### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Structure Functions for a Random Scalar Field
- 3. Structure Functions for a Spaced Antenna Profiling Radar
- 4. Structure Functions for Signals With Noise
- 5. Discussion
- 6. Summary
- Appendix A: Coefficients in Equations (7) - (10)
- Appendix B: Variables θ
- Appendix C: Mean Speed of the Diffraction Pattern on a Ground
- Acknowledgments
- References

[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*_{cor} integral scale of the random processes Ψ_{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*_{cor} and *T*_{cor}. The term “locally” is used throughout this paper in the same sense as in a theory of the fine-scale turbulence [*Monin and Yaglom*, 1975, section 21]. In particular, the “locally stationary” stands for stationary over a time period which is much smaller than *T*_{cor}. 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 [*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.

### 5. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Structure Functions for a Random Scalar Field
- 3. Structure Functions for a Spaced Antenna Profiling Radar
- 4. Structure Functions for Signals With Noise
- 5. Discussion
- 6. Summary
- Appendix A: Coefficients in Equations (7) - (10)
- Appendix B: Variables θ
- Appendix C: Mean Speed of the Diffraction Pattern on a Ground
- Acknowledgments
- References

[112] Theoretical equations in sections 2 - 4 provide the affirmative answer to questions Q1 - Q5 in section 1. Let us consider the physical content of the derived equations.

[113] 1. Contrary to equations for CF and spectra, equations for SF of received signals of any order *p*≥2 can be derived and applied to practical measurements. Spectra are the second order functions, and it seems fairly impossible to derive practically useful equations for CF of received signals at *p*>2 unless too many restrictive assumptions are made.

[114] The higher-order SF provide additional information about a scattering medium to that obtained at *p* = 2. For example, equations (49) and (50) provide estimates of the mean velocity components of a scattering medium supplemental to those from equations (42) and (43). Another example is equation (54) for the fourth-order moment 〈*w*^{4}〉 of the vertical turbulent velocity component *w*(*t*); this value can be estimated from neither spectra, nor CF and SF at *p* = 2.

[115] One can also apply the higher-order SF to a more detailed identification of a scattering medium. For example, one can estimate the diffraction pattern's correlation time and correlation length with FCA/FSA and HAD methods. Applying specific scattering models, one can relate these values to the aspect sensitivity, correlation length of a scattering medium, shape of scatterers, etc. [e.g., *Hocking et al.*, 1989; *Doviak et al.*, 1996]. As shown in section 3, coefficients in equation (34) at *p* ≥ 2 are related to local characteristics of a scattering medium while independent of the medium's motion; the coefficients provide information on the medium itself. The statement is obvious at *p* = 2. One can easily relate coefficients at different to corresponding coefficients in CF (e.g., to coefficients *A*, *B*, and *H* in equation (89) below), and then apply well known relations from the FCA/FSA and HAD methods. However, such straightforward procedure is not constructive. As explained below, equations for SF can be derived at a smaller number of less restrictive assumptions than those for CF and spectra. One can apply directly different models of a scattering medium to deriving equations for SF, and establish relations between “measurable” coefficients in equation (34) and characteristics of the medium. The utility of SF at *p*>2 becomes very clear in this case. Indeed, CF and SF at *p* = 2 provide only three independent parameters such as *A*, *B*, and *H* in equation (89), or in equation (42). A similar equation at *p* = 3 provides four additional parameters, and equation (49) at *p* = 4 provides five more parameters Being used together, these parameters can provide more detailed and reliable identification of scattering medium (type and shape of scatterers, spatial and temporal scales of their distribution, etc.) than only three parameters at *p* = 2.

[117] Equations (53), (64), (65), and (54) for the moments as well as similar equations at *p* ≠ 2, 4 are theoretically exact until assumptions 1 - 5 are satisfied. It is important that the measured moments are not affected by the beam broadening as the spectral width is [e.g., *Hocking*, 1983; *Hocking et al.*, 1989].

[119] However, there are significant differences between equation (89) and equations (6) - (10). Equations (6) - (10) were derived with only two assumptions T1 and T2; the assumptions are satisfied in practically any realistic conditions. Equations (6) - (10) are valid for statistically inhomogeneous and locally stationary diffraction pattern, while equation (89) implies the pattern to be globally statistically stationary and homogeneous. The most restrictive FCA assumption is equation (89) itself, which implies a functional identity between spatial–temporal cross CF and spatial/temporal auto CF. As emphasized by *Briggs* [1984, p. 174], “This assumption, that the spatial and temporal correlation functions have the same functional form, is never likely to be exactly true, and should be tested for each application of the theory to any experimental data.”

[120] 4. Equation (34) at *p* = 2 with coefficients (42), (43), (48), and (60), (62), (63) are similar to those in the HAD method [*Holloway et al.*, 1997, sections 2 and 3; *Doviak et al.*, 1996]. Similarity can be clearly seen after decomposing CF into the Taylor series at τ0, and applying the standard relation between CF and SF at *p* = 2. The decompositions are not presented here because the formulas are slightly different. CF in *Doviak et al.* [1996] and *Holloway et al.* [1997] are derived for complex signals given by equation (19), while SF are derived for the signal power given by equations (21) and (22). The basic similarity between equations for SF in section 3 and equations for CF in the HAD method is in that they directly relate “measurable” parameters of CF and SF to characteristics of the scattering medium.

