Upper Ocean 1-D Seasonal Models
File last modified 16 November 1998
19.5 Tying It All Together
How do we compare the results to "reality"? What features do we look for in making this comparison? One, generic approach is to use some "objective" technique such as least squares minimization between the data and the models. The way you'd do this is to take the observations, and cast them onto the same grid as the model, using some objective mapping or other gridding technique. This process, in itself, is rife with difficulties, as you may have gathered from our earlier discussions on diagnostic techniques (e.g. Kriging). You have to be very careful to make sure that the gridding process does not introduce an implicit model or bias into your "data" product.
However, let's assume that you have gridded your data, and done it in a politically, morally and mathematically correct way. You now are faced with the fact that you may value some aspects or regions of the data (or model) more than others. This may arise because you have better knowledge of the properties in question either because of better sampling density, or better measurements. Thus if you have used some objective mapping scheme which provides you with uncertainties in your gridded fields, you can use that uncertainty to weight your model-data chi squared comparison.
Another reason why you'd value some regions over others relates to the care you have put into characterizing boundary conditions or physicochemical/biological processes in the model. For example, in our oxygen simulations, you may not really care what is going on too far below the euphotic zone (we are, after all only trying to model the photosynthetic oxygen cycle), or more appropriately, too far below the maximum mixed layer depth. As long as we get the approximate behavior of the deeper waters right, their influence on the shallower solutions will be minimal and not too important. So if we were to compare model with observations, we may want to "deweight" the deeper regions.
This may apply to regions in time as well. For example, if we were not too interested in the winter mixed layer behavior (perhaps because we feel that we cannot do a good job modeling the highly non-linear gas injection processes in winter) but just want to get it good enough to initialize for the summer-time development, then we might deweight the winter convection times.
Well, you get the picture. The take-home lesson is that there is no truly objective method of comparison, but you should take as statistical an approach as possible, and you must flavor the comparison based on your goals, the limits of your data, the limits of your model, and what you think are the important features of your data.
Also, you should think in terms of degrees of freedom. What we mean is that your first inclination may be to say you have a model with, say 5 adjustable parameters (e.g., vertical diffusivity, gas exchange rate, air injection rate, trapping fractions, and oxygen productivity). Yet you only have, say, 5 different properties to measure (e.g. temperature, He, Ne, Ar, and O2). Thus you might say that the system is "uniquely determined" (number unknowns = number equations), so that you can get a "unique solution" but cannot test it. Well, that's not in general true.
In the first place, it depends on what the observations (by "observations", we mean both the model generated property distributions and the actual field observations of those properties) are, and how their behavior in the model depends on the parameters. It is always possible that all of the parameters depend on a number of the parameters in the same way, and hence do not yield information independent of one another. In those circumstances, the model may be underdetermined for some parameters. That is, the observations do not adequately constrain the parameters. Not a good situation.
In the second place, since your observables are generally determined over space and time, and are known to exhibit space/time structure you are actually interested in those space-time features as independent observation elements. Thus the number of unique, identifiable features in the data is a better measure of the numbers of degrees of freedom. Certainly, in a spatially-temporally coupled model, different features of the same property (and conversely different properties sharing the same feature) will not be strictly independent. However, there are statistical measures of this independence, and if you use sensible optimization schemes (such as SVD) you will obtain almost automatically a measure of this through the rank of the coefficients matrix and the relative magnitudes of the singular values.
We will proceed by using the Spitzer and Jenkins (1989) example. It is not as complete as it could be, in retrospect, but it should give you an idea of where to go from there. The strategy is quite simple. Find a series of diagnostic features in the data/model and then run the model while varying the relevant parameters. You then construct a linearized equation which represents the behavior of this feature in response to these changes. The features chosen by them were:
Ar
maximum (in the seasonal thermocline)
Ar related
to gas exchange, vertical mixing, air injection
Ar related
to trapping fractions, air injection
He gives
strength of air injection
He
gives wind dependence of air injection
O2 difference
between summer and winter
O2 maximum
The set of linearized constraint equations produces a weighted matrix of coefficients, which is given below:
| Eqn. No. | 1/Kz | Ainj | Gamma | fc | 1/G | Pml | P50 |
| 1 | 9.9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 2.0 | 4.4 | 0 | 0.8 | 5.0 | 0 | 0 |
| 3 | 0 | 6.7 | 0 | 0.8 | 8.0 | 0 | 0 |
| 4 | 0 | 9.0 | 0 | 4.0 | 3.0 | 0 | 0 |
| 5 | 0 | 23 | 0 | 0 | 11.0 | 0 | 0 |
| 6 | 0 | 0 | 5.7 | 0 | 0 | 0 | 0 |
| 7 | 0 | -0.3 | 0 | 0 | 5.7 | 0.2 | 0.1 |
| 8 | 1.3 | -0.2 | 0 | 0 | 0.7 | 0.06 | 0.7 |
Now how do you read this table? Well it's really quite easy. As you go across a row, the larger the coefficient in a box, the more important that parameter is to determining the feature. As you go down a column, the larger the number, the more important the feature is in determining the parameter. Hence, the most important diagnostic of vertical mixing (Kz) is the mixed layer temperature, but quite surprisingly, the summertime subsurface Ar and O2 maxima also play a role in determining vertical mixing (Ar more than O2). We encourage you to stare at this table to get a better idea of how this approach provides you with information about processes in the model, and processes in nature. Further, have a look at the paper to see how things are done.
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