1 Introduction

The method based on the original Whittaker-Robinson (1946) method of smoothing observational data was described by Vondrák (1969,1977) in order to suppress high-frequency noise present in the observations. The method uses unequally spaced input data of different uncertainties of measurement to derive a set of points lying on a smooth curve. The basic idea consists in finding a compromise between "fidelity'' of the searched smooth curve to the observed values on the one hand, and the "smoothness'' of the curve on the other. The method became quite popular in many branches of astronomy; e.g., it is routinely used at the International Earth Rotation Service (IERS) to obtain smooth curves of the Earth Orientation Parameters (EOP), in atomic scale smoothing (Guinot 1988) or in analyses in stellar astronomy (Harmanec et al. 1978; Stefl 1995). The method, whose properties were studied in detail by Feissel & Lewandowski (1984), is further referred to as "original smoothing''.

However, we are very often facing a more general problem when not only the values of the function itself but also its first time derivatives are measured. Quite often, the former is given with larger spacing, and the latter, with a higher time resolution, is then used to make the output series denser. The analytical expression of the measured function is generally unknown, but we can assume that it is continuous and relatively smooth. This occurs, e.g., when the values of Universal Time UT1 (defined by the rotation of the Earth) are measured by Very Long-Baseline Interferometry (VLBI), and at the same time the variations in length of day (l.o.d.) are measured by Global Positioning System (GPS). The former represents the angle between zero meridian and a fiducial point on the celestial sphere, whose value is a non-linear function of time, while the latter is its change during one day. Another example is polar motion that is determined by GPS, and the same technique yields simultaneously also its rate.

Of course, the same estimator can be applied to both series separately and independently, but in such a case the result would be two functions, mutually not fully consistent - the latter would not be exactly the time derivative of the former. Using this approach in practical application would lead to unsatisfactory results, namely in case when less frequent VLBI determinations of UT1 and more frequent GPS determinations of l.o.d. are treated. In this specific case, the separate smoothing of UT1 from VLBI would lack the variations with short periods, and separate smoothing of l.o.d. would suffer from long-periodic instabilities. In order to remove these instabilities, present practice of GPS analysis is to "scale'' the values of l.o.d. against VLBI determination of UT1 on intervals several weeks long.

Here we propose a more general method of smoothing in which the estimation is done from all available data obtained by both techniques of observation. Both data series are combined to yield two smooth curves tied by the constraints assuring that he latter is the time derivative of the former. The first one fits well to the first data series and the second one fits well to the second series. This approach also provides an important test of systematic errors specific for each of the data series that are difficult to reveal had both series be treated separately. The goal is to make use of advantages of both series (long-term stability of the former and higher time resolution of the latter) in one solution. The main ideas of the method are shortly outlined and its use in combining the VLBI and GPS data is demonstrated by Vondrák & Gambis (2000); here we describe the method in a greater detail, and give more examples of application.