Mathematical Modeling

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.

There is often a combination of empirical and physical relationships which make up a given environmental model; however, it is the models which are primarily physically based which are the more adaptable. Moreover, physically-based models provide a means of synthesizing and integrating our current knowledge. Failure of physically-based models generally occurs because of our incomplete understanding of the real world. Model failure can lead to new and improved insights into how the real world functions.

Steps in Model Development and Use

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.

Model Plane Analogy

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