## 1. Introduction

[2] The spaced antenna (SA) methods for measuring wind velocities in the atmosphere have become commonly used radar techniques. The methods have been successfully applied to estimating characteristics of a scattering medium at altitudes from the low troposphere to the high mesosphere and ionosphere [*Fooks*, 1965; *Fedor*, 1967; *Manson et al.*, 1974; *Meek*, 1980; *Vincent and Röttger*, 1980; *Röttger and Larsen*, 1990; *Röttger et al.*, 1990; *Cohn et al.*, 1997; *Riggin et al.*, 1997] (and many others). SA methods have been thoroughly tested by comparing the results with those from the Doppler Beam Swinging (DBS) technique as well as rawinsonde, anemometers, and other independent instruments [*Röttger and Vincent*, 1978; *Röttger*, 1981; *Röttger and Czechowsky*, 1980; *Vincent et al.*, 1987; *Larsen and Röttger*, 1989; *Hocking et al.*, 1989; *Van Baelen et al.*, 1990; *Burrage et al.*, 1993, 1996; *Cervera and Reid*, 1995; *Cohn et al.*, 2001] (and many others). It was well established that SA methods produce reliable estimates of the mean horizontal velocities in the atmosphere. Other parameters, such as turbulence intensity and spatial scales of the refractive index irregularities, can also be measured, although a caution is needed in the interpretation of the results [e.g., *Briggs*, 1984; *Hocking et al.*, 1989].

[3] The basic concept of the SA approach is presented in the classic paper by *Briggs et al.* [1950]. The radar transmitter sends pulses of radio waves vertically upwards into the atmosphere and these are scattered by the refractive index irregularities to form a moving and changing diffraction pattern on the ground. The magnitude and phase of this pattern is sampled with three or more spatially separated receiving antennas and the time-varying signals are analyzed to determine characteristics of a scattering medium.

[4] A diversity of the SA methods has been developed; for reviews, see, for example, *Hocking et al.* [1989], *Fukao and Palmer* [1991], and *Palmer* [1994]. The methods could be divided into two clearly distinguishable groups in accordance with the different approaches to considering received signals from multiple antennas [*Doviak et al.*, 1996], and there are numerous specific techniques within each approach. The first approach is to assume models of the diffraction pattern on the ground without relating the models to the properties of scattering medium. The mostly used techniques in this group are referred to as the Full Correlation Analysis (FCA) in a time domain [*Briggs*, 1984], and the Full Spectral Analysis (FSA) in a frequency domain [*Briggs and Vincent*, 1992]. The second approach is to relate the properties of the refractive index field and its advection flow to parameters of echoes in spaced receivers, therefore, models of scattering medium are assumed. *Liu et al.* [1990] are the first to realize this approach; the Liu et al. theory was generalized and further expanded by *Doviak et al.* [1996]. The approach-based method is described in *Holloway et al.* [1997], and it is referred to below as the Holloway - Doviak (HAD) method.

[5] As emphasized by many authors [e.g., *Briggs and Vincent*, 1992; *Sheppard et al.*, 1993; *Hocking et al.*, 1989], all SA methods are basically similar in that they utilize the same initial information: time series of the amplitude and phase of signals from several receivers. The methods differ by (1) mathematical tools for analyzing multiple signals (auto and/or cross correlation functions; auto and/or cross spectra); (2) parameters of these functions to be estimated (time delay for the maximum and/or fixed values of cross and/or auto correlation function, slope of the cross spectrum phase, etc.); (3) equations for relating these parameters to characteristics of scattering medium; and (4) assumptions which are adopted for deriving the equations.

[6] Although basically similar, all methods produce important information about a scattering medium. As noted by *Sheppard et al.* [1993, p. 593] regarding FCA and FSA, “Most probably, a combination of these methods will provide the optimal wind estimation technique.” We consider “combination” as the key word in this statement. Multiple signals from several receivers provide an enormous amount of raw information. The objective of data analysis in remote sensing is to extract as much useful information as possible about the scattering medium. Each data analysis tool (spectra, correlation functions, wavelets, etc.) extracts only a small part of useful information from multiple random signals. Being used in a combination, different tools increase informational efficiency of measurements by supplementing each other.

[7] The primary objective of the present work is to extract supplemental to other techniques useful information from multiple signals by applying an alternative data analysis tool, structure functions (SF). SF of order *p* ≥ 2 for random scalar processes of arbitrary physical nature Ψ_{1}(*t*) and Ψ_{2}(*t*) can be defined as:

Hereafter *t* is time, τ is the temporal separation, and the angular brackets 〈 〉 denote statistical (ensemble) averages. It follows from the definition that SF are the *p*-th order moments of the difference between signals; the difference Ψ_{1}(*t*) − Ψ_{2}(*t* + τ) is often referred to as the increment [*Tatarskii*, 1971, section 1A]. It is well known that a theory of locally isotropic turbulence has been developed only after (and due to!) introducing by *Kolmogorov* [1941] an adequate mathematical tool, structure functions. SF have been, and still remain the major tool for studying fully developed turbulence; they also found numerous applications in different areas of science and engineering [e.g., *Monin and Yaglom*, 1975; *Tatarskii*, 1971].

