The following paragraphs from an e-mail by John Dykema describes how co-location and coincidence could be formalized. As an example, the measurements of a temperature profile by a sensor on a balloon and a profile retrieved from microwave radiometry are assumed.
Once we have two well-characterized sensors with rigorously assessed and validated uncertainties, then we can make robust recommendations on co-location.
To do this, I would propose we use an extension of the Common Reference Value (CRV) formalism that is in wide usage in the intercomparison of high-accuracy measurements (see for example, for an application to climate-relevant measurements, <!–[endif]–>R. U. Datla, R. Kessel, A. W. Smith, R. N. Kacker and D. B. Pollock, “Uncertainty Analysis of remote sensing optical sensor data – guiding principles to achieve metrological consistency,” Int. J. of Remote Sensing, in Print, 2009). In our case, we would modify the CRV formalism so that there was an uncertainty associated with the atmospheric temperature (eg, measurand) itself, to account for the fact that the spatial and temporal character of the sonde and radiometer measurements are different. We would combine results from radiative transfer calculations (to determine the spatial average kernel), atmospheric models, and fluid dynamics models to determine this uncertainty. In performing these calculations, we would need to specify a set of time and space boundary conditions that would define our co-location criteria. Let the resulting uncertainty, which specifies the uncertainty in our ability to relate the temperature in the space-time region measured by the sonde to the temperature in the space-time region measured by the radiometer, be u. This uncertainty u is caused mainly by the natural variability of temperature in the given space and time boundaries and can be assessed using physical models and measurement data. Let the uncertainty for the radiometer temperature be r, let s be the uncertainty for the sonde temperature, and let d be the difference between the sonde and radiometer temperature, adjusted as necessary according the model calculations. Then, to first order (some assumptions about parent distributions and degrees of freedom need to be specified), if (d2+r2+s2)1/2<(u2+r2+s2)1/2, we confirm that our uncertainty estimates are robust, and the resulting estimate of the temperature of the atmospheric region in question would be (u2+r2+s2)1/2.
I should add that in the spirit of not reinventing the wheel, the starting point for the studies to associate an uncertainty with the co-location criteria would start with work that has already been done in the community. Work by Dave Tobin and Nikita Pougachev comes to mind, and I’m sure there’s more.