FAQ - Frequently asked questions related to OMP analysis

Question: Solving a linear system of equations through least squares minimization requires that all parameters are linearly independent. Does the generally assumed linear relationship between nitrate and phosphate known as the Redfield ratio preclude the simultaneous use of both parameters in OMP analysis?

Answer: It is generally safe to use both nitrate and phosphate in OMP analysis, and in most cases using both parameters rather than only one of them results in information gain. There are a number of reasons for this:

  1. Observations are invariably afflicted with instrumental error. This error may already be enough to produce linear independence of the data sets for the two parameters. If the errors are only small, the OMP algorithm recognises this by placing very low weights on the two parameters.
  2. The Redfield ratio is not constant throughout the world ocean. It varies with depth, and more importantly for OMP analysis it varies between water masses. This guarantees linear indepence of all parameters. (It makes the use of a constant Redfield ratio in the expanded OMP analysis questionable; but the method is still developing, and this question will no doubt be addressed by future users. First experience shows that nutrient weights are usually low, so the method is not very sensitive to the choice of the Redfield ratio.)
  3. The method used in the OMP algorithm for finding the least square minimization tests for linear independence of all equations. It eliminates line duplicates automatically and will therefore exclude either nitrate or phosphate from the solution if by chance the two equations are in fact linearly dependent.

Question: How can I get a feel for the quality of my OMP result?

Answer: OMP analysis nearly always returns a result, but it can only be as good as the information which goes into it.

  1. The first thing to check is the distribution of the mass residual. Most users place a large weight on mass conservation and therefore can expect small mass residuals. If the mass residual is small for most of the data points, the occurrence of larger mass residuals for some data points can indicate a problem with these data. For example: Your analysis uses 5 water types; all mass residuals are small except those associated with data points which contain a very large contribution from water type 3. This may indicate a problem with the definition of water type 3.
  2. You can follow this up by checking the residuals for individual parameters. If you then find that (say) oxygen shows a large residual for those data with a very large contribution from water type 3 and the other parameters behave normally, this indicates a problem with your oxygen definition of water type 3.
  3. If you find that ALL parameters display large residuals in a particular area of the region under investigation, this probably indicates the presence of another water mass which you did not include in your water mass matrix.

Question: How stable is OMP analysis to small perturbations?

Answer: OMP analysis is surprisingly robust, but it is generally good practice to test the result against small variations in the water type matrix and in the data. A good procedure is to generate a new synthetic data set from the original observations by adding white noise with an amplitude equal to one or two standard deviations and run OMP analysis again. This will show how much the mixing contributions change if the observations are randomly changed by the observed (and therefore realistic) data variability. A similar test should be done by adding white noise with an amplitude of the appropriate one or two standard deviations to the water type matrix. See (4 ) for examples.

Question: How is it possible to use the OMP code with n parameters and n+1 water types? Shouldn't this give a fully determined linear system of equations which does not produce residuals?

Answer: The additional constraint that only positive solutions are accepted (non-negativity constraint) gives an additional degree of freedom, so using n parameters for n+1 water types still produces an optimised solution - but you are really pushing the system to its limits.

Question: Why is mass conservation handled in the same way as every other parameter when physical reasoning suggest that mass should be accurately conserved?

Answer: This is a question of personal choice. On one hand, mass conservation is an elementary physical principle and should be satisfied by any model. On the other hand, OMP analysis is based on the assumption that oceanic mixing is a linear process. (In other words, oceanic diffusion of properties is achieved by turbulence). Mixing products from two water masses therefore can only produce temperature-salinity combinations which are located on the straight line connecting the temperature-salinity combinations of the sources.

In case of double-diffusion, linear mixing of all properties is not warranted. Salinity and nutrients may have the same mixing behaviour (McDougall and Ruddick, 1992, DSR #39) but temperature will not follow them. If there is a suspicion that double diffuive processes may occur, one has to be careful by using OMP analysis. One way around the problem is to put different weights on temperature and salinity within this regions, insted of using the same (as usually in OMP analysis).

In any case, there are ways to enforce mass conservation, or at least come close to it. The easiest way is to give mass conservation a much larger weight than any other parameter. This will reduce the mass residuals but not enforce strict mass conservation. The obvious way to enforce mass conservation is to exclude mass conservation from the source water type matrix and replace the unknown x1 by 1 - Sxj, where j = 2, ... n.

© 30 June 1999 OMP analysis user group