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Abstract

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

[1] This paper describes an alternative approach to analyzing multiple received signals for spaced antenna radars. The approach is based on computation and analysis of the different order auto and cross structure functions (SF) for the received signal's power. SF of order p ≥ 2 is the p-th order statistical moment of the difference between signals. Equations for cross and auto SF at p = 2 and 4 are derived for a spaced antenna radar with a vertically directed transmitted beam. SF parameters in these equations are related to the mean horizontal velocity components and turbulence characteristics of a scattering medium. Theoretical limitations on applying the derived equations to a specific radar are presented. Advantages and shortcomings of the proposed approach with respect to the correlation function and spectra-based approaches are discussed as well. The major advantage of the proposed approach is that SF of any order p ≥ 2 can be derived and applied to practical measurements; the higher-order SF provide additional information about the scattering medium.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions for a Random Scalar Field
  5. 3. Structure Functions for a Spaced Antenna Profiling Radar
  6. 4. Structure Functions for Signals With Noise
  7. 5. Discussion
  8. 6. Summary
  9. Appendix A: Coefficients in Equations (7) - (10)
  10. Appendix B: Variables θ
  11. Appendix C: Mean Speed of the Diffraction Pattern on a Ground
  12. Acknowledgments
  13. 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 Ψ1(t) and Ψ2(t) can be defined as:

  • equation image

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 Ψ1(t) − Ψ2(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:

  • equation image

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 Lcor and temporal Tcor integral scale of the random processes Ψ1(t) and Ψ2(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 Lcor and Tcor. 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 Tcor. 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.

2. Structure Functions for a Random Scalar Field

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

[15] Equations for SF of any order p ≥ 2 for a moving random scalar field of arbitrary physical nature are derived in this section. The equations are derived both in the physical (time) domain and in the Fourier (spectral) domain; the latter derivation was suggested by V.I. Tatarskii.

2.1. Derivation in the Time Domain

[16] Consider a random scalar field Φ(⟶, t) of arbitrary physical nature; here Φ can be temperature, pressure, or any other scalar. The field Φ(⟶, t) is defined inside a finite volume Ω of arbitrary shape, and temporal interval T; that is at ∈Ω, and t∈[−T/2,T/2] such as LminLcor, and TTcor. Hereafter, Lcor and Tcor denote the spatial and temporal integral scales of the field Φ(⟶, t), and Lmin is the minimum characteristic size of the volume Ω. The infinite spatial volume and/or temporal interval are special cases at Lmin [RIGHTWARDS ARROW] ∞, and/or T [RIGHTWARDS ARROW] ∞. The Cartesian coordinate system = {x, y, z} with an arbitrary origin is used hereafter and symbols in the brackets {} denote the Cartesian components of a vector.

[17] Consider equation image to be the instantaneous velocity vectors of the scalar field's motion. Let us make two assumptions about the field.

[18] Assumption T1: The values Φ(⟶, t), U(⟶, t), V(⟶, t), and W(⟶, t) are the locally statistically stationary random processes at ∈ Ω.

[19] It follows from assumption T1 that the instantaneous velocities can be presented as:

  • equation image

Hereafter equation image and equation image are the mean and turbulent velocities in location . It also follows from assumption T1 that ensemble averages of any function of Φ(⟶, t) and equation image are independent of time t; argument t is omitted whenever it is not confusing.

[20] Assumption T2: The field Φ(⟶, t) at ∈Ω, t ∈ [−T/2, T/2] belongs to the class equation image it has continuous temporal and spatial derivatives of at least the first order.

[21] It follows from this assumption that Φ(⟶, t) can be decomposed into the spatial/temporal Taylor series in the vicinity of any ∈ Ω and t ∈ [−T/2, T/2].

[22] Consider two arbitrary, while very close spatial locations , + δ⟶ ∈ Ω, and a very small temporal separation τ, such as |δ| ≪ Lcor and |τ| ≪ Tcor. Cross and auto SF of order p ≥ 2 for a pair of random processes Φ(⟶, t) and Φ( + δ⟶, t) are defined as [Tatarskii, 1971, section 1A; Monin and Yaglom, 1975, section 13]:

  • equation image

From now on, the asymptotic limits |δ| [RIGHTWARDS ARROW] 0 and τ [RIGHTWARDS ARROW] 0 are considered. Using assumption T2, one can present the process Φ( + δ⟶, t + τ) as:

  • equation image

where equation image is the gradient vector of the scalar field Φ(⟶, t) in location at instant t. Hereafter, a symbol • denotes the scalar product of two vectors and the standard notations for the partial derivatives are used as:

  • equation image

The derivatives Φt(⟶, t), Φx(⟶, t), Φy(⟶, t), and Φz(⟶, t) have a very simple physical meaning. They describe the temporal and spatial rates of changes of the field Φ(⟶, t) in location at instant t. One can consider rates of temporal and spatial changes and turbulent velocities equation image as statistically independent random processes because they describe physically different characteristics of the field Φ(⟶, t).

[23] One should apply the total derivative dΦ/dt in decomposition (3) because Φ(⟶, t) is the spatially and temporally random, as well as the randomly moving field. To estimate the derivative, one can use the following equation:

  • equation image

At a small while finite temporal separation Δt, this equation is referred to as the “local Taylor hypothesis” [Tatarskii, 1971, section 29]. In the asymptotic limit Δt [RIGHTWARDS ARROW] 0, the equation is exact with accuracy O(Δt2). Using equation (4), the total derivative can be presented as:

  • equation image

[24] Combining equation (1), (3), and (5) with definitions (2), one can present cross and auto SF in the following form:

  • equation image

with the coefficients:

  • equation image
  • equation image
  • equation image
  • equation image

The expansion of equations (7) - (10) in the Cartesian coordinate system at p = 2 is presented in Appendix A. Similar expansions at p ≥ 3 can be easily obtained from equations (7) - (10) although they are too lengthy for being presented in the paper.

2.2. Derivation in the Spectral Domain

[25] To simplify the equations, the second order cross and auto SF for a one-dimensional field are presented. Let us consider a moving random scalar field F(x, t) of an arbitrary physical nature. The field is defined at x ∈ [−L/2, L/2] and t ∈ [−T/2, T/2], such as LLcor, and TTcor. One can present F(x, t) as the Fourier series, for example as:

  • equation image

where Ai, Bi, and ui are the amplitudes and the velocity for the Fourier component with the wave number ki = 2πi/L. The random amplitudes Ai, Bi are statistically independent of random velocities ui, and all these numbers are independent of x and t. Using equation (11), one can present cross SF for the field F(x, t) at p = 2 as follows:

  • equation image

The prime superscript is used in this and the following subsections for distinguishing SF and their coefficients derived in the spectral domain for a one-dimensional field F(x, t) from those derived in the time domain in section 3.1 for a three-dimensional field Φ(⟶, t).

[26] Assumption F1: The field F(x, t) is statistically stationary and homogeneous at x ∈ [−L/2, L/2] and t ∈ [−T/2, T/2], and the velocities ui are statistically stationary random values.

