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<p>Society-induced changes to the environment are altering the effectiveness of existing
management strategies for sustaining natural and agricultural ecosystem productivity.
At the watershed scale, natural and agro-ecosystems represent complex spatiotemporal
stochastic processes. In time, they respond to random rainfall events, evapotranspiration
and other losses that are spatially variable because of heterogeneities in soil properties,
root distributions, topography, and other factors. To quantify the environmental impact
of anthropogenic activities, it is essential that we characterize the evolution of
space and time patterns of ecosystem fluxes (e.g., energy, water, and nutrients).
Such a characterization then provides a basis for assessing and managing future anthropogenic
risks to the sustainability of ecosystem productivity.</p><p>To characterize the space
and time evolution of watershed scale processes, this dissertation introduces a mean
field approach to watershed hydrology. Mean field theory (also known as self-consistent
field theory) is commonly used in statistical physics when modeling the space-time
behavior of complex systems. The mean field theory approximates a complex multi-component
system by considering a lumped (or average) effect of all individual components acting
on a single component. Thus, the many body problem is reduced to a one body problem.
For watershed hydrology, a mean field theory reduces the numerous point component
effects to more tractable watershed averages resulting in a consistent method for
linking the average watershed fluxes (evapotranspiration, runoff, etc.) to the local
fluxes at each point.</p><p>The starting point for this work is a general point description
of the soil moisture, rainfall, and runoff system. For this system, we find the joint
PDF that describes the temporal variability of the soil water, rainfall, and runoff
processes. Since this approach does not account for the spatial variability of runoff,
we introduce a probabilistic storage (ProStor) framework for constructing a lumped
(unit area) rainfall-runoff response from the spatial distribution of watershed storage.
This framework provides a basis for unifying and extending common event-based hydrology
models (e.g. Soil Conservation Service curve number (SCS-CN) method) with more modern
semi-distributed models (e.g. Variable Infiltration Capacity (VIC) model, the Probability
Distributed (PDM) model, and TOPMODEL). In each case, we obtain simple equations for
the fractions of the different source areas of runoff, the spatial variability of
runoff and soil moisture, and the average runoff value (i.e., the so-called runoff
curve). Finally, we link the temporal and spatial descriptions with a mean field approach
for watershed hydrology. By applying this mean field approach, we upscale the point
description with the spatial distribution of soil moisture and parameterize the numerous
local interactions related to lateral fluxes of soil water in terms of its average.
With this approach, we then derive PDFs that represent the space and time distribution
of soil water and associated watershed fluxes such as evapotranspiration and runoff.</p>
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