Radio Science

Local and global statistics of clear-air Doppler radar signals

Authors


Abstract

[1] A refined theoretical analysis of the clear-air Doppler radar (CDR) measurement process is presented. The refined theory builds on the Fresnel-approximated (as opposed to Fraunhofer-approximated) radio wave propagation theory, and turbulence statistics like locally averaged velocities, local velocity variances, local dissipation rates, and local structure parameters are allowed to vary randomly within the radar's sampling volume and during the dwell time. A local version of the moments theorem and the random Taylor hypothesis are used to derive first-principle formulations of all higher moments of the Doppler cross-spectrum. The mth moment is written as a convolution integral of a spectral sampling function and a generalized, mth-order refractive-index spectrum or, alternatively, as a convolution integral of a lag-space sampling function and a spatial cross-covariance function of the local refractive-index fluctuations and their local mth-order time derivatives. Closed-form expressions of the first three moments (i.e., m = 0, 1, 2) of the Doppler spectrum for the monostatic, single-signal case are derived. This refined theory, or “local sampling theory,” enables one to correctly interpret CDR observations that are collected under conditions where the applicability of the traditional “global sampling theory” is questionable. The commonly used global sampling assumptions (Bragg-isotropy, homogeneity, and stationarity of all turbulence statistics within the sampling volume and during the dwell time) may be invalid for small-scale intermittency in the mixed layer, for refractive-index sheets corrugated by gravity waves or instabilities, and for layered turbulence in the stably stratified atmosphere.

1. Introduction

[2] Ground-based, phase-coherent clear-air radars, also known as clear-air Doppler radars (CDRs), UHF/VHF radars, or radar wind profilers, have been established as standard instruments for remote sensing of winds, waves, and turbulence in the atmospheric boundary layer (ABL), the free troposphere, and the lower stratosphere for both research and operational purposes [e.g., Woodman and Guillén, 1974; Gage and Balsley, 1978; Balsley and Gage, 1982; Gossard and Strauch, 1983; Weber et al., 1990; Mead et al., 1998; Steinhagen et al., 1998]. The state-of-the-art of CDR science and technology until around 1990 was reviewed by Gage [1990], Gossard [1990], Röttger and Larsen [1990], and Doviak and Zrnić [1993]. More recent developments were summarized by Luce et al. [2001a], Gage and Gossard [2003] and Fritts and Alexander [2003].

[3] CDRs transmit electromagnetic pulses into the atmosphere and detect their echoes, which are caused by scatter or reflection from small-scale spatial perturbations of the instantaneous refractive-index field [Tatarskii, 1961; Doviak and Zrnić, 1993]. In contrast to lidars and sodars, CDRs are phase-coherent, such that the phase information is not lost between subsequent pulses. Also in contrast to lidars and sodars, the CDR Doppler shifts are not obtained by spectral analysis of a single echo but by spectral analysis of a sequence of echoes, where the delay time (between transmitting and receiving of a pulse) is kept fixed. A typical pulse repetition period of a CDR is 100 μs, and a typical “dwell time” (the length of a signal time series from which a Doppler shift is computed) is 10 s. That is, a single Doppler shift is typically estimated from a sequence containing on the order of 105 echoes, whereas weather radars can make these estimates using far shorter dwell times (e.g., shorter than 0.1 s) and with far fewer samples (e.g., 50). There are two main reasons for this. First, clear-air reflectivities are usually much smaller than reflectivities from hydrometeors. Second, weather radars are horizontally scanning radars operating typically at wavelengths of 10 cm, while CDRs have usually near-vertical beam-pointing directions and operate at longer wavelengths. As a result, compared to weather radars, CDRs are typically operated at much smaller signal-to-noise ratios (SNRs) and with much longer signal correlation times. This requires a much larger number of samples and therefore much longer dwell times for CDRs than for weather radars. May and Strauch [1989] show that meaningful Doppler velocities can be retrieved even if the CDR is operated at an SNR as small as −35 dB.

[4] Through temporal changes of the echoes' amplitudes and phases, CDRs can “see” mean and turbulent motion in the optically clear atmosphere, like an observer looking down from a bridge across a river can recognize speed, direction, and turbulence intensity of the water flow by evaluating the spatiotemporal patterns of light reflected and scattered from the water surface. In contrast to the observer on the bridge, however, CDRs can detect echoes not only from a discrete surface but from refractive-index irregularities that quasi-continuously populate the atmospheric boundary layer and also the overlying free atmosphere. This offers the possibility to use CDRs to monitor the optically clear atmosphere quasi-continuously in time and height. The first three moments of the signal spectrum can be used to track the refractive-index variability, the mean air motion, and the turbulence intensity [Woodman and Guillén, 1974].

[5] Since Woodman and Guillén's [1974] pioneering experiments at the Jicamarca VHF radar, CDR research and technology has benefited from major advancements in several directions: (1) CDR sensitivity has been improved through the development of better antennas, better receivers, and more powerful transmitters; (2) UHF CDRs have been developed, which have become known as boundary layer profilers and have been used to probe the atmospheric boundary layer (ABL) at altitudes down to less than 200 m AGL [Ecklund et al., 1988; Wilczak et al., 1996]; (3) interferometric and imaging techniques have been developed that take advantage of multiple receivers [e.g., Briggs et al., 1950; Röttger, 1981; Doviak et al., 1996; Chau and Balsley, 1998; Palmer et al., 1998; Mead et al., 1998], multiple carrier frequencies [e.g., Kudeki and Stitt, 1987; Chilson et al., 1997; Muschinski et al., 1999a; Palmer et al., 1999; Luce et al., 2001b; Chilson et al., 2003], or both [e.g., Yu and Palmer, 2001] to overcome resolution limitations set by a finite pulse length and a finite angular beam width; (4) advanced signal processing allows the atmospheric signal to be separated from ground and sea clutter, intermittent echoes from birds and aircraft, radio interference, and external and internal noise [e.g., Lehmann and Teschke, 2001, and references therein]; (5) the Fresnel approximation [Doviak and Zrnić, 1984], which retains the quadratic phase term as function of the transverse spatial coordinates and is therefore more accurate than the traditional Fraunhofer approximation, serves as a unifying theoretical framework for Bragg scatter, Fresnel scatter, and Fresnel reflection; (6) advancements in computational fluid dynamics [e.g., Werne and Fritts, 1999; Smyth and Moum, 2000a, 2000b] and in airborne in situ sensor and platform technology [e.g., Dalaudier et al., 1994; Balsley et al., 1998; Muschinski and Wode, 1998; Luce et al., 2001a; Muschinski et al., 2001; Muschinski and Lenschow, 2001; Siebert et al., 2003; Frehlich et al., 2003; Balsley et al., 2003] have contributed to more realistic simulations of and to a better observational accessibility to the fine-structure of atmospheric velocity and scalar fields in the atmosphere far from the ground; (7) realistic numerical simulation of CDR signals in the atmospheric boundary layer has become possible by combining the large-eddy simulation (LES) technique with first-principle radio-wave propagation physics for forward scatter [Gilbert et al., 1999] and backscatter [Muschinski et al., 1999b].