[27] It follows from this assumption that:

  • equation image

where δij is the Kroneker symbol; Ui and vi denote the mean and turbulent velocities of the Fourier component with the wave number ki. It also follows from assumption F1 that equation image is independent of x and t.

[28] Applying equations (13) to equation (12), one can derive in the asymptotic limit δx [RIGHTWARDS ARROW] 0, τ [RIGHTWARDS ARROW] 0 the following expression:

  • equation image

[29] Assumption F2: All Fourier components of the field F(x, t) have the same mean velocity and turbulence intensity; that is:

  • equation image

[30] Applying equation (15) to equation (14), one can present cross SF equation image in the form:

  • equation image

with the coefficients:

  • equation image

[31] The auto SF of the field F(x, t) in any location x ∈ [−L/2, L/2] is a particular case of the cross SF (16) at δx = 0:

  • equation image

where equation image

2.3. Physical Content of Equations for Structure Functions

[32] Three features of the derived equations for SF of a moving random scalar field are relevent to the subject of this paper.

  1. 1. Each coefficient in decompositions (6), (16), and (18) of cross and auto SF describes specific characteristics of the random scalar field.
  2. Coefficients ap(, δ) at p ≥ 2 in equation (6), and equation image in equation (16) depend solely on the local parameters of a scalar field such as the p-th order moments of the instantaneous gradients (see Appendix A), while independent of the field's motion.
  3. Coefficients bp(, δ) at p ≥ 2 in equation (6), and equation image in equation (16) depend on the mean velocity of the field's motion, while independent of turbulence.
  4. Turbulence affects only coefficients cp(, δ), dp,auto() at p ≥ 2 in equation (6), and equation image in equations (16) and (18).
  5. 2. Derivation of equations for cross and auto SF in time and spectral domains provides similar results, equations (6), (16), and (18). However, derivation in a time domain is simpler and more straightforward than that in a spectral domain. It is important that derivation in a time domain provides more general results at less restrictive assumptions. Indeed, equations (6) -(10) for SF of any order p ≥ 2 are applicable to a statistically inhomogeneous scalar field. Only two realistic assumptions T1 and T2 about local stationarity and continuous first-order derivatives were sufficient for deriving the equations in the time domain. At the same time, the global stationarity and homogeneity of both the field F(x, t) and the speed ui was required to be assumed for deriving equations (16) - (18) in the spectral domain (assumptions F1 and F2). Furthermore, we do not see a possibility of deriving equations for SF at p > 2 in a spectral domain without adopting too restrictive assumptions.
  6. Advantages of applying structure functions in a time domain have been well recognized in turbulence research [e.g., Monin and Yaglom, 1975, chapter 8]. Following the pioneering paper by Kolmogorov [1941], all major results on the small-scale turbulence have been first obtained in a time domain and some of the results were then reproduced in a spectral domain. For example, the Kolmogorov's [1941] “two-thirds law” in a time domain has been reproduced by Oboukhov [1941] in a spectral domain; the latter is usually referred to as the “five-thirds law.” For this reason, all equations in this paper will be derived in a time domain.
  7. 3. Equations (6) - (10) are applicable to a moving random scalar field of arbitrary physical nature. Consider the diffraction pattern on a ground from the refractive index irregularities in the atmosphere illuminated by the radar's transmitter. The power of such a pattern is a moving random scalar field which satisfies assumptions T1 and T2 in practically any condition. Random processes Φ(⟶, t) and Φ( + δ⟶, t) in equations (2) - (10) can be considered as time series of the signal's power collected by two receiving antennas with centers and . Therefore, equations (6) - (10) relate the SF parameters ap(, δ), bp(, δ), cp(, δ), and dp,auto() in equation (2) to characteristics of the diffraction pattern, such as its mean velocity components and turbulence intensity.

[33] On the other hand, no one is interested in characteristics of the diffraction pattern itself. The objective of remote sensing is to obtain information about the scattering medium; neither equations (6) - (10), nor equations (16) - (18) provide such information.

[34] One can, however, expect that for some types of remote sensors, and for some types of scattering media, the SF parameters ap(, δ), bp(, δ), cp(, δ), and dp,auto() in equation (6) can be related to some characteristics of the medium. According to V.I. Tatarskii, this statement can be considered as a “believable assumption.” If one accepts this assumption, equations (6) - (10) provide a strong motivation for analyzing SF of received signals for multiple antenna remote sensors. An example of such analysis is presented in the next section.

3. Structure Functions for a Spaced Antenna Profiling Radar

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

[35] All existing operational and research SA atmospheric radars are mainly used in the profiling mode with vertically directed transmitted beam. As typical examples, one can refer to the NCAR MAPR [Cohn et al., 1997]; the ESRAD [Chilson et al., 1999]; and the MU radar [Fukao et al., 1985a, 1985b]. Traditional methods of data analysis for SA radars (FCA, FSA, and HAD) have been developed for the profiling mode as well [Briggs, 1984; Briggs and Vincent, 1992; Holloway et al., 1997]. Therefore, the SA profiling radar is a natural choice for the first test of the SF-based approach.

[36] Equations for the second and fourth order cross and auto SF of the received signal power for an atmospheric SA profiling radar are derived in this section. The SF parameters are related to the mean horizontal velocity components and turbulence characteristics of the scatterer's motion. Pure received signals from a scattering medium with no noise are considered; signals with noise are analyzed in section 4.

3.1. Received Signal Power for a Bistatic Radar

[37] Consider a schematic SA profiling radar of quite a general configuration as in Figure 1. The radar's phased array antenna consists of ne × ne identical elements with a size d. The elements could be Yagis as in the MU radar and the ESRAD, or panels as in the NCAR MAPR. Consider ne to be sufficiently large; ne = 12 is illustrated in Figure 1a. Let all equation image elements be used as a pulsed wave transmitter T with the vertically directed beam, the transmitting antenna size is D = ned. Consider a Cartesian coordinate system = {x, y, z} with vertically directed z axis, and x and y axes in a horizontal plane. The origin of the coordinate system is placed at height R above the transmitter's center as in Figure 1b; the center's coordinate is T = {0, 0, −R}. Hereafter R is an arbitrary measurement center-range.

image

Figure 1. Schematic of a phased array profiling radar. (a) Top view: the transmitting antenna T with a size D consists of equation image identical elements with a size d. The bullet indicates the transmitter center T. (b) Side view: the origin of the Cartesian coordinate system is located at height R above the transmitter's center. A shaded area illustrates the illuminated volume with the center-height R.

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[38] Consider Na groups of na × na elements each, na < ne, to be operating as Na spaced receiving antennas Ra,l, l = 1, 2, … , Na; Na = 4 and na = 10 is illustrated in Figure 2. The receiving antennas (receivers) are identical in shape, have the same size Da = nad, and collect signals from the same range R. Square-shaped antennas are shown in Figures 1 and 2 as an example; the transmitting and receiving antennas can be of any shape. Receivers in Figure 2 overlap each other; such configuration provides a much higher signal-to-noise ratio (SNR) than adjacent and separated receivers. Let us denote by equation image coordinates of the receiver centers as in Figure 2. Each transmitter-receiver pair equation image is a bistatic radar.

image

Figure 2. A schematic depicting equation image receiving antennas equation image Each antenna with a size Da consists of equation image elements. The receiver centers equation image are indicated by bullets.

