Rating Curves, Blind Men, an Elephant, and the Goldilocks Principle

Rating Curve Blind Man Elephant & Goldilocks - Hydrology Corner Blog

While there must be an underlying true relation between water level at a given place and time and the corresponding discharge, our experience of that truth is limited to gauging observations from which we must infer the totality of the relationship.

It is generally true that if you give the same set of data to “n” different hydrographers they will produce “n” different discharge hydrographs. There is no assurance that any of the hydrographs are actually true. Each hydrographer is making inference about what they believe to be true based on a relatively few gaugings.

The parable of the blind men and the elephant has often been called on to explain what can happen when inference is made from too little information. In this story a group of blind men each touch a different part of an elephant and then describe what an elephant is. The man who touched the leg says it is a pillar; the man who touched the tail says it is a rope; the man who touched the belly says it is a wall.

A partial viewpoint deludes people with a truth they do understand to deny aspects of the truth they don’t understand.

Inference can result in an overly simplistic perception of the rating shape, hence there is a greater risk of under-fitting the curve. Alternatively, an overly complex inference results in a greater risk of over-fitting the curve. The Goldilocks principle tells us that between any two opposing extremes there is a place that is ‘just right’.

In principle, all hydrometric practitioners have policies, procedures, and training in place to guard against rating curves that are either under- or over-fit to the available gaugings. In practice, I am not convinced these protocols are adequate.

What I see in retrospectives of ratings from agencies all over the world is that a priori determination of rating curve form is a rare skill.  If you look at gauges with stable controls with the benefit of hindsight you can see that many agencies take many years to converge on a unique curve through a scatter of uncertainty. We should celebrate the inference-making skill of highly effective hydrographers and learn from their successes to share for global benefit.

There is a global tendency is to over-fit the curve to gaugings.

When new gaugings don’t agree with the over-fit curve the deviations are ‘explained’ as a change in control, resulting in the evolution of many curves before the true form of the curve begins to emerge out of the gauging uncertainty. This is particularly the case when the model is given too many degrees of freedom. In other words, a model that is too powerful can be very good at fitting to measurement imprecision.

Given that this is the case for stable controls, it doesn’t give you a warm and fuzzy feeling for retrospective analysis of gauges with unstable controls, where there might be only one gauging to represent the state of the control at a given point in time.

There are many different endings to the story of the blind men and the elephant. Collectively, we get to choose the ending we want to be a benchmark for a global hydrometric best practice. Let us be as discerning as Goldilocks.


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3 responses to “Rating Curves, Blind Men, an Elephant, and the Goldilocks Principle”

  1. Ferdinand Quiñones July 27, 2016 at 3:37 pm

    Stu, you speak in a forqued tongue because your perception of rating curves and stream gaging is over-complexed to the relative simplicity that it is. You need to study the development of rating curves by the chinese going back several thousand years, and not much has changed. It is an overkill for most countries to try to dwelve in complex mathematival correlations for something that is simple and works. Streamgagers worldwide do not need to elaborate complex rating curves as long as calibration measurements are made periodically. The gain in accuracy from all these extreme complex methods is not warranted.

    • Hi Ferdinand,
      It is certainly true that you can produce high quality discharge with a simple approximation if you gauge the stream at a very high frequency, say up to several times a day for a flashy stream. In that case the majority of the error in discharge derivation would be measurement error. However, for the case of less frequent gaugings some inference has to be made about what was happening in between gaugings. In these cases a highly skilled and experienced hydrographer will do a much better job of producing an accurate discharge hydrograph than an unskilled, inexperienced, hydrographer. You only need to look at the difference between the highly skilled and the unskilled to understand the importance of inference.

      I am very interested in what you know about the Chinese producing rating curves several thousand years ago. I was unaware that anyone was even measuring discharge, much less deriving it from rating curves, that long ago. I am aware of the Nilometer in Pharaonic Egypt but I believe that was for water level only. If you have any relevant literature citations that you could share I would be most grateful.

  2. Stu and Ferdinand, I am not sure about the Chinese, but i do recall hearing something of the Koreans and their level of hydrological acuity. They were pretty on to it from memory, but that could have been rainfall mainly. I’d also be interested in any evidence – from any culture – of long term rating development and some historical application of that (I never thought I’d be interested in that…)

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