[6] There are a number of problems, however, that have been puzzling the CDR community for many years. What are the reasons of the biases that are consistently found in VHF and UHF vertical-velocity observations and that cannot be attributed to instrumental deficiencies [Nastrom and VanZandt, 1994; Muschinski, 1996b; Angevine, 1997; Worthington et al., 2001; Lothon et al., 2002]? What is the meaning of spectral moments estimated from signal time series as short as 1 s [e.g., Pollard et al., 2000], i.e., short compared to the “renewal time” (the time needed for an air parcel to be advected across the CDR's resolution volume)? Down to what length and timescales, and in what sense, are signal contributions from different locations within the radar's resolution volume and different instants during the dwell time localizable? What is the correct interpretation of spectral moments estimated from long signal time series measured in an intermittent or statistically nonstationary atmosphere?

[7] One reason why there is no consensus of how to address these problems is the lack of a unifying theory that allows the effects of variations of the local and instantaneous turbulence characteristics within the resolution volume and during the dwell time to be examined both generally and specifically. Such a theory is presented in the following. The theoretical development builds on Fresnel-approximated radio-wave scattering theory [Doviak and Zrnić, 1984, 1993], and it allows turbulence statistics like local velocity variances, local energy dissipation rates, and local structure parameters, which in classical turbulence theory are deterministic variables, to vary randomly within the CDR's resolution volume and during the dwell time. In the fluid mechanics community, the treatment of local turbulence statistics as random flow variables has long been common [e.g., Oboukhov, 1962; Kolmogorov, 1962; Kuznetsov et al., 1992; Praskovsky et al., 1993; Peltier and Wyngaard, 1995; Sreenivasan and Antonia, 1997; Wang et al., 1996, 1999; Wyngaard et al., 2001].

[8] The paper is organized as follows. In Section 2, equations for instantaneous covariances equation image12(m)(t) ≡ 〈I*1(t)I2(m)(t)〉 are developed, where I1(t) and I2(t) are two phase-coherently measured, complex CDR signals, where I2(m)(t) ≡ ∂mI2(t)/∂tm is the mth time derivative of I2(t), and where t is time. (Here, I1 and I2 could be two phase-coherent signals measured with two different transmitting and/or receiving antennas, at two different carrier frequencies, at two different delay times, or “range gates,” with two different transmitted pulse durations, or with two different receiver bandwidths.) Then it is shown that equation image12(m)(t) can be written as a convolution product over six-dimensional x-r space (x is the location vector, r is the spatial lag vector):

equation image

where the local cross-covariance function

equation image

is a random function of its arguments, G12(x, r) is a deterministic instrument function, and equation image is the refractive-index fluctuation with respect to the sampling-volume average of the refractive index [Doviak and Zrnić, 1993, p. 427].

[9] Throughout the paper, a tilde over a symbol stands for a variable that is allowed to vary randomly within the CDR's resolution volume and during the dwell time, and whose local mean value is not necessarily zero.

[10] In section 3, the Fresnel approximation is used to derive a model for the single-signal (I1(t) ≡ I2(t)) sampling function G11(x, r) for a monostatic radar. It is shown that equation image11(m)(t) can be written as a weighted x-space integral over contributions equation imagenn(m)(x, −equation imageB(x), t), where equation imageB(x) is the local Bragg wave vector, and equation imagenn(m)(x, k, t) is the local spectrum associated with equation imagenn(m)(x, r, t).

[11] In section 4, explicit models for equation imagenn(1)(x, k, t) and equation imagenn(2)(x, k, t) are derived based on the assumptions that velocity and generalized potential refractive index are conserved quantities and that the viscous terms may be neglected.

[12] In section 5, equations for the zeroth, first, and second moments of the instantaneous Doppler spectrum in the monostatic, single-signal case are derived. A discussion follows in section 6, and a summary and conclusions are given in section 7.

[13] Throughout the paper, only the statistics of atmospheric signals are analyzed. That is, any difficulties associated with the estimation and removal of any nonatmospheric components in the CDR measurements (i.e., receiver noise, cosmic noise, ground clutter, radar interference, aircraft echoes, etc.) are not discussed.

2. Two-Signal Statistics: Basic Theory

2.1. Instantaneous Covariances of Signals and Signal Time Derivatives

[14] Consider the scattering integrals for the two signals I1(t) and I2(t) measured at the same time t:

equation image

and

equation image

[Doviak and Zrnić, 1993, equation (11.115) on p. 456], where equation imagea is the fluctuating (with respect to the resolution-volume average) part of the actual refractive index. Note that no averaging is involved at this point, neither time averaging nor spatial averaging nor ensemble averaging. The sampling functions G1(x′) and G2(x″) are deterministic functions of location. Here we assume that the two signals are measured at fixed delay times, such that G1 and G2 do not depend on the time t. If equation imagea is a random variable somewhere in the resolution volume, or sampling volume (the volume within which the respective sampling function has nonnegligible weight), then the signals I1 and I2 are random variables. However, if equation imagea is a deterministic variable everywhere in the sampling volume, then the signals are deterministic variables.