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[39] The transmitted waves are scattered by the refractive index irregularities within the illuminated volume. These irregularities are referred to as scatterers independent of their nature and type of scattering. Among the various available scattering models we select, as did Liu et al. [1990] and Doviak et al. [1996], the volume-scattering model to describe the scattering medium.

[40] A portion of scattered waves reaches the receivers and induces signals in the receiving channels. The current induced in the internal resistance Rint of the matched-filter antenna Ra,l can be presented in the standard complex form as [Doviak et al., 1996]:

  • equation image

where Ia,l(t) and Qa,l(t) are the in-phase and quadrature components of the induced signal; the subscript i identifies the i-th scatterer; Ns is the number of scatterers; Δn(i, t) is the magnitude of the fluctuation in the refractive index field (the refractive index irregularity) produced by a scatterer in the location equation image at instant t; equation image, and equation imagea,l(i) are the range weighting function, the transmitting antenna gain function, and the receiver's field of view gain function for the i-th scatterer, respectively; rT(i) and ra,l(i) are distances to the scatterer from the transmitter and receiver; λ is the wavelength of the transmitted signal; and equation image The constant C is given by:

  • equation image

where PT is the average power of transmitted pulse. Equation (19) is valid for the far zone rT ≥ (2D2/λ) and equation image To satisfy the far zone requirements, sufficiently large range R is considered below, such as:

  • equation image

where σr defines the width of the range weighting function in equation (23) below.

[41] For a pulsed wave radar, Ia,l(t) and Qa,l(t) are discrete time series with the sampling time interval δt = NCI / PRF. Hereafter, PRF and NCI denote the transmitter pulse repetition frequency and the number of coherent integrations. Very small δt [RIGHTWARDS ARROW] 0 is considered below, and Ia,l(t) and Qa,l(t) are interpreted as continuous functions of time t; limitations on finite δt are discussed in section 3.5.

[42] The instantaneous signal power is defined as:

  • equation image

The received power Pa,l(t) from scatterers in the illuminated volume with no contribution from noise, clutter, or other contaminants is defined as the pure received signal from range R in the receiving antenna Ra,l with center a,l; it is denoted below as S(a,l, t).

[43] Combining equations (19) and (21), one can get:

  • equation image

An argument t is omitted whenever it is not confusing, for example, i(t) is denoted as i in equations (19) and (22).

3.2. Signals From a Pair of Receiving Antennas

[44] Consider any pair of receivers Ra,l1 and Ra,l2, (l1 ≠ l2) = 1, 2, …, Na, for SA radar in Figures 1 and 2. Let us specify equation (22) for signals from these receivers.

[45] Following Doviak et al. [1996] and taking into account inequalities (20), one can approximate the range weighting function, the transmitting antenna gain function, and the receiver's field of view gain function in the chosen coordinate system as:

  • equation image

where σr depends on the width of the transmitted pulse. The transmitter's beam width σ and the receiver's field of view width σa can be estimated as:

  • equation image

where γ, γa ≈ 0.5 are constant for a given SA radar; in general γa ≠ γ.

[46] It follows from equations (20) and (24) that

  • equation image

at l, l1, l2 = 1, 2, …, Na, and i = 1, 2, …, Ns. Hereafter δl2,l1 = a,l2a,l1 = {δxl2, l1, δyl2,l1, 0} is a separation between the receiver centers.

[47] Combining equations (20) and (22) - (25), one can present a signal S(a,l1,t) as:

  • equation image

where

  • equation image

and equation image The argument equation image in equation image is used to emphasize that Δni is the reflectivity of a scatterer in location i(t) at instant t for the receiving antenna Ra,l1 with center a,l1. The insignificant small terms are neglected hereafter.

[48] To describe a signal S(a,l2, t + τ), one should take into account random motion and changes in the reflectivity of each scatterer during the interval [t, t + τ] at τ [RIGHTWARDS ARROW] 0. Hereafter, the temporal separation τ is equation image Choosing a reasonable value of kmax≈ 2 - 5, one can always satisfy the condition τ [RIGHTWARDS ARROW] 0 because the asymptotic limit of δt [RIGHTWARDS ARROW] 0 is considered.

[49] Let us denote by equation image the mean velocity vector of i-th scatterer over the interval [t, t + τ]. In the asymptotic limit of τ [RIGHTWARDS ARROW] 0, one can consider equation image as the instantaneous velocity vector of i-th scatterer at instant t for i = 1, 2, …, Ns.

[50] Let us make the key assumption of the derivation.

[51] Assumption 1: The instantaneous velocity components U(i, t), V(i, t), and W(i, t), as well as all other random characteristics of each scatterer, are the locally statistically stationary random processes.

[52] It follows from this assumption that the random vector equation image can be presented as:

  • equation image

where equation image and equation image describe the mean and turbulent motion of i-th scatterer. For the locally statistically stationary random processes, the ensemble averages are independent of time t. In general, the mean values equation image can differ for different scatterers. Neglecting the terms O2), one can define the location of i-th scatterer at instant t + τ as:

  • equation image

Neglecting the terms O(|δl2,l1|2, τ2) at small separation δl2,l1 and τ [RIGHTWARDS ARROW] 0, one can present the reflectivity of i-th scatterer in location i(t + τ) at instant t + τ for the receiving antenna Ra,l2 with center a,l2 as follows:

  • equation image

where Δni = Δn(a,l1, i, t) and i = i(t). The derivative ∂Δni/∂t characterizes the instantaneous temporal rate of change in the reflectivity of i-th scatterer at instant t for the antenna Ra,l1 with center a,l1. This rate depends on the rate of change in the scatterer's shape, size, and/or orientation. In the asymptotic limit τ [RIGHTWARDS ARROW] 0, the term τ(∂Δni/∂t) is very small and is neglected below.

[53] The spatial derivatives in equation (30) are taken with respect to the center a,l1 of antenna Ra,l1. These derivatives characterize the instantaneous spatial rate of change in the reflectivity of i-th scatterer at instant t with the antenna Ra,l1 being moved in x or y direction. These rates depend on the scatterer's shape, size, and orientation as well as on the scatterer's location in the illuminated volume at instant t. It is easy to show that the terms with spatial derivatives are of the order of |δl2,l1|/R, and they are neglected below.

[54] Using equations (20), (22) - (25), (29), and (30), and neglecting the terms O2), one can present signal S(a,l2, t + τ) as:

  • equation image

where

  • equation image

The scatterer's location i and the velocity vector equation image are taken at the same instant t in equations (26) and (27) for signal S(a,l1, t) and equations (31) and (32) for signal S(a,l2, t + τ). The reflectivities Δni in equations (26) and (31) are taken at the same instant t, and for the same antenna Ra,l1.

