. The SSR is a
non-monotonic function of YIG thickness. A diffusive model for magnon transport
demonstrates how these data give evidence for the existence of two distinct
length scales in thermal spin transport, a spin diffusion length and a magnon
energy relaxation length.