[15] The mth time derivative of I2(t) is

equation image

where equation imagea(m)(x″, t) ≡ ∂mequation imagea(x″, t)/∂tm is the mth local time derivative of equation imagea at the location x″. In the remainder of the paper, we take advantage of the fact that the generalized potential refractive index, for (the fluctuations of) which we use the symbol equation image, is a conserved quantity, in contrast to the actual refractive index, which is not conserved [Ottersten, 1969]. By definition, equation image = equation imagea at an altitude that is of specific interest (here the altitude of the center of the resolution volume). In an altitude region around this reference level, relative pressure changes are much smaller than 1 as long as that altitude region (here the vertical extent of the resolution volume) is small, which we assume. Therefore, in the scattering integrals for I1 and I2, and in the expressions for their time derivatives, equation imagea may be replaced with equation image [see also Doviak and Zrnić, 1993, p. 466]. A very similar argument was made by Lumley and Panofsky [1964, p. 62] for the relationship between temperature fluctuations and potential temperature fluctuations.

[16] Now, we introduce the instantaneous covariance of I1(t) and I2(m)(t) as

equation image

In practice, the instantaneous ensemble average (over a large number of realizations at the same time t) has to be replaced by a time average over the temporal evolution of a single realization of the pair {I1(t), I2(t)}. CDR signals have usually very short integral timescales (or “correlation times” or “fading times”), such that statistically meaningful signal covariances can be measured (quasi-) instantaneously. An excellent discussion on the relationships between spatial averages, time averages, and ensemble averages of turbulent variables can be found in Monin and Yaglom [1975, pp. 205 ff].

[17] Using the relationships given above, we find

equation image

where

equation image

[18] Now, we follow Tatarskii [1961, p. 52ff.] and introduce sum and difference coordinates. We call

equation image

the location and

equation image

the spatial lag. Furthermore, we define the “signal covariance sampling function”

equation image

and the local and instantaneous, spatial cross-covariance function of the (generalized potential) refractive index and its mth local time derivative:

equation image

We obtain the very general result

equation image

Note that the covariances equation image12(m)(t) are defined as instantaneous covariances, such that the equation image12(m)(t) are allowed to be a random function of time.

[19] Equation (13) generalizes equation (11.121a) in Doviak and Zrnić [1993, p. 458] in several respects: first, (13) describes both the two-signal case and the single-signal case; second, equation imagenn(m) is allowed to vary randomly within the resolution volume and during the dwell time, while Doviak and Zrnić [1993] assume equation imagenn(m) to be independent of x and t); third, (13) describes also the higher-order (m > 0) Doppler moments, while Doviak and Zrnić [1993] consider only the zeroth moment (m = 0); and fourth, the instrument function G12(x, t), which we have shown to be the same for all m, is left unspecified in (13). Doviak and Zrnić [1993, p. 458] mention that one “could separate R(r, r′) [equation imagenn(0)(x′, x″, t) in the notation in this paper] into a product of two functions [of position and of spatial lag, respectively]… This would allow the variance of Δn to be spatially dependent, a feature that would be especially useful if scattering irregularities did not fill the resolution volume.” The assumption of separability appears to be an unnecessarily strong assumption, and Doviak and Zrnić [1993] do not justify it. Therefore, in the following we make no attempt to separate equation imagenn(m)(x, r, t) into products. Instead, we allow equation imagenn(m)(x, r, t) to vary with all of its arguments simultaneously.

[20] Often it is more convenient to work in wavenumber space rather than in lag space. Applying the correlation theorem (see Appendix B) to (13) leads to

equation image

where

equation image

and

equation image

[21] In the remainder of this paper, we will consider the instantaneous covariances equation image12(m)(t) as the primary CDR observables. In the following subsection, we will see that, via the moments theorem, the equation image12(m)(t) may be interpreted as, and estimated by, the moments of instantaneous Doppler (cross-) spectra.

2.2. Instantaneous Doppler Spectra and Instantaneous Spectral Moments

[22] Now, consider two signals, I1(t) and I2(t′), measured at two different times t and t′, where t′ = t + τ and where τ is the time lag. The instantaneous cross-covariance function is

equation image

In general, CDR signals are not statistically stationary, but often the nonstationarity can be ignored within narrow time windows. We refer to such processes as quasi-stationary.

[23] In the case of quasi-stationarity we may define the instantaneous (frequency) cross-spectrum equation image12(t, ω) as (2π)−1 times the Fourier transform of equation image12(t, τ) with respect to τ:

equation image

We call equation image12(t, ω) the instantaneous Doppler cross-spectrum. In practice, estimates of equation image12(t, ω) are meaningful if the dwell time (the length of the signal time series from which a spectrum is estimated) contains at least a few signal correlation times. For UHF clear-air radars operating in the mixed layer, the correlation times are typically on the order of a tenth of a second, such that dwell times as short as 1 s usually lead to meaningful estimates of the instantaneous Doppler spectrum [e.g., Muschinski et al., 1999b; Pollard et al., 2000], provided the signal-to-noise ratio is sufficiently high. A quantitative discussion of the dependencies of the correlation times on the various parameters characterizing the CDR and the micrometeorological conditions in the sampling volume is beyond the scope of this paper.

[24] The integral of equation image12(t, ω)ωm over all frequencies is the mth moment of the instantaneous Doppler spectrum, or the instantaneous mth spectral moment:

equation image

(Here and in the following, omission of the integration limits means that the integration is to be performed from −∞ to +∞.)

[25] Now, consider the mth-order τ-derivative of equation image12(t, τ) at zero time lag:

equation image

which leads to

equation image

[26] If the signals are statistically stationary or quasi-stationary, then equation image12(m)(t, 0) is directly connected with equation image12(m)(t) through the moments theorem, which is derived in Appendix A:

equation image

That is, the instantaneous covariances equation image12(m)(t, 0) are (apart from the factor im) identical to the instantaneous spectral moments equation image12(m)(t) in the case of quasi-stationarity. In the remainder of this paper, we consider only the case τ = 0. Therefore, in the following we omit the argument “0” in equation image12(m)(t, 0), as in the previous subsection.

[27] An interesting alternative approach to the analysis of CDR signals has recently been put forward by Praskovsky and Praskovskaya [2003]. Instead of analyzing auto- and cross-covariance functions at zero time lag and the moments of Doppler spectra and Doppler cross-spectra, as described in this paper, they have investigated properties of the auto- and cross-structure functions of CDR signals. It is not clear at this point to what extent the theory presented here and the one developed by Praskovsky and Praskovskaya [2003] are consistent with each other, and what their relative advantages and disadvantages are.