3.3. Structure Functions of Received Signals

[55] The p-th order cross and auto SF for a pair of signals S(a,l1, t) and S(a,l2, t) can be defined in a nondimensional form as [Tatarskii, 1971, section 1A; Monin and Yaglom, 1975, section 13]:

  • equation image

The hat symbol ˆ is used hereafter to emphasize that functions equation image and equation image are nondimensionalized by σp,l. One can see from equations (27) and (32) that Cs, Bi, ϕij, hi, φij, and ψij do not depend on a,l1 at |a,l1T|/σ≪1, and neither are functions equation image and equation image.

[56] Results of section 2 indicate that cross and auto SF can be presented in the form of equation (6) as:

  • equation image

Coefficients âp, equation image, ĉp, and equation image in equation (34) are nondimensional values.

[57] Combining equation (33) with equations (26) and (31), one can get the following expression for cross SF of received signals S(a,l1, t) and S(a,l2, t) at sufficiently small |a,l1T|/σ, |δl2,l1|/σ ≪ 1 and τ [RIGHTWARDS ARROW] 0:

  • equation image

Auto SF of p-th order is also described by equation (35) at δl2,l1 = 0.

[58] Equations (35), (27), and (32) contain seven random variables which characterize each scatterer i at instant t: the coordinates xi, yi, zi; the velocity components Ui, Vi, Wi; and the reflectivity Δni. According to assumption 1 in section 3.2, all these random variables are considered as locally statistically stationary random processes; the ensemble average of any their function is independent of time t. Each one of the random variables describes a physically different characteristic of the scatterer; therefore, they can be considered as statistically independent random values. Indeed, a scatterer in a location with coordinate xi at arbitrary instant t can have any coordinates yi and zi, reflectivity Δni, and velocity components Ui, Vi, Wi; the same is obviously valid for different scatterers i and j. However, the velocity components can be correlated both for the same scatterer, and for different scatterers.

[59] Let us begin from the second order cross SF. At p = 2, equation (35) can be presented as:

  • equation image

[60] One can see from definitions (27), (32) that random values equation image differ significantly in magnitude. Indeed, equation image are very large, |φij| < 1 are small for the overlapping receivers in Figure 2 such as |δl2,l1|/D ≪ 1, and equation image are very small. Each term in the RHS of equation (36) is a product of a positive function, such as equation image and a sign changing, fast fluctuating function, such as equation image The average of a product of positive function and the sign changing, fast fluctuating function is negligibly small compared to the average of the positive function itself. Therefore, the only significant terms in the RHS of equation (36) are those at equation image containing no sines and cosines of ϕij, or ϕkm. Applying the standard trigonometric relations, decomposing exponents and trigonometric functions into the Taylor series and neglecting the insignificant terms, one can simplify equation (36) as follows:

  • equation image

where equation image The expression for θ00 is given by equation (B1) in Appendix B.

[61] Taking into account that ψi = O(τ), one can present equation (37) in the form of equation (34) with coefficients:

  • equation image

Combining these equations with definitions (27) and (32), one can derive the following expressions for the coefficients equation image

  • equation image
  • equation image
  • equation image

[62] It follows from equation (38) that coefficient equation image depends on the mean square of the reflectivity equation image and spatial location i(t) of the individual scatterers in the illuminated volume, while independent of the scatterer's motion. Coefficient equation image in equation (39) depends on the power-weighted mean velocity components equation image of individual scatterers, while independent of turbulence. The weighting is proportional to equation image and also depends on the scatterer's location in the illuminated volume. Coefficient equation image in equation (40) depends on the power-weighted second moments of the velocity components of individual scatterers, such as equation image and so on. The weighting for the second moments is the same as that for the mean components.

[63] Equations (39) and (40) cannot be used for measuring the mean horizontal velocity components and turbulence intensity of the scattering medium because they contain too many unknown values: the first and second moments of the velocity components equation image of individual scatterers for equation image To get practically fruitful equations for the mean horizontal velocity components, another assumption is necessary.

[64] Assumption 2: All scatterers in the illuminated volume move with the same mean horizontal velocity components:

  • equation image

[65] Using equation (41), equations (38) and (39) can be presented as follows:

  • equation image
  • equation image

Expressions for equation image are given by equations (B2) and (B3) in Appendix B.

[66] However, assumptions 1 and 2 are insufficient for deriving a practically fruitful equation for the turbulence intensity. At least three more assumptions should be adopted, e.g., assumptions 3, 4, and 5 below.

[67] Assumption 3: Spatial distributions of scatterers inside the illuminated volume are statistically symmetric with respect to the volume center for an arbitrary direction in a horizontal plane.

[68] It follows from this assumption that

  • equation image

where equation image are the probability density distributions of the i-th scatterer location in x and y directions. One can see from definition (27) that equation image is a symmetric function of its random arguments xi and yi. An average of the product of symmetric and anti-symmetric functions is zero; therefore, all terms in the RHS of equation (40) with the averages such as equation image become zero if assumption 3 is adopted.

[69] Assumption 4: Statistical characteristics of turbulent motion are the same for all scatterers in the illuminated volume:

  • equation image

[70] Assumption 5: The spatial integral scale Lw of the vertical turbulent velocity component wi for all scatterers satisfies at least one of the following conditions:

  • equation image

It follows from this assumption that a sum with terms equation image is small compared to that with terms equation image

[71] Applying assumptions 2 - 5 and equation (28) to equation (40), and taking into account that equation image for VHF and UHF atmospheric radars, one can derive the following expression:

  • equation image

[72] It follows from equations (42), (43), and (47) that the second order auto SF equation image can be presented in the form of equation (34) with the coefficient equation image as follows:

  • equation image

One can see that equation image.

[73] Equations (34), (42), (43), (47), and (48) relate the SF parameters of received signals at p = 2 for SA atmospheric profiling radar in Figures 1 and 2 to the mean horizontal velocity components and turbulence intensity of a scattering medium. SF of any order p > 2 can be derived in a similar way. For example, the fourth-order SF of received signals equation image can be presented in the form of equation (34) with the coefficients:

  • equation image
  • equation image
  • equation image
  • equation image

Expressions for the variables θ in equations (49) - (52) are given by equations (B4) - (B6) in Appendix B. Equations (49) and (50) are derived by applying assumptions 1 and 2, and equations (51) and (52) are derived by applying assumptions 1 - 5 at equation image

3.4. Mean Speed and Turbulence of a Scattering Medium

[74] Equation (34) for SF with coefficients (42), (43), (47), (48) at p = 2 and coefficients (49)(52) at p = 4 contain the mean horizontal velocity components equation image and the turbulence characteristics equation image of a scattering medium. The LHS terms in these equations are “measurable” coefficients equation image of cross and auto SF at p = 2 and 4.

