needs to be developed to address model selection in this and other more complex
models.
3.7 Dynamic spatio-temporal models
In this report, we emphasize the integration of diverse data sources, process-
based or theoretical models, and parameter models, which include parameters arising
by embedding the process models in a stochastic framework, and quantities (e.g.,
coefficients) directly incorporated in the process models. The process models are where
the science resides, and their development is facilitated expert knowledge and existing
information. Within a probabilistic framework that combines available data sources and
the process models, the process model(s) may be viewed as describing the underlying
latent quantities that we would like to have observed (e.g., the “true” process), but
cannot directly observe due to, for example, measurement error, instrument inaccuracy,
instrument drift, or other sources of observation uncertainty. In some cases, the process
model may be viewed as yielding the true process exactly, without error, but in many
cases, since no model is perfectly correct, the process model may be viewed as
describing the expected process. The true, latent process would be given by the
“expected process plus process error” (see Section 3.1). In the context of understanding,
quantifying, and forecasting elements of sustainability, we suggest that such process
models must be able to accommodate spatial and temporal dimensions.
The evaluation of sustainability involves comparisons between base-line (or
current) quantities and predicted (future) quantities, and measures of change will be key
to defining and evaluating metrics of sustainability. That is, change with respect to space
and/or time (e.g., as may be quantified by analytical or numerical derivatives), and thus
divergence from the baseline(s), will be critical to evaluating the degree to which
sustainability has been achieved or not. Thus, the underlying process models must be
able to accommodate temporal dynamics to obtain predictions and/or to evaluate rates
of change. Of course, these quantities may vary over space due to, for example,
heterogeneity in drivers or initial conditions (e.g., land-use, climatic conditions,
population density). This implies that dynamic, spatio-temporal models (Cressie and
Wikle, 2011) ⎯such as might be encapsulated by partial differential equations (PDEs) or
stochastic PDEs, but certainly not limited to these types of models⎯are critical for
advancing sustainability science, and the mathematical sciences can contribute greatly
to the development and evaluation of such models. There are many existing spatio-
temporal process models that may be useful for quantifying and forecasting
sustainability in different contexts, including models of biogeochemical cycles (e.g.,
transformations, storage, and fluxes of elements such as carbon and nitrogen),
hydrological processes (e.g., river flow, flooding, groundwater movement and
extraction), atmospheric chemistry (e.g., production, degradation, distribution, and
concentration of pollutants), climate (e.g., temperature, precipitation, cloud formation),
epidemiology (e.g., spread of infectious diseases, vaccination strategies), population
dynamics (e.g., animals, plants, or humans), and economics (e.g., financial stability,
price indices).
None of the above modeling examples are particularly new, and some are often
only applied to quantify temporal dynamics (e.g., many may lack an explicit spatial
component). What will be important for advancing our understanding and ability to
quantify sustainability is the coupling of multiple spatial-temporal models, and the