3. Single-Signal Sampling Functions for a Monostatic CDR

3.1. General Theory

[28] If the angle between the axes of the transmitting and receiving beams is small, the two-way sampling function G(x) for a signal I is given by

equation image

where FT(x) and FR(x) are the one-way electrical field weighting patterns of the transmitting antenna and the receiving antenna, respectively, and φT(x) is the phase change that the transmitted pulse undergoes along the straight path from the center of the transmitting antenna to the location x within the radar's resolution volume. Correspondingly, φR(x) is the phase change that the backscattered pulse undergoes along the straight path from the location x to the center of the receiving antenna.

[29] Let us consider here a single signal measured with a monostatic radar, where the same antenna is used for transmitting and receiving, such that FT(x) = FR(x) ≡ F(x). In this case,

equation image

where A(x) ≡ F2(x) is the two-way amplitude weighting function and

equation image

is the two-way phase factor. Here,

equation image

is the Bragg wavenumber, and r0 is the range vector, which points from the origin of the x coordinate system to the antenna center; see Figure 1. We choose the x coordinate system such that its origin coincides with the center of the sampling volume, and that the z-axis points in the beam direction (opposite to the direction of r0). Expanding the length ∣−r0 + x∣, which is the distance between the antenna center and the scattering point x, in a Taylor series up to second order yields

equation image

Doviak and Zrnić [1984, p. 328] point out that the first-order theory [Tatarskii, 1961], which neglects quadratic and higher-order terms in x and y (Fraunhofer approximation), is invalid for some relevant applications. We follow Doviak and Zrnić [1984] and retain the quadratic terms in (27). That is, we use the Fresnel approximation.

Figure 1.

Schematic sketch of the sampling geometry of a clear-air Doppler radar (CDR). The origin of the coordinate system coincides with the center of the CDR's sampling volume. A local averaging volume within the CDR's sampling volume is depicted. Note that the sketch is not on scale; the beam width is greatly exaggerated.

[30] According to Doviak and Zrnić [1984, equation (6)] in the case of narrow beams and small pulse widths, A(x) may be written as

equation image

where g is the antenna gain, Pt is the transmitted power, fΘ2x, Θy) is the angular two-way electrical field pattern of the antenna, Wr(z) is the combined weighting of the transmitted pulse and the receiver bandwidth, and R is the receiver resistance. The beam pattern is normalized such that fΘ2(0, 0) = 1. For a narrow-beam antenna, the gain g is given by

equation image

Inserting (27) and (28) into (24) yields

equation image

The value of equation image for which the phase of G(x) changes by π (compared to x = y = 0 at the same z) is called the radius f of the first half-period Fresnel zone [e.g., Jenkins and White, 1976, p. 380] or, shorter, the radius of the first Fresnel zone [e.g., Doviak and Zrnić, 1993, p. 459]. From (30) we find

equation image

in agreement with Jenkins and White [1976, p. 381, equation (18b)] and Doviak and Zrnić [1993, p. 459]. Some authors [Jenkins and White, 1937, p. 175] define the Fresnel zones based on the one-way (as opposed to two-way) phase differences. This leads to a Fresnel-zone radius of equation image, [Jenkins and White, 1937, p. 182], which is by equation image smaller than f as it is defined here.

[31] Now, let us write the covariance sampling function in (13) for the single-signal case (i.e., G12(x, r) ≡ G11(x, r) ≡ G*(xr/2) G(x + r/2)) as a product of an amplitude factor A11(x, r) and a phase factor P11(x, r):

equation image

where

equation image

and

equation image

The second-order expansion of P(x) given in (27) yields

equation image

where rx, ry, and rz are the components of the lag vector r.

[32] From (28) we obtain

equation image

Therefore,

equation image

[33] This equation is very general. No specific assumptions about the antenna pattern fΘ2x, Θy) and the range weighting function Wr(z) have been made so far. In particular, we have not assumed that fΘ2x, Θy) and Wr(z) are Gaussian, which was assumed by Doviak and Zrnić [1984] from the beginning. The Fourier transform of G11(x, r) is

equation image

and from (14) we obtain a very general equation for the instantaneous single-signal (I1 = I2) covariances:

equation image

[34] There are two paths along which one can proceed from here. The first path is to assume homogeneity of equation imagenn(m)(x, k, t) within the sampling volume. Then equation imagenn(m)(x, k, t) does not depend on x and may be taken out of the integral over x:

equation image

where

equation image

This is the traditional path, which was explored by Doviak and Zrnić [1984], but only for m = 0.

[35] The second path takes advantage of the fact that H11(x, k) may be interpreted as a local (with respect to x) spectral sampling function, such that for a specific location x within the sampling volume the integral over k can be evaluated. Then equation image11(m)(t) may be written as a spatial integral over local contributions equation image11(m)(x, t):

equation image

where

equation image

This second path does not require homogeneity, such that equation imagenn(m)(x, k, t) is allowed to vary with x within the sampling volume. In practically all CDR applications, H11(x, k) samples only a small region within k-space, a region that we will refer to as the “local Bragg window,” such that H11(x, k) may be approximated by a delta function in k-space. This delta function has its peak at the local Bragg wave vector. In the following, we will refer to the first path as global sampling and to the second path as local sampling.

3.2. Local Sampling

[36] In order to derive a specific model for H11(x, k) and h11(k), we now follow Doviak and Zrnić [1984] and assume that the antenna pattern and range weighting functions are Gaussian:

equation image

and

equation image

We will see that for Gaussian beam patterns the angle

equation image

is a natural measure for the beam width. In the following, we will refer to Θ0 as the “characteristic beam width,” as an alternative to the commonly used half-power beam width Θ1.

[37] For a Gaussian beam pattern the gain is

equation image

The effective antenna area Ae is defined by

equation image

[Doviak and Zrnić, 1993 p. 45]. For a rotationally symmetric antenna, an effective antenna diameter De can be defined via Ae = π(De/2)2, which yields

equation image

With (47), we obtain for the effective antenna diameter of a Gaussian antenna:

equation image

[38] From (37) we obtain

equation image

After taking the Fourier transform with respect to lag coordinates and after elementary rearrangements we obtain

equation image

That is, for a given location x, the sampling function H11(x, k) gives maximum weight to the wave vector

equation image

where Θxx/r0 and Θyx/r0 are angular coordinates. We call equation imageB(x) the “local Bragg wave vector” and the k-space region around equation imageB(x) within which the magnitude of H11(x, k) for a fixed x is nonnegligible the “local Bragg window.” The local Bragg wave vector points from the scattering point x to the antenna center.