[75] Let us consider radar in Figures 1 and 2. The centers of receiving antennas equation image are shown in Figure 3. One can see that Na = 4 receivers provide six baselines equation image such as neither two baselines are parallel to each other. Using signals from four receivers in Figures 2 and 3, one can measure six cross SF equation image of the second order and six cross SF equation image of the fourth order. One can then estimate six sets of coefficients equation image from measured cross SF. One can also measure equation image auto SF equation image at p = 2 and 4 for estimating coefficients equation image

image

Figure 3. Four receiving antennas with centers equation image provide six receiver pairs (baselines). Neither two baselines are parallel to each other.

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[76] The RHS of equation (42) contains a linear combination of three unknown variables equation image therefore, any three coefficients equation image for nonparallel baselines equation image provide a unique estimate of the variables. One can apply equation image to the linear equation (43) for obtaining unique estimates of equation image from three coefficients equation image Six baselines in Figure 3 provide formally equation image sets of three nonparallel baselines each; therefore, one can get up to 20 estimates of 〈U〉 and 〈V〉. However, only two sets are fully independent in the sense that none of the baselines are used in both sets. For example, the sets of baselines (1,2), (1,3), (1,4), and (2,3), (2,4), (3,4) are fully independent of each other.

[77] Similarly, equation (49) is a linear equation with five unknown variables; any five coefficients equation image for nonparallel baselines provide unique estimates of equation image Applying the variables to equation (50), one can get unique estimates of equation image for any three coefficients equation image

[78] Equations (42), (43), (49), and (50) contain only two radar parameters, separation between centers of receiving antennas δl2,l1, and the sampling time interval δt; these values are always known exactly in practical measurements.

[79] After variables equation image and the mean velocity components equation image are estimated with equations (42) and (43) for any three baselines in Figure 3, one can obtain several estimates of the turbulence intensity 〈w2〈 from these baselines using equations (47) and (48) as:

  • equation image

where equation image One can also obtain several estimates of equation image with equations (49) - (51) from coefficients equation image for any set of five baselines.

[80] After the variables equation image mean velocities equation image and intensity 〈w2〉 are estimated with equations (42), (43), and (47) - (50), one can obtain several estimates of the fourth order moment 〈w4〉 of the vertical turbulent velocity component w(t) with equation (52) as:

  • equation image

Equations (53) and (54) contain only two radar parameters λ and δt; these values are always known exactly in practical measurements.

[81] Therefore, cross and auto SF (34) with coefficients (42), (43), (47), (48) at p = 2, and (49) - (52) at p = 4 can be used for measuring the mean horizontal velocity components and turbulence characteristics of a scattering medium by atmospheric SA profiling radars. Equation (54) clearly demonstrates practical importance of SF at p > 2. The equation provides the fourth order moment 〈w4〉; this turbulence characteristic cannot be measured using the second order functions such as spectra, SF and CF at p = 2. The moments 〈wp〉 can be obtained for any p > 2 in a similar way by applying the p-th order auto and cross SF.

[82] Equations (42), (43), and (47) - (52) were derived in the asymptotic limit equation image This condition is satisfied only for highly overlapping receiving antennas such as in Figure 2; it can be easily satisfied in practical measurements with the MU radar. However, most existing SA radars have adjacent or separated receiving antennas, e.g., the NCAR MAPR and the ESRAD. One can still create overlapping receivers for these radars by combining received signals; the technique is discussed in the paper on the UCAR-STARS method. To apply equation (34) directly to received signals for SA radars with adjacent or separated receivers, equations for the coefficients should be derived for separation between the antenna centers equation image of the unity order.

[83] Let us consider second order cross SF (36) for a pair of receivers equation image with separation between their centers Δl2,l1. Notation Δl2,l1 for the separation |Δl2,l1|/D = O(1) is used for distinguishing it from a very small separation |δl2,l1|/D ≪ 1. Without loss of generality, one can consider x axis of the Cartesian coordinate system in Figure 1 to be directed along the baseline Δl2,l1; in this case Δl2,l1 = {Δx, 0, 0}. At |Δl2,l1|/D = O(1), equation (37) in the asymptotic limit τ [RIGHTWARDS ARROW] 0 can be replaced by the following, more general expression:

  • equation image

where equation image are given by equations (B1), (27), (32), and:

  • equation image

[84] Taking into account that ψik = O(τ), one can present equation (55) in the form (34) with the coefficients:

  • equation image

Applying assumptions 2 and 3, and equations (B1), (27), (32), and (56) to equation (57), one can present coefficients equation image as follows:

  • equation image

One can see that sums in the RHS of equation (58) for equation image depend on the scatterer's location xi in a different way; therefore, one cannot estimate 〈U〉 from these coefficients. To obtain practically fruitful equations, one should replace assumption 3 with a more specific assumption about spatial distribution of scatterers. For example, one can specify assumption 3 in the following way.

[85] Assumption 3a: Spatial distribution of scatterers inside the illuminated volume is statistically uniform for an arbitrary direction in a horizontal plane.

[86] It follows from this assumption that:

  • equation image

Using assumption 3a, one can directly calculate the ratios equation image in equation (58). Applying equation (59) to equations (58) and (B1), one can get:

  • equation image

Combining equations (60), one can obtain the following expression for a projection of the mean velocity of a scattering medium 〈U〉 on the baseline Δl2,l1:

  • equation image

One can estimate the mean horizontal velocity components of scattering medium with SA profiling radar by applying equation (61) to any pair of nonparallel baselines, for example (2,3) and (1,4) in Figure 3.

[87] Similar to equations (42) and (43), equation (61) depends on the only radar parameters Δl2,l1 and δt. Equation (61) looks more effective than equations (42) and (43) because it directly provides the projection 〈U〉 on a single baseline Δl2,l1. Therefore, one needs two nonparallel baselines for estimating 〈U〉 and 〈V〉 with equation (61), while equations (42) and (43) require three nonparallel baselines. On the other hand, equations (42) and (43) were derived by using assumptions 1 and 2, while derivation of equation (61) required one more assumption 3a.

[88] By applying assumptions 2, 3a, 4, and 5, one can present coefficient equation image in equation (57) at equation image as:

  • equation image

It follows from equation (62) that coefficient equation image in equation (34) at equation image is:

  • equation image

Superscript ′ is used in equation (63) to distinguish equation image from equation image in equation (48) at equation imageEquations (62) and (63) show that equation image One can see that expression (48) for equation image differs from equation (63) although auto SF is independent of a separation between receivers. The difference results from different expressions for equation image

[89] Let us consider SF for any two nonparallel baselines, for example, the orthogonal baselines (2,3) and (1,4) in Figure 3. Combining equations (62) and (63) with equation (60), one can get:

  • equation image
  • equation image

Equations for other baseline pairs in Figure 3 can be easily derived from equations (62), (63), and (60). Similar to equation (53), equations (64) and (65) depend on the only radar parameters λ and δt. Equations (64) and (65) require two nonparallel baselines for estimating 〈w2〉, while equation (53) requires three nonparallel baselines. On the other hand, more restrictive assumption 3a was applied for deriving equations (64) and (65).