[8] SF and correlation functions (CF) describe fluctuations of random processes and fields in a time domain at different scales. CF of order *p* ≥ 2 can be defined as:

It is important that CF have been used in SA methods only at *p* = 2 (*i* = *j* = 1), and spectra are second order functions as well. It seems fairly impossible to derive practically useful equations for CF of received signals at *p* > 2.

[9] CF is used to reveal the similarity between random signals at large spatial and temporal separations while SF is used to reveal the difference between the signals at small separations. CF describes fluctuations at all scales but mainly at large ones of the order of the spatial *L _{cor}* and temporal

*T*integral scale of the random processes Ψ

_{cor}_{1}(

*t*) and Ψ

_{2}(

*t*). CF is applicable to the globally statistically stationary and/or homogeneous random processes and fields. SF describes the locally statistically stationary and/or homogeneous random processes and fields at very small scales, much smaller than

*L*and

_{cor}*T*. The term “locally” is used throughout this paper in the same sense as in a theory of the fine-scale turbulence [

_{cor}*Monin and Yaglom*, 1975, section 21]. In particular, the “locally stationary” stands for stationary over a time period which is much smaller than

*T*. Another commonly used term with the same meaning is a random process with statistically stationary temporal increments. It is important that any statistically stationary random process or field is always the locally statistically stationary and not otherwise. Real physical processes are almost never statistically stationary while practically any real process can be safely considered as that with statistically stationary increments [

_{cor}*Tatarskii*, 1971, section 1A;

*Monin and Yaglom*, 1975, section 13].

[10] An outstanding performance of SF in turbulence theory and their applicability to real random processes stimulated us to apply this tool to analyzing multiple signals from several closely located receivers. The present work was intended to address the following questions: (Q1) Can parameters of SF of received signals be unambiguously related to characteristics of the diffraction pattern? (Q2) Can parameters of SF of received signals be unambiguously related to characteristics of a scattering medium? (Q3) Are there characteristics of scatterers that can be related to parameters of SF more rigorously than to parameters of CF and spectra? (Q4) Are there characteristics of a scattering medium that can be estimated with SF but which cannot be estimated with CF or spectra? (Q5) Can the necessary conditions for applying SF to SA radars be satisfied in practical measurements?

[11] While this work has been ongoing for more than five years, we did not publish any results except for three brief conference presentations [*Praskovsky et al.*, 1998; *Praskovskaya and Praskovsky*, 2001; *Praskovsky and Praskovskaya*, 2001]. We were not confident enough to share our beliefs and hopes with the world's scientific community until the questions Q1 - Q5 were answered affirmatively, beyond a shadow of doubt. During this time, theoretical background of the SF-based method for analyzing received signals for SA radars was developed and is presented in this paper. Both the first and second approaches (models for the diffraction pattern and scattering medium) are considered in sections 2 and 3, respectively. Analysis in sections 2 and 3 is executed for pure received signals without noise while the effect of noise is considered in section 4. Advantages and shortcomings of SF with respect to CF and spectra are discussed in section 5.

[12] We developed a practical technique for measuring the mean horizontal velocity components and turbulence characteristics of a scattering medium by SA profiling radars. The technique is called UCAR-STARS which stands for the “University Corporation for Atmospheric Research - STructure function Analysis of Received Signals.” The name was coined by the NCAR/RAP director, Brant Foote. A detailed description of the UCAR-STARS method will be presented in the next paper. The UCAR-STARS practical application to SA profiling radars will be illustrated with simulated signals; the signals are generated using the *Holdsworth and Reid* [1995] simulation technique. The measurement error analysis and data rejection criteria will be presented and discussed.

[13] UCAR-STARS was intensively tested with simulated data over a wide range of atmospheric conditions. It was also tested for real signals from the NCAR Multiple Antenna profiler (MAPR); good agreement with sonic anemometer data, FCA, and HAD methods was found. UCAR-STARS measurements for the Esrange MST radar (ESRAD) were found in reasonable agreement with FCA measurements. The results of these tests will be published in a separate paper together with R.D. Palmer, T.-Y. Yu, S.A. Cohn, W.O.J. Brown, P.B. Chilson, and V. Barabash.

[14] A comprehensive test of the UCAR-STARS was accomplished for the Middle and Upper Atmosphere (MU) radar. Different configurations of receiving antennas were analyzed and good agreement with DBS and HAD methods was found. The results will be published in a separate paper together with G. Hassenpflug, M. Yamamoto, and S. Fukao.