Mathematical models are a way to represent
"reality" through the development of mathematical
relationships. This conceptualization of reality
has been attempted for things such as the modeling
of stock market behavior, population growth, weather
forcasting, etc., and can be applied (with varying
success) to any process which can be decribed by
a mathematical expression.

However, models are not the "real thing" and always
involve some degree of simplification and
approximation. Exclusion of processes or elements known
to be functioning in a system is often excused by assumptions
that these things have negligible or insignificant effects
on the system as a whole. The development of mathematical
models also involves the assumption that people understand
the various processes well enough to be able to translate
system behavior into mathematical expressions.

Because models are developed from mathematical expressions,
there are parts of the equation which must be known (input), and
parts which are being solved (output). *Parameters* are the
elements which the user must input into the model and can
be universal constants, measurements, etc. To *parameterize
* a model often means choosing input which
will best describe the given system. *Variables* are the
elements which the model calculates and turns into output.
It is possible for a variable to become a parameter when
feedback cycles are included in the model.

Hydrologic models are simply a variation on this
theme. Mathematical expressions are developed by
scientists to describe the relationships between
various hydrologic parameters and variables.
One could argue that the water balance equation, which
is based on the conservation of mass principle, is itself a
hydrologic model. Yet, although this equation
is at the heart of many hydrologic models, we often
associate "modeling" with more complex mathematical
equations which require computer processing capabilities
to solve them.

There are basically two classes of environmental models:

- Emprically based
- Empirically based models are built to fit a very specific situation and are tailored with system specific relationships, parameters, and variables.

- Physically based
- Physically (or process) based models are built using more universal relationships, i.e. physical laws.

- Problem identification, eg rate of snowmelt
- Conceptual development
- Discretization into fundamental elements
- What are the processes involved?; or
- What are the empirical relationships involved?

- Formulate; develop equations
- Put all the formulas together into a model, eg SNTHERM
- Calibrate the model; solve the formulas using a data set
- Verifty the model: Test the model in the real world.
- Evaluation of model results:
- Reject the model; or
- Accept the model; or
- Modify the model based on new insights.
- Most models use an iterative approach, which means that they are continually refined (changed and hopefully improved) with use.

- Problem identification: how to fly, improve flight performance
- Conceptual development: build a glider
- Discretization: physics of flight
- Formulate: blueprints for glider based on physics of flying
- Put everything together: build the model
- Calibrate: test the glider in the lab
- Verification: fly the glider outside
- Model sensitivity: move wings forward, backwards, etc.

Results of physically based model: "Chaotic Advection":

Contact Mark Williams at **markw@snobear.colorado.edu** with questions or comments.

This page was constructed by:

Mindia Brown