[90] Using assumptions 1, 2, and 3a, one can derive equations for coefficients equation image in equation (34) at equation image for any p > 2. One can also derive equations for the coefficients equation image by adding assumptions 4 and 5; equations at p > 2 are too lengthy to be presented in the paper.

3.5. Theoretical Limitations on Applying the Derived Equations

[91] Equations for cross and auto SF for atmospheric SA profiling radars were derived in sections 3 and 4 under several limiting conditions. The conditions impose theoretical limitations on applying the equations to a specific radar. These limitations are discussed below while practical limitations such as measurement errors due to low SNR and finite time of averaging are discussed in the paper on the UCAR-STARS method.

[92] Let us begin from a limitation on the sampling time interval δt. Equations for SF in sections 3.3 and 3.4 were derived in the asymptotic limit τ [RIGHTWARDS ARROW] 0. The asymptotic condition allows us to consider equation image as very small values. One can see from the derivation that conditions equation image are sufficient but not necessary. Equations (42), (43), (47) - (52), and (60) - (63) are reasonably accurate under less restrictive conditions such as:

  • equation image

It follows from equations (32), (24), and (20) that equation image

[93] One needs at least three values of τ to estimate coefficients equation image with equation (34) for cross SF equation image for example equation image (it is shown in section 4 that τ = 0 cannot be used safely for signals with noise). One needs at least two separations, for example τ = δt and 2δt to estimate coefficient equation image in equation (34) for auto SF equation image for signals with noise (equation (83) in section 4.2). Therefore, the temporal separation equation image is to be used for applying the derived equations for SF. Taking this into account and combining equation (66) with definition (32), one can get the following limitations:

  • equation image

where equation image denote the maximum values of the horizontal and vertical velocities of scatterers during a measurement.

[94] Let us consider two typical examples. For the NCAR MAPR, λ = 0.33 m, D = 1.84 m, and γ ≈ 0.37 [Cohn et al., 1997]. At rather strong winds, such as equation image 50 m/s and equation image 3 m/s, equations (67) are satisfied at equation image For the MU radar, λ = 6.45 m, D = 103 m, and equation image [Fukao et al., 1985a, 1985b]. At rather extreme conditions equation imageequations (67) are satisfied at δt ≤ 16 ms. By definition, δt = NCI / PRF while typical PRF is approximately 10 - 40 kHz for the MAPR and 1 - 2.5 kHz for the MUR. Therefore, limitations (67) can be easily satisfied by choosing a proper NCI for the MAPR, MUR, and any other existing atmospheric SA radar.

[95] Another small parameter in the derivation of equations for SF is the spatial separation between the receiver centers. Equations (42), (43), and (47) - (52) are valid in the asymptotic limit equation image which is applicable only to highly overlapping receiving antennas. Such antennas can be directly realized, for example at the MU radar with remarkably flexible data acquisition parameters. Most atmospheric SA radars have adjacent or separated receivers; equations (60) - (63) should be applied to such radars. One can see that equation (61) can be used only under the following condition:

  • equation image

Using the standard relation between SF and CF at p = 2, one can present equation (68) as follows:

  • equation image

where

  • equation image

is the correlation coefficient between the signal power for receivers equation image are the minimum and the maximum values of the required correlation between a pair of signals.

[96] Limitation (69) has a clear physical meaning: centers of receiving antennas should be sufficiently close to ensure positive correlation between signals equation image while they should not be too close to ensure a “measurable” difference between the signals. Using equations (68) and (60), one can specify limitation (69) as:

  • equation image

[97] A choice of equation image in equations (69) and (71) is always somewhat arbitrary. Our experience shows that the SF-based approach still works at ρmin as low as 0.03, and ρmax as high as 0.99. For example, equation image 0.34 and 0.12 for different antenna pairs of the NCAR MAPR and varies from approximately 0.6 to 0.04 for different antenna pairs of the ESRAD. The most effective range of equation image for applying the SF-based approach is approximately 0.5 ≤ equation image One should note that a limitation on equation image is more demanding for SF at p > 2; the issue is considered in the paper on the UCAR-STARS method.

4. Structure Functions for Signals With Noise

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

[98] Equations for SF in section 3 were derived for pure signals from atmospheric scatterers in the illuminated volume. The RHS terms in the equations contain the mean horizontal velocity components and turbulence characteristics of a scattering medium that are intended to be measured by an atmospheric SA radar. The LHS terms contain SF of pure signal power with no contribution from electronic noise, or other contaminants. Real signals always contain electronic noise and may also be affected by other contaminants such as ground clutter. Futhermore, the equations in section 3 were derived for signals from identical receiving antennas which is not always the case in practical measurements. For the equations in section 3 to be applicable to real SA radars, one should relate the cross and auto SF for pure signals equation image in equations (33) and (34) to those of received signals with noise; the relations are derived below.

4.1. Basic Relations for Signals With Noise

[99] Real signal power equation image from a receiving antenna with center a can be presented as:

  • equation image

where equation image is the pure received signal power from atmospheric scatterers in the illuminated volume; equation image is the noise component with a large temporal correlation lag (much larger than δt), e.g., ground clutter, point targets; equation image is the noise component with a zero temporal correlation lag, e.g., electronic noise, atmospheric thermal noise; and equation image is the effective gain factor for a signal equation image for estimating the p-th order SF during a measurement (to be defined below). Without loss of generality, one can consider equation image To proceed further, one should make the following assumption.

[100] Assumption 6: Pure received signal power equation image from atmospheric scatterers in the illuminated volume and noise components equation image are not correlated and neither are the components equation image

[101] It follows from this assumption that signals from two receivers with centers equation image satisfy the conditions:

  • equation image
  • equation image
  • equation image

Equations (73) - (75) are also valid at equation imageequation (74) is valid for separated, adjacent, and overlapping receivers.

[102] One can derive from equations (72) and (73) for p ≥ 2 and any τ that:

  • equation image

The first RHS term in this equation represents cross SF equation image for pure signals and equation image Note that SF equation image is not nondimensionalized by equation image as in equation (33).

[103] Applying equations (73) and (74) to the third RHS term in equation (76), one can derive the following equation for the noise component with zero temporal correlation lag at p ≥ 2 and τ ≠ 0:

  • equation image

[104] As shown in section 4.3 below, one can always determine the gains equation image for a given measurement in such a way as:

  • equation image

The gain factors that satisfy equation (78) will be called the effective gain factors. So defined, the effective gains are random values which are constant for each signal pair during a measurement but may vary from one measurement to another, and from one pair of signals to another. One can get from equations (75) and (78) that at p ≥ 2, sufficiently small equation image

  • equation image

[105] Combining equations (76), (77), and (79), one can derive at p ≥ 2, sufficiently small equation image (but τ ≠ 0) that:

  • equation image

Hereafter equation image denotes the p-th order cross SF for real signals with noise which is defined as:

  • equation image

where equation image is the relative gain factor for signal equation image with respect to signal equation imageEquation (80) shows that SF are not significantly affected by the noise component with a large temporal correlation lag. By definition, equation image therefore equation (81) defines the p-th order auto SF for signals with noise as:

  • equation image

[106] Equation (80) relates cross and auto SF equation image for pure signal power from atmospheric scatterers to SF equation image for signals with noise; the latter can be calculated from received signals for real atmospheric SA radars. However, the equation contains new unknown variables: the moments of noise equation image and the effective gain factors equation image relations for estimating these variables are presented in the following subsections.

