## 1. Introduction

[2] The amplitude of seasonal and diurnal variations of meteorological variables is generally much larger than weather-related fluctuations. As such, the instruments and analytical techniques used for climatic studies are often different than those used for studying short-term localized weather events. From the very beginning, meteorologists used monthly averages to approximate the seasonal cycle, maximum/minimum temperatures to describe the diurnal cycle of temperature, and other simplifications. Techniques were developed to avoid excessive computations and to fix the calendar-related problems of leap years and unequal length of months.

[3] In addition to diurnal and seasonal variations, much smaller amplitude climatic trends are an important component of meteorological variables, and they can also display diurnal and seasonal cycles. Modern climatologists mainly consider the long-term trend to be linear in time rather than a higher order polynomial. For relatively short time intervals, trend estimates are always contaminated by trend-like and episodic (El Niño events and volcanic eruptions) components of natural climate variability. This is why such trend analysis can only be used as a diagnostic tool and not for extrapolation of climatic data into the future. To address this problem and filter out short-time random variations in climatic trend, trend analysis must use very long time periods.

[4] Although seasonal and diurnal variations in multi-year averages of surface air temperature have been analyzed in the past [e.g., *Fassig*, 1907], we know of no attempts to determine the trend of such variations. *Polyak* [1975] investigated different techniques for approximating the seasonal cycle in multi-year averages of meteorological variables. He estimated multi-year averages for five-day periods and than approximated their seasonal variation using Fourier harmonics, polynomials, or smoothing of the pentad averages with statistically optimal numerical filters. The most recent global analysis of the geographical pattern of amplitude and phase of the first two harmonics of the diurnal cycle in observed temperature for two seasons, winter and summer, revealed the importance of the semidiurnal component of the diurnal cycle [*Dai and Trenberth*, 2004].

[5] The goal of this paper is to introduce a simple technique of approximating both the diurnal and seasonal cycles as well as the climatic trend using a limited number of Fourier harmonics. The main advantage of this technique is that it can be applied to data with changing and arbitrary observation times. Changing observation times is well known in the history of meteorological observation in different countries. The same problem also arises with satellite observations of surface and atmospheric variables. Here we use long-term surface air temperature observations at a few regular meteorological stations to illustrate our technique. Simpler versions of this technique can be found in our recent papers [*Vinnikov and Robock*, 2002; *Vinnikov et al.*, 2002a, 2002b; *Cavalieri et al.*, 2003]. This exact technique was used by *Vinnikov and Grody* [2003], but they did not have room to describe the complete technique. It is the purpose of this paper to describe the general technique for others to use.