*By Daniel Collins*

Back in June I introduced you to a model we use, called TopNet, and I talked about modelling on the ‘Ever Wondered?’ episode back in September. I’d now like to give you a richer picture of what models are in general.

Models feature widely in hydrology. Not the fashion models usually, but the mathematical or computer variety. In fact, to be pedantic, everything in hydrology is based on a model in one way or another, right down to the measurement of river flow. So to understand hydrology properly, you need to understand modelling. But be warned: modelling cannot be explained in one blog post. Whole books are written on the subject. Whole careers are built on building models. Instead, what I will try to do here is share some of the fundamentals of the philosophy of modelling as I see them.

The first rule to remember is that all models are to some extent wrong, but some models are useful.

Models are simplifications or idealisations of reality with the purpose of serving a particular function. This goes for fashion models, who are idealisations of the human form and whose job it is to make clothing look good. Or model or toy aeroplanes, which are easy-to-build miniatures that you can play with. Or mathematical models, which attempt to explain some phenomenon as a calculable function of defined variables.

The same goes for hydrological models. Mostly they are of the mathematical variety, but sometimes they can take on aspects of the fashion or toy models.

A mathematical model is basically an algebraic relationship with one or more variables, like Einstein’s e = mc^{2}. You can add randomness, make it self-reflexive, or combine a suite of individual equations together if you want, but it’s still a mathematical model. One of the strengths of mathematical models is that they typically allow for more reproducibility, one of the hallmarks of science. And as soon as you write it into computer code, it becomes a computer model too. We take this step whenever it’s too burdensome to calculate the equation(s) by hand.

The model could be based on empirical observations, like the Rational Method for peak flood flow:

Q_{peak} = C I A

Or on physical principles, like Richards equation for water movement in unsaturated porous media:

But no matter what the model looks like, it would have been developed over the course of much time and research. For it to become an accepted model of reality, it will have been seeded by observation, shaped by imagination, potentially tweaked by calibration, and will have successfully jumped through the hoops of validation and either verification or corroboration.

And as modellers build their models, they should remember (though we don’t always) that models should be as simple as possible, but no simpler.

This basically means that models shouldn’t include features that don’t help them serve their purpose, but that they must still be useful somehow (in contrast to a spherical cow). For example, water temperature won’t improve Manning’s equation of river flow noticeably, so it’s left out, even though a liquid’s viscosity is temperature-dependent. While too few parameters in a model can lead to huge errors, too many parameters can be overly burdensome in terms of data collection or cause the model to lose all generality.

Now remember that hydrological models can sometimes resemble fashion or toy models?

If you’re trying to explain an idea, such as the water cycle, you need a conceptual model that makes sense – a model that makes people want to buy an idea. This is much like a fashion model who is used to sell clothing. And if you’re trying to explore the implications of existing scientific theories, such as the role of carbon fertilisation on catchment water yield, you need a toy mathematical model whose variables you can experiment or play with.

In the end, of course, we must remember that while good models serve a purpose, the results will be no better than the veracity of the model. It takes skill to use a model properly, and to interpret its results. And no matter how sophisticated our modelling skills, we should never lose sight of the observational data that feed our models and the questions that drive them.