[121] The major difference between the methods is in the assumptions under which the equations were derived. All equations in sections 3 and 4 were derived with assumption 1 about the local stationarity of the scatterer's characteristics while CF/spectra-based methods require the global stationarity and global homogeneity of a scattering medium [e.g., *Doviak et al.*, 1996, p. 161; *Liu et al.*, 1990, p. 552]. Equations (42) and (43) for SA profiler with highly overlapping receivers utilize one more assumption 2; the assumption is taken for granted in the FCA/FSA and HAD methods [e.g., *Briggs*, 1980, pp. 824 and 830; *Doviak et al.*, 1996, p. 163]. Equation (60) for a separation implies additional assumption 3a about statistically uniform distribution of scatterers in a horizontal plane; the assumption is used in the HAD method as well [*Doviak et al.*, 1996, p. 161]. To derive equations (48), (62), and (63) for the turbulence intensity, additional assumptions 4 and 5 have been applied. These assumptions are also used in the HAD method in more restrictive formulation. In addition, the HAD method requires an assumption about a specific form of CF or spectrum of the refractive index irregularities e.g., the Gaussian one [*Doviak et al.*, 1996].

[122] 5. Therefore, equations (6) - (10) for SF of the diffraction pattern, and equations (34), (42), (43), (48), (60), (62), and (63) for SF of the scattering medium were derived at a smaller number of less restrictive assumptions than the corresponding equations in FCA/FSA and HAD methods. That makes equations for SF of received signals more universal than those for CF and spectra. Equations for SF are asymptotically exact at sufficiently small separations they contain neither empirical functions, nor constants.

[124] Another example is a relation between the mean speed of the diffraction pattern on a ground and that of the scattering medium The FCA/FSA method considers a fixed ratio for all atmospheric SA profilers [e.g., *Briggs*, 1980; *Larsen and Röttger*, 1989; *Hocking et al.*, 1989]. *Doviak et al.* [1996] have shown that the ratio can be any value between 2 and 1. However, the Doviak et al. result for two scatterers in the illuminated volume cannot be applied to intrinsically statistical SA methods; the methods are theoretically substantiated only at Using equations for SF in sections 2 and 3, one can derive an exact statistical relation between for any specific SA profiler; the derivation is presented in Appendix C. It is shown that the physical mean speed of the diffraction pattern is always twice larger than equation (C5). At the same time, the measured mean speed heavily depends on the radar parameters: it can be any value from 2, and smaller (equation (C8) and Figure 4).Equations (C9) and (C11) can be used for estimating from in practical measurements, for example, with the FCA/FSA methods.

[125] 6. Equations for SF of received signals are fully “controllable.” Starting from equation (35) which implies the only assumption 1, one can derive equations for the coefficients in equation (34) for any specific SA radar by choosing sufficient number of more or less restrictive assumptions. For example, equations (42) and (43) for very close receivers were derived with assumptions 1 and 2, while derivation of equation (60) for a separation required one more assumption 3a.

[127] These examples illustrate the mathematical elegance of the SF-based approach. One keeps derivation of the equations under complete control, and physical content of the resulting equations is always clearly defined.

[128] 7. The optimal SA radar configuration for applying SF is one with highly overlapping receiving antennas. More universal equations (42), (43), (53), and (54) can be used for estimating the mean horizontal velocity components and turbulence characteristics of a scattering medium for such a configuration. Applying highly overlapping receiving antennas as in Figure 2, one can significantly increase SNR of received signals.

[129] Another practical advantage of the SF-based approach can be seen from equations (34), (42), (43), (48), (61), (64), and (65). The equations depend on the only radar parameters while are independent of the receiver's size and shape. Therefore, the equations can be applied to receivers with arbitrary size and shape.

[130] 8. Equations for real received signals with noise in section 4 were derived for two scalar signals of arbitrary physical nature; only two realistic assumptions 1 and 6 were used in the derivation. The equations are applicable not only to a SA profiler as those in section 3 but to any multiple receiver remote sensor. It follows from equations (80), (84), and (88) that noise affects only a technique for calculating SF in the LHS of equations in section 3. Therefore, one can derive equations for any specific SA radar by considering pure signals from atmospheric scatterers. Noise can be taken into account in practical measurements by applying equations (80), (84), and (88) to received signals before applying the equations as in section 3 to the coefficients of SF for pure signals; the procedure is described in the paper on the UCAR-STARS method.

[131] Equation (80) illustrates a remarkable feature of SF: they are not very sensitive to noise with a large temporal correlation lag, for example ground clutter or point targets. This feature has a simple physical reason. SF are statistical moments of the difference between signals. The noise component with a large temporal correlation lag can be produced only by objects which remain in the illuminated volume or within the side lobes for a relatively long time interval without significant changes in their size and shape, e.g., buildings, cars, airplanes, etc. The statistical difference between signals from such objects in closely located receivers is quite small and it does not significantly affect SF.

