Linear Modelling in Climatology - A Local Scale, Minimum Temperature Model for Great Britain

Dan Cornford and J. E. Thornes
School of Geography, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom

Abstract
Climate is both a spatial and temporal phenomenon. In general climate data users require the climate information at unsampled locations. This presentation addresses the question of how to optimally interpolate temperature data within a linear modelling framework, with the aid of Geographic Information Systems (GIS). There have been previous works investigating this problem, which has been addressed from two angles. Firstly a spatial model of surface temperature using a 3D numerical model, with some detailed surface parameterisations. This approach is possible where there is a large amount of very detailed information on the variables required by the model at the surface (boundary conditions), but is only computationally feasible over small domains, and depends critically on good parameterisation.

Secondly statistical models have been produced however these have either been regional or fit data at fairly coarse resolution. This work seeks to take these attempts further, by combining statistical models, with physical understanding of the processes that determine minimum temperatures. Using our knowledge of the physics of the atmosphere the GIS can generate several coverages, potentially very useful for the prediction of minimum temperatures. Simple factors such as altitude and distance to the coast could be included, but more complex ones such as a 'cooling potential' could be generated using typical specific heat capacities of landuse types and a landuse coverage. These factors can then be simplified using principal components analysis on the terrain variables.

A statistical framework to interpolate minimum temperatures was then proposed. The components that were to be included in the model were:

large scale variation: different 'air masses' and modification thereof.
local scale variation: interaction of the 'terrain' with the 'synoptic' flow.
micro scale variation: sub-grid scale features; trees, house, etc.

These components could then be related to numerical techniques, but as with all real world problems, there is no unique partition. The mathematical models chosen were:

Spline / Polynomial Surface: smooth, large scale changes.
Local Linear Regression: spatially varying impact of the interaction of terrain with 'airflow'
Geostatistics: 'micro-climate', but also might compensate for any terrain variables omitted from the model

The methodology used to combine these models was an additive, linear modelling approach, summarised in Figure 1.

Figure 1: Flowchart of the modelling approach

One of the most important aspects of the work was model validation. This is achieved by splitting the initial data into two sets - the fit and the validation set. By using resampling techniques successive small validation sets are removed, allowing a good estimation of the models predictive performance to be made. The results indicate that even the complex spatial distribution of daily minimum surface air temperatures is well predicted by the model (root mean squared errors generally in the range 1 - 1.5 oC). It is believed that much of this can he attributed to site micro-climate which will never be successfully resolved at the scale of this study (500 metres). It is particularly important that validation is objective for this work since the aim is prediction rather than 'understanding'.

Aspects of the performance of the model are shown, and a physical explanation is offered. It is demonstrated that this type of linear, additive modelling is robust to changes in the model parameters, by presenting results of sensitivity analysis. This also revealed that despite the apparently variable nature of the interaction of the terrain with the 'airflow', a simple country-wide regression model was found to give very slightly higher root mean squared errors. This is suggested to result from the locally differing terrain effects being modelled in the geostatistical phase. It is shown that the variograms of the residuals vary with synoptic type, this being linked to the relative impact of micro-climate and terrain factors omitted from the regression model.

It must be recalled that this is an additive model whereby the polynominal surface and regression equations are computed before the spatial correlation of the variables is taken into account. This is not an ideal situation, since the correlation in the terrain and temperature variables will tend to produce some bias in the estimation of the regression coefficients. An iterative model fitting approach, although computationally more expensive might improve the modelís predictive power, and produce better estimates of the correlation structure of the minimum temperature field. However for the applications for which this model was designed the current performance was felt to be adequate.