[39] If we write the Gaussian terms in standard form, i.e., like expequation image, we obtain the transverse and radial widths of the local Bragg window,

equation image

and

equation image

For practically all CDR applications, the relative transverse width, κlt/kB = 1/equation imager0Θ0kB = De/8r0, is very small compared to 1. Also the relative radial width, κlr/kB = 1/2σrkB = λ/8πσr, is very small compared to 1. Therefore, within the local Bragg window, changes of equation imagenn(m)(x, k, t) with respect to k can usually be neglected, and we may approximate the local spectral sampling function H11(x, k) by a delta function that peaks at −equation imageB(x):

equation image

Therefore,

equation image

where we have introduced

equation image

as the local Bragg component of equation imagenn(m)(x, k, t).

[40] Ultrawideband radars [e.g., Dvorak et al., 1997] operate with pulse lengths so short that the bandwidth and the carrier frequency are of the same magnitude. Although in such cases the spectral sampling function cannot be longer approximated as a delta function, local sampling theory is still applicable.

3.3. Global Sampling

[41] Now, let us assume homogeneity across the sampling volume, such that equation imageB(m)(x, t) does not depend on x. In this case, we may integrate (52) over x-space and obtain the global spectral sampling function in the Fresnel approximation:

equation image

Here we have introduced the near-field parameter

equation image

where

equation image

By definition, the sampling volume lies in the far field of the antenna if r0 is large compared to rc. If that is the case, then q = 1 is a good approximation, such that in (59) the term 2σr2/(qr02/rc2) may be neglected in the far field. Hence, the global spectral sampling function in the far field is

equation image

Obviously, the global Bragg wave vector is

equation image

and is simultaneously the sampling-volume average of the local Bragg wave vector.

4. Generalized Spatial Refractive-Index Spectra

[42] In order to express the instantaneous covariance equation image11(m)(t), as given in (57), explicitly in terms of local turbulence statistics, specific models for the “generalized spatial spectra” equation imagenn(m)(x, k, t) are required. In the following, we derive models for equation imagenn(1)(x, k, t) and equation imagenn(2)(x, k, t). We use the compact notation suggested by Lumley and Panofsky [1964, p. 5f.]: ∂ui/∂xjui,j, ∂ui/∂tui,t, ∂n/∂xin,i, etc.

4.1. Spatial Cross-Spectrum of Refractive-Index Fluctuations and Their First Local Time Derivatives

[43] In order to derive models for equation imagenn(1)(k), we consider the spatial cross-covariance function

equation image

where equation image is the generalized potential refractive index fluctuation with respect to the resolution-volume average [Doviak and Zrnić, 1993, p. 427], which we assume to be a conserved quantity. Neglecting molecular dissipation and compressibility effects, we have

equation image

where

equation image

is the velocity written as the sum of a local average Ui and a randomly fluctuating part ui. Correspondingly,

equation image

Note that N is several orders of magnitude smaller than the refractive-index itself, which is close to 1 in the optically clear atmosphere.

[44] Using the more compact notation N(x) ≡ N, N(x′) ≡ N′, Ui(x″) ≡ Ui, ∂Ui(x″)/∂xiUi,j, etc., we find

equation image

By definition, the local averages Ui, N, and N,i are constant within the local averaging volume centered at x ≡ (x′ + x″)/2, such that Ui(x′) = Ui(x″) = Ui(x) etc. Therefore,

equation image

Now, we evaluate the second term, −Uinn,i〉, which, as we will see in section 5.2, leads to the commonly assumed proportionality between Doppler shift and radial wind velocity. (The fourth term, −N,inui〉, was first identified by Muschinski [1998]. It was discussed in detail and called the “correlation velocity” term by Tatarskii and Muschinski [2001].)

[45] The covariance 〈nn,i〉 can be evaluated by using the Fourier-Stieltjes representation of n:

equation image

Therefore,

equation image

and (see Appendix B)

equation image

(Here we have taken advantage of the fact that the refractive index is a real quantity, such that n′ ≡ n′*.) We obtain

equation image

where equation imageB(m)(x, t) is defined in (58) and where

equation image

is the local and instantaneous radial wind velocity. A positive Vr means that the air moves away from the radar. Equation (73) means that the ratio equation imageB(1)(x, t)/equation imageB(0)(x, t) is proportional to the local radial wind velocity.

[46] We have derived (73) from (69) by neglecting all terms in (69) except the second term. Therefore, (73) is in general invalid if any of the other four terms in (69) is nonnegligible.

4.2. Spatial Cross-Spectrum of Refractive-Index Fluctuations and Their Second Local Time Derivatives

[47] Consider the spatial cross-covariance function of equation image at x′ and its second local time derivative, equation image(2), at x″:

equation image

The second local time derivative of equation image is

equation image

Here, we have assumed

equation image

that is, compared to equation (2.8) in Lumley and Panofsky [1964, p. 61] we have neglected the pressure term, the buoyancy term, and the viscosity term. With this simplification, we obtain

equation image

Each of the two terms contains four factors, each of which is the sum of a local average and a random fluctuation. Therefore, expanding equation imagenn(2)(x′, x″) leads to an expression that contains 32 terms. Out of these 32 terms, eight are of the form NUjUj,in,i〉; they are zero by definition. That is, there are 24 terms that are not necessarily zero. It is beyond the scope of this paper to systematically analyze the relative importance of these 24 terms.

[48] Let us consider the simplified case that the refractive-index fluctuations are statistically independent of the velocity fluctuations, such that

equation image

Neglecting NN,i leads to

equation image

Note that 〈ujuj,i〉 ≡ 〈ujuj,i〉 is a single-point covariance, while 〈equation image〉 is a two-point covariance. This distinction is important because two-point statistics depend on both x and the lag vector rx″ − x′, while single-point statistics do not depend on r. Then, after neglecting NN,ij in 〈equation image〉, we obtain

equation image

[49] In the previous subsection, we have shown

equation image

Differentiation with respect to xj″ leads to

equation image

Therefore,

equation image

This leads to

equation image

[50] The second spectral moment of a single-signal (auto-) spectrum is real; therefore only the real part of equation imagenn(2)(k) can contribute, and we may ignore the imaginary part. (Note that the imaginary part must not be ignored in the two-signal case.) Let us further assume that 〈uiuj〉 is negligible compared to UiUj. Then we have

equation image

where Vr(x, t) is the radial wind velocity defined in (74).

