  Turbulent flows over a large surface area (S) covered by n obstacles
experience an overall drag due to the presence of the ground and the protruding
obstacles into the flow. The drag partition between the roughness obstacles and
the ground is analyzed using an analytical model proposed by Raupach (1992) and
is hereafter referred to as R92. The R92 is based on the premise that the wake
behind an isolated roughness element can be described by a shelter area A and a
shelter volume V. The individual sizes of A and V without any interference from
other obstacles can be determined from scaling analysis for the spread of
wakes. To upscale from an individual roughness element to n/S elements where
wakes may interact, R92 adopted a background stress re-normalizing instead of
reducing A or V with each element addition. This work shows that R92's approach
only converges to a linear reduction in A and V for small n/S where wakes have
low probability of interacting with one another. This probabilistic nature
suggests that up-scaling from individual to multiple roughness elements can be
re-formulated using stochastic averaging methods proposed here. The two
approaches are shown to recover R92 under plausible conditions. Comparisons
between R92 and available data on blocks and vegetation-like roughness elements
confirm the practical utility of R92 and its potential use in large-scale
models provided the relevant parameters accommodate certain features of the
roughness element type (cube versus vegetation-like) and, to a lesser extent,
their configuration throughout S.