4.2. Moments of Noise

[107] The moments of noise component with a zero temporal correlation lag equation image in equation (80) can be estimated from auto SF of real signals with noise. Using equations (82) and (72) - (75), one can present auto SF of even order p = 2, 4, … at equation image as:

  • equation image

Similar to equation (80), this equation is valid at τ [RIGHTWARDS ARROW] 0 but τ ≠ 0. Combining equation (83) with equation (34), and neglecting the terms equation image one can derive at τ [RIGHTWARDS ARROW] 0 but τ ≠ 0 that:

  • equation image

[108] Equation (84) describes the p-th order auto SF for real signals with noise. The limit of these functions at τ [RIGHTWARDS ARROW] 0 determines the moments equation image Applying equation (84) at equation image one can determine the moments of the noise component with a zero temporal correlation lag in equation (80) for any even p. The moments equation image for odd p = 3, 5, … are discussed in the paper on the UCAR-STARS method.

4.3. Effective Gain Factors

[109] One notices that neither equation (80) after nondimensionalization with σp,k nor equation (81) includes the gains themselves but rather their ratio equation image the ratio is determined below.

[110] Using equations (72) - (74), one can derive for a real signal with noise that:

  • equation image

The receivers of a real SA radar might differ in size and shape and the receiving channels might be set up at different amplifications. To take into account possible differences, one can impose the following condition on a pair of pure received signals:

  • equation image

The condition corresponds to pure signals which would be received by closely located identical receivers from atmospheric scatterers inside the same illuminated volume. The condition is similar to a definition of the effective gain factors (78) which equalizes the moments of noise with a large temporal correlation lag for a pair of receivers. The sum of equations (78) and (86) provides the following condition for signals from two receivers with centers equation image

  • equation image

Combining equation (85) at equation image and equation image with equation (87), one can derive for any p≥2 that:

  • equation image

[111] This equation provides the relative gain factor equation image for a pair of real signals with noise equation image and equation image at p ≥ 2 and sufficiently small equation imageEquation (88) shows that relative gains equation image depend on p.

5. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions for a Random Scalar Field
  5. 3. Structure Functions for a Spaced Antenna Profiling Radar
  6. 4. Structure Functions for Signals With Noise
  7. 5. Discussion
  8. 6. Summary
  9. Appendix A: Coefficients in Equations (7) - (10)
  10. Appendix B: Variables θ
  11. Appendix C: Mean Speed of the Diffraction Pattern on a Ground
  12. Acknowledgments
  13. 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 〈w4〉 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 equation image 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 equation image at different equation image 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 equation image 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 equation image in equation (42). A similar equation at p = 3 provides four additional parameters, and equation (49) at p = 4 provides five more parameters equation image 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.

[116] 2. The SF-based approach provides unambiguous, theoretically substantiated turbulence characteristics of scattering medium. For VHF/UHF radars with a narrow beam equation image one can estimate the moments 〈wp〉 of the vertical turbulent velocity component at any p ≥ 2 (e.g., equations (53) and (54)). For MF radars with a broad beam equation image one can estimate the moments equation image at p ≥ 2 for each turbulent velocity component separately. Neither spectra nor CF-based technique can provide turbulence characteristics at p > 2.

[117] Equations (53), (64), (65), and (54) for the moments equation image 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].

[118] 3. Equations for SF of a moving random scalar field in section 2 describe the diffraction pattern's power on a ground from the refractive index irregularities in the atmosphere illuminated by the radar's transmitter. Equations (6) - (10) at p = 2 are similar to those in the FCA/FSA methods. Indeed, the basic FCA equation [Briggs, 1984, equation (13)] can be presented as:

  • equation image

where equation image is the second order CF for a pair of received signals; f and A−H are the unknown function and coefficients to be determined from experimental data. One can easily present equations (6) and (A1) - (A3) in the form (89). The most important similarity between equation (89) and equations (6) - (10) is in that they describe the diffraction pattern on a ground. Equation (89) does not relate “measurable” parameters AH of CF to characteristics of scattering medium [Briggs et al., 1950; Briggs, 1984; Briggs and Vincent, 1992; Liu et al., 1990; Hocking et al., 1989], and neither do equations (6) - (10) for SF.

[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 τ[RIGHTWARDS ARROW]0, 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 equation image 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 equation image 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 equation image they contain neither empirical functions, nor constants.

[123] Being more universal and asymptotically exact, equations for SF of received signals can provide a deeper insight and more rigorous theoretical background for the SA methods. For example, the SF-based approach provides a relation between mean speed and turbulence of scatterers and those produced by the SA methods; the relation has not been established with CF/spectra-based approaches. Indeed, equation (39) shows that coefficient equation image in equation (34) directly depends on the mean velocity components equation image of individual scatterers in the illuminated volume power-weighted by the squared scatterer's reflectivity equation image and the transmitter/receiver functions equation imageEquation (40) shows that coefficient equation image in equation (34) directly depends on the power-weighted second moments of the velocity components of individual scatterers; the second moments are weighted in the same way as the mean components. The weighting for the first and second moments of the velocity components of individual scatterers in equations (39) and (40) is completely identical to that for the radial velocity in the Doppler method [Doviak and Zrnić, 1993, section 5.2]. Equations (39) and (40) are rather universal; they were derived with the only assumption 1 that is satisfied in practically any condition. SF and CF at p = 2 are linearly related to each other, therefore the result is valid for any SA method. Equations (39) and (40) provide a rigorous theoretical confirmation to the Briggs [1980, p. 832] conclusion: “… the Doppler radar and the spaced antenna radar techniques are basically the same, both for determination of wind and of random turbulent velocities.”

[124] Another example is a relation between the mean speed of the diffraction pattern on a ground equation image and that of the scattering medium equation image The FCA/FSA method considers a fixed ratio equation image 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 equation image Using equations for SF in sections 2 and 3, one can derive an exact statistical relation between equation image for any specific SA profiler; the derivation is presented in Appendix C. It is shown that the physical mean speed of the diffraction pattern equation image is always twice larger than equation imageequation (C5). At the same time, the measured mean speed equation image 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 equation image from equation image in practical measurements, for example, with the FCA/FSA methods.

image

Figure 4. The ratio ωm for different beam width θT at α2 = 1.25.

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[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 equation image were derived with assumptions 1 and 2, while derivation of equation (60) for a separation equation image required one more assumption 3a.