5. Single-Signal Statistics for a Monostatic CDR

[51] In this section, we apply the general theory presented above to the first three spectral moments of a single signal measured with a monostatic CDR.

5.1. Zeroth Spectral Moment: Radar Equations and VHF Aspect Sensitivity

[52] The instantaneous power equation imager(t) received at the antenna is proportional to the instantaneous signal variance equation image11(0)(t):

equation image

[e.g., Doviak and Zrnić, 1984, p. 327]. From local sampling, (57), we obtain the following radar equation:

equation image

From global sampling, (62), we find an alternative radar equation:

equation image

Note that global sampling assumes homogeneity across the radar's sampling volume, which is why we have omitted the argument x in the integrand in (89).

5.1.1. Isotropic Turbulence

[53] In the case of Bragg-isotropic and homogeneous turbulence, both radar equations lead to

equation image

[54] Probably (90) is the most concise formulation for the radar equation that is possible. An important property of (90) is that also in the Fresnel approximation the integral length scale of the refractive-index perturbations does not enter in the radar equation in the case of Bragg-isotropy, in agreement with traditional theory [Tatarskii, 1961], which relies on the Fraunhofer approximation.

[55] Since the radar equation (90) looks very different from the clear-air radar equations found elsewhere in the literature, a consistency check is in order. Doviak and Zrnić [1993, p. 75] present the radar equation in the following form:

equation image

where

equation image

[e.g., Doviak and Zrnić, 1993, equation (11.93) on p. 450] is the volume reflectivity, and

equation image

is the loss due to finite receiver bandwidth (where τp is the length of the transmitted pulse and c is the speed of light). Inserting the inertial-range spectral density at the Bragg wavenumber, ΦB(0) = 0.033Cn2kB−11/3 [Tatarskii, 1961, equation (3.25), p. 48], provides the well-known equation η = 0.379Cn2λ−1/3. The relationship (47) between gain and beam width, g = 2/Θ02 = 16 ln 2/Θ12, is confirmed by Doviak and Zrnić [1993, p. 478]. After elementary rearrangements one finds that (91) is consistent with (90).

[56] That in the case of a Bragg-isotropic refractive-index field the Fraunhofer approximation and the Fresnel approximation lead to the same radar equation seems to be at odds with by Tatarskii [2003], who concludes that the Fraunhofer approximation “can never be used in the theory of scattering by distributed scatterer (such as turbulence) in the far zone…,” a statement that is in opposition to his earlier view [Tatarskii, 1961].

5.1.2. Bragg-Anisotropy and VHF Aspect Sensitivity

[57] Consider the simple Gaussian model for the spatial autocovariance function of the refractive-index, as suggested by Doviak and Zrnić [1984]:

equation image

Here, L is a correlation length, and σn2 is the variance of the refractive index. That is, the refractive-index perturbations are assumed to be delta-correlated in the z-direction. The Wiener-Khintchine relation,

equation image

provides

equation image

Now, we tilt the plane of the laminae by replacing ky with (kycos Ψ − kzsin Ψ), where Ψ is the angle between the plane of the refractive-index laminae and the x-y plane (i.e., the plane perpendicular to the beam axis). We obtain

equation image

In the case of horizontal laminae, Ψ is the off-zenith angle of the beam direction. Here, we use the small-angle approximations cos Ψ = 1 and sin Ψ = Ψ and obtain

equation image

Inserting into (88) leads, after elementary rearrangements, to the radar equation

equation image

where

equation image

For L large compared to the critical length

equation image

we obtain Θs = Θ0. A convincing empirical verification of this asymptotic behavior of Θs can be found in Figure 4 in Hocking et al. [1986], where an observed vertical profile of Θs is shown. The smallest values of Θs were reached in the very stably stratified lower stratosphere, i.e., at altitudes between about 12 km and 20 km AGL. In that region, Θs = 2.3° was found. The SOUSY radar, which was used for these observations, had Θ1 = 5°, corresponding to Θ0 = 1/equation image × 5° = 2.1°. That is, the observed minimum of Θs is close to Θ0, as predicted by (100).

[58] Using (50), it can be shown that the critical length Lc amounts to one quarter of the effective antenna diameter De:

equation image

That is, Lc is not directly related to the Fresnel length f. This result is in agreement with Gurvich and Kon [1992] but is contrary to common belief, as will be documented in section 6.

5.2. First Spectral Moment: Doppler Velocity

[59] We define the instantaneous Doppler shift as the ratio of the first and the zeroth instantaneous spectral moments:

equation image

From (52) follows

equation image

where A(x) is the amplitude weighting function defined in (28). The common definition of the Doppler velocity as vD = −ωD/kB leads naturally to the definition of the instantaneous Doppler velocity:

equation image

such that

equation image

Note that this formulation is very general.

[60] Inserting (73), which neglects the correlation velocity term in equation imageB(1)(x, t), leads to

equation image

where

equation image

is the instantaneous power-weighted radial velocity, i.e., the local and instantaneous, radial velocity weighted with the normalized product of A2(x) and equation imageB(0)(x, t).

[61] An equation similar to (107) has been derived earlier [e.g., Doviak and Zrnić, 1993, p. 110] for a sampling volume populated with point scatterers, and it has been used heuristically also for scatter from continuous refractive fields [e.g., Frisch and Clifford, 1974; Hocking et al., 1986; Muschinski et al., 1999b; Görsdorf and Lehmann, 2000; Johnston et al., 2001].

5.3. Second Spectral Moment: Spectral Width

[62] Of particular interest for turbulence measurements is the instantaneous second central moment, normalized by the instantaneous zeroth moment:

equation image

The moments theorem (Appendix A) leads to

equation image

Defining the instantaneous second central moment in velocity units,

equation image

leads to the very general formulation

equation image

[63] The simplified models for equation imageB(1)(x, t) and equation imageB(2)(x, t) given in (73) and (86), respectively, yield

equation image

This simplified relationship has been derived earlier for point scatterers [e.g., Doviak and Zrnić, 1993, p. 110]. Frisch and Clifford [1974] anticipated that this relationship is valid also for clear-air backscatter and used it to retrieve energy dissipation rates from equation imagev2 measurements in the atmospheric boundary layer. A discussion of the Frisch and Clifford [1974] approach can be found in Doviak and Zrnić [1993, p. 408f.].

