Many long-term records of climate variables have missing data or have had changes in their times of observation. Here we present a technique to analyze such inhomogeneous records. We assume that the underlying climatic processes are nonstationary, where the observations contain a long-term trend superimposed on periodic shorter time seasonal and diurnal cycles. The seasonal and diurnal variations are approximated using a limited number of Fourier harmonics, while the trend is represented by a monotonic function of time whose amplitude can also vary seasonally and diurnally. A least squares method is used to estimate the unknown Fourier coefficients. As an example of the technique, we present an analysis of multi-decadal hourly observations of surface air temperature obtained from several meteorological stations within the United States.