## 1. Introduction

There are different ways to calculate daily mean temperature (DMT) at a station from data collected at different times of the day. In many countries, the approach is to average the minimum and maximum temperature observed, although this may be the minimum and maximum hourly readings or the actual minimum and maximum obtained from minimum and maximum thermometers or other devices (see WMO, 2008, for details about temperature measurements). In other countries, a linear combination of measurements taken at different times of the day is used, sometimes including the minimum and maximum as well. For example, the Scandinavian countries each have a different linear combination of data, depending on the frequency of recorded observations (Nordli *et al.*, 1996, Appendix II). When using temperature data for climatological purposes, such as calculating the uncertainty of estimates of global mean temperature, it is important to take into account the variability of the method used to calculate DMT. The purpose of this paper is to present a case study quantifying the difference between some DMT estimators.

There has not been much work on comparing the different approaches to estimating DMT. Hovmöller (1960) discusses how to adjust historical Icelandic data so as to be useful for climatological purposes, based on a variety of different observational schedules.

Weiss and Hays (2005) compare hourly average (taken as ground truth), 3-hourly average, average of min and max, a weighted average, and a method used in the CERES crop simulation program that uses a cubic interpolation between min and max. The goal of their paper was to see the effect of different DMT computations when used as input to a highly nonlinear algorithm. The 3 h average performed best in their context. However, few synoptic networks report that frequently.

Reicosky *et al.* (1989) looked at five different ways of computing the diurnal hourly temperature curve based on observing only the minimum and the maximum. They found that such methods worked better on clear than on cloudy days.

Here, we will look at different ways of combining synoptic temperature measurements to estimate the DMT. We will mainly focus on Swedish measurements at a few stations in the SMHI synoptic network (http://www.smhi.se/klimatdata/meteorologi/dataserier-for-observationsstationer-1961-2008-1.7375). Figure 1shows the locations of the stations.

The standard Swedish approach dates back at least to 1914 (Ekholm, 1914), and in its current form has been in use since 1947 (Nordli *et al.*, 1996). It is called the Ekholm–Modén (EM) formula, and is a linear combination of the daily minimum, the daily maximum, and measurements at 6, 12, and 18 h UTC. The maximum and minimum both correspond to the time period 18 h UTC the previous day until 18 h UTC the current day. Swedish time is UTC + 1 in the winter, UTC + 2 in the summer (last Sunday of March through last Sunday of October). We then can write the formula, using observations at Swedish standard time, as *T*_{mean} = *aT*_{07} + *bT*_{13} + *cT*_{19} + *dT*_{max} + *eT*_{min}. The coefficients of the linear combination depend on month and longitude, although the longitude dependence is relatively small, and can be found in Alexanderson (2002) or online at http://www.smhi.se/kunskapsbanken/meteorologi/koeffi-cienterna-i-ekholm-modens-formel-1.18371. They were essentially derived by least squares (LS) fitting to stations with hourly data available (Ekholm, 1914; Modén, 1939). It is interesting that the coefficients are restricted to sum to one. Apparently this originates in numerical work in the early part of the 20th century, where this constraint stabilized the LS calculations (Ekholm, 1914). Also, the coefficient *d* for the maximum daily temperature is always set to 0.1, regardless of month and longitude. We have not found any reason for this constraint in the literature. In Ekholm's and Modén's original papers, the maximum temperature was not included (i.e. *d* = 0).

We begin in Section 2 by looking at the accuracy and precision of various estimates of DMT compared to an estimate from a high resolution (1 min) data set. We do not have access to 1 min data at any of the Swedish synoptic stations, although hourly data are available at some of them. In Section 3, we compare the EM formula to various linear combinations of the minimum and maximum, and note that the latter generally are substantially more variable. Section 4 is devoted to comparing the EM formula to hourly averages for two stations. We discuss our findings in the final Section 5, and describe some future research.