6. Discussion

[64] The three common assumptions in the literature on CDR theory are that refractive-index and velocity perturbations are (1) statistically homogeneous within the radar's sampling volume, (2) statistically isotropic at the Bragg wavenumber, and (3) statistically stationary during the dwell time. There is a wide consensus, however, that one or more of these assumptions are usually not fulfilled, neither in the free atmosphere [e.g., VanZandt et al., 1978; Gage et al., 1981; Woodman and Chu, 1989; Gage, 1990; Fairall et al., 1991; Dalaudier et al., 1994; Chilson et al., 1997; Muschinski, 1997; Muschinski and Wode, 1998; Luce et al., 2001a] nor in the atmospheric boundary layer [e.g., Gossard et al., 1984; Gossard, 1990; Eaton et al., 1995; Muschinski et al., 1999b; Pollard et al., 2000; Wyngaard et al., 2001; Balsley et al., 2003; Chilson et al., 2003].

[65] The theory developed in this paper allows the robustness of CDR retrieval techniques against violations of these three assumptions to be analyzed both specifically and generally. In the following discussion, a number of implications of the refined theory are addressed in an illustrative, rather than in a systematic manner.

6.1. Why Local Sampling?

[66] There are basically two different mechanisms for CDR backscatter. First, scatter from locally Bragg-isotropic, fully developed refractive-index turbulence [Tatarskii, 1961]; second, scatter from refractive-index discontinuities (“sheets”) that are thin compared to the radar wavelength but whose extent in the directions transverse to the radial direction is much larger than the radar wavelength [Metcalf and Atlas, 1973; Gage and Green, 1978; Röttger and Liu, 1978; Doviak and Zrnić, 1984; Woodman and Chu, 1989; Tsuda et al., 1997a, 1997b; Worthington et al., 1999; Luce et al., 2001a].

[67] The first mechanism is usually referred to as “isotropic scatter,” “Bragg scatter,” or “turbulence scatter.” The second mechanism has been termed “partial reflection,” “quasi-specular reflection,” “Fresnel scatter,” or “Fresnel reflection.” It is commonly accepted terminology that Fresnel scatter is scatter from a population of parallel sheets with random relative spacing, while Fresnel reflection is scatter (or reflection) from a single sheet. That is, in contrast to Fresnel reflection, Fresnel scatter requires a statistical approach [Doviak and Zrnić, 1984].

[68] As Gossard et al. [1984, p. 1532] pointed out, “under conditions when partial reflection can be important, the return is from only a tiny fraction of the resolution cell of the radar.” The case study by Gage et al. [1981] corroborates this statement. Using a narrow-beam, vertically pointing 50-MHz CDR, they observed specular reflection from sheets whose tilt angles varied because of gravity-wave motion. When the “specular point” moved out of the CDR's angular field of view, the echo intensity dropped dramatically. Within minutes, echo intensity variations of up to 40 dB were observed.

[69] More recently, Chau and Balsley [1998] and Palmer et al. [1998] used interferometric and imaging techniques to track the angular position of the specular point as a function height and time. Hocking et al. [1986] pointed out that the position of the specular point relative to the beam axis leads to a bias in the retrieved radial wind velocity and suggested a correction formula. Muschinski [1996b] suggested a simple statistical model of this bias in the case of Fresnel scatter from laminae at the edges of a train of Kelvin-Helmholtz billows, and he obtained biases as large as a few tens of centimeters per second for moderate beam widths and for wind speeds typical for upper-tropospheric jet streams. The model predicts a decrease of the bias magnitude with decreasing beam width. In a recent case study on VHF CDR observations in the upper troposphere over Indonesia, Yamamoto et al. [2003] report quasi-persistent KHI and good quantitative agreement with the Muschinski [1996b] model, both with respect to the sign and magnitude of the vertical-velocity bias and the sign of the streamwise echo-intensity imbalance.

[70] Woodman and Chu [1989] discussed a conceptual model for backscatter from a turbulent shear layer embedded in a stably stratified atmosphere. Figure 2 is a modification of Figure 4 in Woodman and Chu [1989]. They presumed that the upper and lower edges of the turbulent layers are usually marked by steep vertical temperature gradients that persist longer than the randomly oriented gradients in the interior of the turbulent layer. If these edges, or interfaces, are thin compared to λ/2 and not too rough (also relative to λ/2), then the echo intensity from the edges may well exceed the echo intensity from the more or less isotropic turbulence within the layer. Recent high-resolution in situ observations of turbulent layers in the lower free troposphere [e.g., Muschinski and Wode, 1998; Muschinski et al., 2001; Balsley et al., 2003] and advanced direct numerical simulations [e.g., Werne and Fritts, 1999; Gibson-Wilde et al., 2000] have provided supporting evidence for the existence and persistence of these sharp edges. Figure 2, in contrast to Figure 4 in Woodman and Chu [1989], emphasizes the roughness of the layer edges.

Figure 2.

Sketch of a turbulent layer extending across the CDR's sampling volume, adapted from Figure 4 in Woodman and Chu [1989]. The thickness of the layer may or may not be larger than the Bragg wavelength. The layer “edges” may or may not be smooth with respect to the Bragg wavelength. The small diagram on the right-hand side of the main sketch shows schematically an instantaneous and local vertical profile (thin line) of the refractive-index fluctuations across the layer. Note the steep gradients in the mean vertical profile (bold line, schematic) at the upper and lower edges of the layer.

[71] The scattering theory presented in this paper can be used to analyze, in a quantitative fashion, the effects of various characteristics of turbulent layers on mean values and fluctuations of the moments of the CDR spectra. Refined models of turbulent layers could include parameters like the shear layer thickness, the time elapsed since the onset of the turbulence, the “background” vertical gradients of horizontal velocity, potential temperature, and specific humidity, the edge regions' roughness statistics, and the larger-scale layer tilt statistics. In a further step, the single-layer scenario could be generalized by allowing the CDR sampling volume to be filled by a population of such turbulent layers, as outlined by VanZandt et al. [1978] and Fairall et al. [1991].