[132] 9. It has been well recognized in turbulence research that SF represent a very powerful tool for theoretical analysis of highly correlated signals. The results in sections 2 - 4 illustrate that SF can also be useful in remote sensing for analyzing signals from closely located receiving antennas. Simplicity, relative universality, and mathematical elegance of equations for SF result from the presence of two small parameters: temporal separation and spatial separation

[133] 10. The presence of two small parameters is a source of the SF's advantages and, at the same time, of their shortcomings. As any differential tool, SF are affected by the noise component with a zero temporal correlation lag much stronger than CF. Such noise affects CF only at τ = 0 while SF are affected at any

[134] To ensure sufficient correlation between the signals for applying SF, the receiver centers must be so close to each other as to satisfy limitation (71). The limitation can be satisfied for all existing atmospheric SA profilers either directly, or by combining signals from the receivers. However, this limitation can become a serious obstacle for applying SF to some SA remote sensors.

[136] The most significant shortcoming is that SF can be applied only to real values such as the signal's power or amplitude. One cannot utilize advantages of coherent radars by analyzing the in-phase and quadrature components in equation (19). This inevitably leads to losing information on the radial velocity unless strong oversampling along the beam is accomplished. If the gate separation of the order of λ is applied, one can estimate with SF although such data acquisition strategy is not practical. For a realistic gate separation, one can estimate the radial velocity from the Doppler spectrum. One cannot, however, estimate it with SF in the direct way as that with CF and spectra.

### 6. Summary

- Top of page
- Abstract
- 1. Introduction
- 2. Structure Functions for a Random Scalar Field
- 3. Structure Functions for a Spaced Antenna Profiling Radar
- 4. Structure Functions for Signals With Noise
- 5. Discussion
- 6. Summary
- Appendix A: Coefficients in Equations (7) - (10)
- Appendix B: Variables θ
- Appendix C: Mean Speed of the Diffraction Pattern on a Ground
- Acknowledgments
- References

[137] This paper presents the SF-based approach to analyzing received signals for SA radars. Parameters of cross and auto SF of any order *p* ≥ 2 are related to characteristics of the diffraction pattern on the ground. The equations at *p* = 2 are similar to those for CF in the FCA method. Parameters of cross and auto SF of received signals at *p* = 2, 4 are related to characteristics of the scattering medium for atmospheric SA radars with a vertically directed transmitted beam. The equations at *p* = 2 are similar to those for CF in the HAD method.

[138] Several advantages of the SF-based approach to analyzing signals from closely located receivers with respect to the spectra and CF-based approaches are demonstrated. For example, relations between parameters of SF and characteristics of the diffraction pattern and the scattering medium can be derived with a smaller number of less restrictive assumptions than the corresponding relations for CF and spectra. More universal and asymptotically exact equations for SF provide a deeper insight and more rigorous theoretical background for the SA methods. Equations for SF of received signals are fully “controllable”: one can derive the equations for any specific SA radar by choosing a sufficient number of more or less restrictive, while always clearly specified assumptions. Turbulence characteristics can be related to the SF parameters more rigorously than to parameters of CF and spectra. One can increase SNR of received signals with the SF-based approach by using highly overlapping receivers. SF are not very sensitive to noise with a large temporal correlation lag, e.g., ground clutter and point targets such as airplanes.

[139] V.I. Tatarskii emphasized the major advantage of the SF-based approach: equations for SF of received signals of any order *p* ≥ 2 can be derived and applied to practical measurements. On the contrary, only the second order CF can be dealt with and spectra are the second order functions as well. The higher-order SF provide additional turbulence characteristics of scattering medium, such as the higher-order moments of the vertical turbulent velocity component; the characteristics cannot be measured by any other technique. SF at *p* > 2 can also provide supplemental information to that at *p* = 2 about the scattering medium, such as type and shape of scatterers, spatial scales of their distribution, etc.

[140] On the other hand, SF are more strongly affected by noise with a zero temporal correlation lag, and practical application of the SF-based approach imposes more restrictive limitations on the SA radar than CF and spectra-based approaches. In particular, SF require the receiver centers to be quite close to each other, and the sampling interval to be small enough. The limitations can be easily satisfied for any existing atmospheric SA profiler, although they could become a serious obstacle for applying SF to some remote sensors. SF can be applied only to scalar processes such as the signal's power or amplitude which results in losing information on the radial velocity.

[141] In general, the SF-based approach to analyzing signals from closely located spaced receivers seems to be a promising technique that merits further study. The near-future work is to consider other types of measurements such as those at low elevation angles, and with HF and MF atmospheric SA profilers. The use of SF at *p* ≥ 2 for identification of a scattering medium is the largest challenge.

[142] It is natural to expect that, depending on measurement conditions, characteristics of a scattering medium can be estimated more effectively with either spectra and CF, or SF. The SF-based approach can become a useful alternative to the CF and spectra-based methods and a combination of several techniques may be optimum.