[126] For the sake of clarity, several insignificant simplifications have been made in section 3. The real reflectivity equation image was considered, although it is in general a complex value. Equations for SF for the complex equation image are exactly the same as those for real equation image except for interpretation of equation image; here * denotes complex conjugation. The narrow-beam UHF and VHF radars with equation image were considered in section 3. This simplification does not affect equations for the mean velocity components but does affect equations (53), (54), (64), and (65) for turbulence characteristics. Equations for turbulence characteristics at equation image can be derived in a similar way by modifying assumptions 4 and 5.

[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 equation image 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 equation image and spatial separation equation image

[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 equation image

[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.

[135] Limitation (67) on the sampling time interval is more demanding. As shown in section 3.5, this limitation can be easily satisfied for any existing atmospheric SA radar with a vertically directed transmitted beam. However, the second limitation (67) might become a very serious problem for the radar operations at a low elevation angle where the radial velocity equation image can reach large values. This limitation could be crucial for radars with a small wavelength. One can significantly weaken limitations (67) by considering equation image Derivation of equations for such a case will be similar to that in section 2.4 for equation image One more assumption, a specific statistical distribution of the scatterer's instantaneous velocity components equation image will be needed. Even strongly weakened, limitation on δt will still be demanding for some remote sensors. For example, one could never realize sufficiently small δt for applying equations for SF as in section 3 to atmospheric lidars with the wavelength of the order of several microns.

[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 equation image unless strong oversampling along the beam is accomplished. If the gate separation of the order of λ is applied, one can estimate equation image 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

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

Appendix C: Mean Speed of the Diffraction Pattern on a Ground

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

[145] Consider a pair of receiving antennas equation image for the radar in Figures 1 - 3. Let us x axis to be directed along the baseline equation image

[146] The mean speed of scattering medium 〈Usc〉 can be derived using equations (42) and (43). To simplify the equations, let us adopt assumption 3 in section 3.3. It follows from a definition (B2) that θxy = 0 in this case, and one can present equations (42) and (43) as follows:

  • equation image

It follows from equation (C1) that

  • equation image

[147] The mean speed of the diffraction pattern on a ground 〈Udp〉 is described by equations (6) - (8) with coefficients (A1) and (A2) at p = 2. One can derive that equation image when assumption 3 is adopted. Using relation τ = τˆδt in equation (34), one can present equations (A1) and (A2) as follows:

  • equation image

One can get from equation (C3) that

  • equation image

[148] Combination of equations (C2) and (C4) provides the well-known result: the mean speed of the diffraction pattern is twice larger than that of the scattering medium [e.g., Briggs, 1980; Larsen and Röttger, 1989; Hocking et al., 1989]:

  • equation image

Equation (C5) is derived with the only assumptions 1 - 3 that are satisfied in practically any realistic conditions. The equation is valid for a large number of scatterers Ns [RIGHTWARDS ARROW] ∞ in the illuminated volume. That corresponds to typical measurement conditions in the atmosphere and consistent with a statistical nature of the SA methods. Equations (42) and (43) are derived at very small separations equation image which is consistent with definition of a speed in physics.

[149] On the other hand, most existing atmospheric SA radars have adjacent, or separated receivers with equation image The measured speed of the diffraction pattern on a ground equation image for such radars can differ from the physical speed 〈Udp〉. Using equations (C2) and (C5), one can estimate 〈Udpmeas for a receiver pair with separation equation image as:

  • equation image

where equation image are defined by equation (34) at p = 2. Combining equation (61) for the mean speed of the scattering medium at equation image with equation (C6), one can derive that:

  • equation image

Applying equation (60) for equation image to equation (C7), one can relate the ratio ωm to the radar parameters:

  • equation image

where σθ = γλ/D is the width of the principal lobe of the one-way angular pattern of power density incident on the scatterers. A common measure of the beam width, the one-way half-power angular width θT of the transmitted power density pattern is related to σθ as equation image [Doviak et al., 1996]. Dependence of ωm on separation Δx/λ for varying beam width θT at α2 = 1.25 is illustrated in Figure 4. The results are plotted for equation image 0.3 because equations for SF are theoretically reliable only at statistically significant correlation between the received signals; the correlation coefficient equation image is given by equation (70).

[150] One can see that the ratio ωm heavily depends on the radar parameters; it can be any value from 2, and smaller. It is well known that the FCA/FSA methods provide mean speed of the diffraction pattern rather than that of the scattering medium [e.g., Briggs, 1984; Liu et al., 1990]; the latter is estimated from the former with equation (C5). Equation (C8) and Figure 4 clearly demonstrate that such procedure can cause significant underestimation of equation image One can apply equation (C8) to the FCA/FSA methods for correcting 〈Usc〉 although it does not seem to be an effective technique. The reason is that equation (C8) contains radar parameters γ, α, and σθ which may not be known exactly in some cases. Using equations (34), (60), (61), and (70), one can present equation (C7) as follows:

  • equation image

Equation (C9) contains no radar parameters; the correlation coefficient equation image can be calculated from the received signals. One should note that equation image is defined by equation (70) for the signal power. The FCA/FSA methods are mainly applied to complex signals equation image defined by equation (19) [e.g., Briggs and Vincent, 1992]. Using the standard relation between SF and CF at p = 2, and combining equation (60) for equation image with equation (89) in Doviak et al. [1996], or equations (2) - (6) in Holloway et al. [1997], one can derive the well-known relation [e.g. Briggs, 1980, equation (16)]:

  • equation image

where equation image is the correlation coefficient between complex signals equation image defined as:

  • equation image

Here * denotes complex conjugation. Combining equations (C9) and (C10), one can derive the following equation for estimating mean speed of the scattering medium from the measured mean speed of the diffraction pattern at equation image

  • equation image

This equation can be directly applied to the FCA/FSA methods although a caution is needed at low correlation, say equation image The major reason for being cautious is that the basic FCA/FSA assumption (89) is valid only at sufficiently small equation image [Briggs, 1984].

Acknowledgments

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

[151] The National Center for Atmospheric Research is sponsored by the National Science Foundation. This work was sponsored by the NCAR Director's fund, the NCAR/RAP Director's fund, the UCAR fund, and the UCAR Foundation's fund. The authors are sincerely grateful to Dave Parsons, Dave Carlson, Brant Foote, Rich Wagoner, Bob Serafin, Tim Killeen, and Rich Anthes for their support and encourangement. The authors thank Bob Serafin, Dick Doviak, Dušan Zrnić, Bob Palmer, Jeff Keeler, Dick Strauch, Chuck Frush, and Paul Herzegh for their constructive criticism and useful comments. Significant progress in the research was made during our work for the Radio Science Center for Space and Atmosphere (RASC) at the Kyoto University, Japan. We are deeply grateful to the RASC director Shoichiro Fukao for his kind invitation and support. We are also thankful to Shoichiro Fukao, Mamoru Yamamoto, Gernot Hassenpflug, Toru Sato, Hubert Luce, Richard Worthington, and Robert Vincent for fruitful discussions and valuable comments. Our understanding of a theoretical background for the SF-based approach was finally shaped up during intensive and productive discussions with Valeryan Tatarskii, to whom the authors are especially grateful.

References

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