What is a model and what is modelling

(The only person who believes a model is the modeler, while the only person who doesn’t believe the data is the data collector)

When the term modeling is used casually, what most people are thinking about is the nuts and bolts of taking a well-defined problem, applying some fancy numerical technique and arriving at an elegant solution with pretty graphics. There are many available “canned” models out there that will do this for you.

Unfortunately, in the Geo-Sciences, most of our problems are not well-defined, many of the realities we would like to understand are very large and complex, which is the nature of our field and of research science. As a result, before we can model in this casual sense, we need to take several underappreciated steps and may not ever get to this nice neat solution.

Making a model. The first step is always to simplify and abstract any real scientific question, and then ask several fundamental questions (I refer to this as problem abstraction):

· Do we understand the problem enough to simplify it?

· What level of abstraction will lead to a solvable AND useful problem?

· Is our available data sufficient?

· What solution technique to use?

· Can we validate the result?

· What is the cost (time and effort) vs benefit (confidence in the result)?

I can’t overestimate the value and necessity of thinking about this abstraction. Your time spent early in this process will pay huge benefits. Thus it is worthwhile taking a little time to investigate the abstraction process that we need to apply to a geosciences problem, before we can really start to “model” (what I am really trying to point out, is that in the geosciences, a large part of the ‘art’ of modeling occurs before we get to page 1 in most books on “modeling”). To make this discussion more concrete, consider the physics problem of a weight on a spring, and the geosciences problem of isotacy of a mountain on a continent. Abstracting the physics problem is relatively straightforward, and without too much thinking or work, we can reduce or abstract the physics problem down to a single simple differential equation. We could then model the abstracted equation using the panoply of numerical or other modeling methods. Our model (our abstraction and numerical model) will not be exact. After all, maybe we should have included air friction, or maybe we didn’t get the balls weight accurately. However; in this well defined problem we have some right to expect reasonable accuracy, and we will learn a lot if our model fails! However, modeling isotacy is a qualitatively different exercise: the list of abstractions is infinitely long, and we probably don’t even know what parts of the problem we can leave out or include. We don’t know if we can include or leave out: thermal density effects, flexural rigidity of the crust, non-linear rheology, phase/melt effects under the mountain, mantle flow, plumes, the stress state in the lithosphere, past history etc etc. This difficulty in most geosciences problems needs to be recognized, and I have found most students vastly underestimate the difficulty of this initial abstraction step, or the huge level of understanding that is required to make a viable abstraction.

Complex problems and complex models. For many problems we can get a scientifically useful answer by ‘back of the envelope’ calculations, if we are smart enough to abstract the problem into simple enough terms. I urge you to spend enough time in the abstracting process to try to search for simple cases and solutions, they are hugely valuable in learning about your problem (the book by Turcotte and Schubert is a classic illustration of this approach). At the other level of abstraction, we can end up with a problem that is very hard and complex to solve. As computer power increases with time, there is a growing tendency to model complex abstractions. Although tempting, this is a dangerous route since the number of errors in a model grows with the complexity. As a result, large complex abstractions lead to large complex modeling exercises that require large complex debugging and validation exercises.

Some meta-thoughts on modeling. At some meta-level, most of what we experience is a model. After all, we each construct our own version of reality, and modeling is essentially the process of abstracting (mimicking or simplifying) a reality that is of interest to us, but can’t be directly manipulated. If we directly manipulate the reality, then we perform an experiment, if we abstract the reality in some way, we create a model and we are then modeling. So I define a model as something that abstracts (mimics a simplified) reality and allows us to manipulate inputs, outputs and transfer functions that convert inputs into outputs. You can make virtually any type of model, from physical models, to analog models, to numerical models to whatever you feel like. And of course you can model just about anything.

In this class: we are going to focus on abstractions that lead to problems that are somewhat more difficult to solve than ‘back of the envelope’ problems, but which are not so complex that we can’t validate our results (to some level). In other words, we will investigate problems that can be abstracted to the level that can be attacked with some of the basic numerical solution techniques; namely Finite Difference, Finite Element and possibly some other techniques such as Cellular Automata. We will make a strong distinction between models that allow a strong degree of validation and those that don't. Models of stock market behavior, lotto etc. are outside this course because they are difficult (impossible) to check and besides they violate the first rule. (Numerical experiments can’t usually be validated either, but the actual results are only of passing interest, mainly you are interested in the process. However, you shouldn’t put too much faith in experiments either).