Averaging Multiple Temperature Records

Author: Dr. Michael Chase

Introduction

This is the first in a series of posts with the general theme of “Do it yourself temperature homogenisation“. The full series of posts will outline a simple but effective procedure for turning raw instrumental temperature data (aided by any available station history and data on rainfall) into reconstructions of the background temperature history for any area with a “sufficient” number of weather stations, the sufficient number depending mostly on the quality and extent of the data.

The overall procedure involves visual detection of inhomogeneities, followed by averaging and integration of interannual temperature differences in which the inhomogeneous data is omitted. Effectively, large inhomogeneities allow visual detection, followed by removal from the data, small ones are suppressed by the averaging process used to obtain regional histories.

Averaging Multiple Records

This first post in the series deals with the final step of the procedure, the method by which multiple temperature records are combined to give a regional average temperature history. The following figure gives a schematic picture of what typical temperature data looks like:

The figure above illustrates the following:

• Black data: Historical urban warming, followed by a station move (typically to an airport or other out-of-town site), followed by a switch to automatic sensors
• Red Data: A good rural station, but with some missing data
• Blue data: A station with a transient perturbation, possibly of non-climatic origin, or possibly due to a period of localised heavy cloud/rainfall

It is assumed for the purposes of this post that the previous stage of the overall procedure has identified the inhomogeneities, marked all periods of “transition” within a computer program, and that the analyst has corrected the data for any large residual inhomogeneities at transition boundaries and at all ends (more on that last issue below and in subsequent posts). The computer program, under analyst control, then does the following:

• Either infills missing raw data or leaves gaps if there is doubt about the station history within a gap. Infilling can also be done manually.
• Computes all valid differences in temperature, separately for each month, between years N and (N-1). Valid differences are those that do not cross, or lie within transitions
• Extrapolate temperature differences for all stations with missing years using the average of valid temperature differences

The following figure illustrates the resulting full coverage of temperature differences:

The temperature differences can now be averaged again with the important feature that each station has a constant weight in the averaging process (1/3 for each station in the example shown in the figure above). Finally, the average temperature differences can be integrated forwards and backwards in time from any desired reference year/temperature to obtain average temperature histories for each month.

Error Analysis

The reason for extrapolating temperature differences for all stations can be seen from the example of the blue data in the figure above. In some cases the dip in the blue data will be deemed to be associated with a period of heavy rainfall, i.e. a genuine climatic effect. This genuine climatic effect will only produce the correct impact on the regional average (i.e. only within the period of rainfall) if the weighting of the blue data is constant in time, which is what results from the extrapolation of temperature differences for the red data.

Using interannual temperature differences avoids the need to estimate and correct most inhomogeneities, but there is a small price to pay for that substantial benefit, the problem of residual inhomogeneities at boundaries, illustrated in the following figure:

The figure above shows an example of a temperature record with perturbations from the regional average. The perturbation within the record is not really a problem, because the downward shift in temperature is matched by the exact reverse in later data, with the constant weighting of each record ensuring that the perturbation will not influence the end-to-end shift in average temperature.

The problem illustrated in the figure above arises from the inhomogeneous ends of the record (more generally from any boundary, including ones created by defining “transitions”), which can distort the end-to-end variation of average temperature if the inhomogeneities are not detected and corrected, which can be done manually (see a later post).