Synthesising information about continuous variables

Scope

Concentrations fields are examples of very compex variables that have several dimension (longitude, latitude, time), are continuous in these dimensions, and are highly varying (i.e., not smooth). Also this kind of variables should be expressed as a random sample fron the probability distribution. A problem is the huge amount of data that is needed to express even one model run with a reasonable grid size. When this method works, it is a generalised way to make grid transformations, because the description is continuous and any grid can be used to extract information from the description.

How could the continuous variable be expressed efficiently in such a way that

• a result can be extracted from the description to any location within the variable boundaries,
• optimally, the fractal-like nature of the result is not lost in the description., ie. the true variability between locations is expressed and not just some linear interpolation,
• the description is deterministic and the randomness comes to the variable through the random sample of the result database.

Definition

Input

In principle, infinitely precise field of result values in one or more dimensions; but still uncertain in which of the possible fields is true. In practice, some kind of grid or Index is used and values are available only for some locations.

Output

A random sample of parameter values that can be used to reconstruct a continuous result field for the variable in all of the dimensions that the input had.

Rationale

It must be based on a random sample in the result database. For each row of the sample, it must be possible to create a function that produces a deterministic result for any location along the dimensions based on the information stored in the result database. It is impossible to store all the actual concentrations, so we must develop some kind of function that describes the actual concentration, and the input parameters of the function are stored in the result database. Which are efficient functions for 3D data? Can mpeg video compression format used? It is something that stores a two-dimensional figure efficiently with some resolution depending on the input data, and it can be extracted from the mpeg file for any screen resolution that is used to view the data. In addition, it has the time dimension, and it efficiencly utilises the fact that the variable result in one location at a particular time is correlated with that at the next time point.

Optimally, there should be some external information about the functional form of the variable. The function would then be used to analyse the data about the variable.

Result

Possible methods for describing the variable:

• Mpeg video compression system
• Fourier analysis