[72] Spatial variability of CDR reflectivity and radial wind velocity, particularly in the atmospheric boundary layer, can be directly observed with imaging CDRs like the Turbulent Eddy Profiler of the University of Massachusetts [Mead et al., 1998; Pollard et al., 2000].

6.2. Why Instantaneous Sampling?

[73] For a statistically meaningful measurement of the Doppler shift, the CDR signal time series must contain a sufficient number of independent samples. In other words, the dwell time must be long compared to the correlation time τc. If the sampling volume is filled with fully developed turbulence, τc is approximately

equation image

where σt is the standard deviation of the radial velocity fluctuations within the resolution volume. For a 915-MHz CDR (λ = 0.33 cm), τc is as short as 0.25 s even if σt is only 0.1 m s−1. Therefore, a dwell time of 1 s is usually long enough to provide statistically meaningful spectral moments, which is in agreement, e.g., with the simulations by Muschinski et al. [1999b]. One second is short compared to typical “renewal time” scales, i.e., the time needed for an air parcel to be advected across the radar's sampling volume.

[74] Frisch and Clifford [1974] introduced a technique to retrieve the energy dissipation rate ε from the spectral width. This method requires that the observed spectral width represents the instantaneous spatial variability of Vr within the sampling volume and that the effect of the temporal variability of Vr during the dwell time may be neglected. Therefore, the Frisch and Clifford [1974] method requires the dwell time to be short compared to the renewal time. The effects of “temporal spectral broadening” associated with dwell times comparable with or longer than the renewal timescale were quantitatively discussed by White et al. [1999].

6.3. Taylor Hypothesis and Random Taylor Hypothesis

[75] Heuristic CDR theory often assumes that within the CDR's sampling volume and during the dwell time, refractive-index perturbations are statistically stationary and homogeneous and are advected with the mean wind vector, in agreement with Taylor's frozen turbulence hypothesis [Taylor, 1938]. This assumption neglects any correlations between velocity fluctuations and refractive-index fluctuations from the outset. Frisch and Clifford [1974] made one step further by allowing the velocity to randomly vary within the CDR's sampling volume but they still assumed statistical homogeneity and statistical stationarity.

[76] The CDR theory developed in this paper allows mth-order moments of instantaneous Doppler spectra to be expressed in terms of generalized spectra, equation imagenn(m)(x, k, t), which were defined in (16). Models for the generalized spectra's Bragg components, equation imageB(m)(x, t) ≡ equation imagenn(m)[x, −equation imageB(x), t], for m = 1 and m = 2 were developed in section 4. These models rely on (65), i.e., on the assumption that for the local rates of change of the refractive index the local and instantaneous advection dominates and that compressibility effects and molecular dissipation may be neglected. This assumption is also known as the “random Taylor hypothesis” [e.g., Tennekes, 1975; Tsinober et al., 2001] or the “sweeping decorrelation hypothesis” [e.g., Praskovsky et al., 1993].

6.4. Velocity Biases Resulting From Incoherent Averaging of Doppler Spectra

[77] Often, dwell times much longer than the renewal time have to be used in order to separate the atmospheric part of the Doppler spectrum from system or cosmic noise. For example, data of the NOAA Profiler Network are routinely processed with a dwell time of about 1 min [Barth et al., 1994].

[78] Now, consider the biases that may result from using time-averaged spectral moments, i.e., moments computed from “incoherently averaged” Doppler spectra. The Doppler shift estimated from time-averaged moments is

equation image

Obviously, ΩD is in general different from the time average equation image of the instantaneous Doppler shifts:

equation image

Only if Vr(x, t) and equation imageB(0)(x, t) are uncorrelated, then ΩD = equation image. On the other hand, if the joint statistics of Vr(x, t) and equation imageB(0)(x, t) are known, then specific expressions for the difference between ΩD and equation image can be derived, which may be used to remove Doppler-velocity biases that result from using incoherently averaged Doppler spectra.

[79] Two different mechanisms that lead to a correlation between Vr(x, t) and equation imageB(0)(x, t) in the free atmosphere have been suggested. According to Nastrom and VanZandt [1994], vertically propagating gravity waves are associated with a correlation between Vr(x, t) and equation imagen2(x, t). Muschinski [1996b] pointed out that in shear regions populated with Kelvin-Helmholtz billows, instantaneous Doppler velocities and equation imageB(0)(x, t) are correlated because the local tilting angles of quasi-specular refractive-index interfaces have a skewed probability density function.

[80] Vertical-velocity biases have been found not only in VHF CDR observations of the free atmosphere but also in UHF CDR observations in the daytime ABL. Angevine [1997], Beyrich et al. [1998], and Lothon et al. [2002] report downward biases in the daytime ABL of up to 30 cm s−1. Because the daytime ABL is unstable, gravity waves and Kelvin-Helmholtz instability do not occur there. Therefore, the scenarios suggested by Nastrom and VanZandt [1994] and Muschinski [1996b] do not apply for the daytime ABL. In order to understand the biases reported by Angevine [1997] and Lothon et al. [2002], further research on the joint statistics Vr(x, t) and equation imageB(0)(x, t) in the ABL is needed.

6.5. Aspect Sensitivity, Fresnel Scatter, and Asymptotic Specularity

[81] In section 5, we have shown that the antenna diameter De, not the Fresnel length f, is the relevant length scale for aspect sensitivity. This is in agreement with Doviak and Zrnić's [1984] analytical result that the shape of the global sampling function h11(k) is independent of range. If f were relevant, then h11(k) would have to depend on range because f increases proportional to the square root of the range.

[82] In section 5.1.2, we have shown that for a refractive-index spectrum that is Gaussian in the transverse directions and flat, or white, in the radial direction, the width Θs of the angular pattern of backscattered power depends on the Bragg wavenumber kB, the transverse refractive-index correlation length L, and the characteristic beam width Θ0 in a way that Θs approaches Θ0 if L is large compared to a critical value Lc. We have shown that Lc amounts to one quarter of the effective diameter and is therefore independent of range. In particular, Lc is not related to f.

[83] The result that a minimum of Θs is set by the beam width Θ0 is in contrast to Hocking and Röttger [2001, p. 941]: “In principle, specular reflectors can have any value of Θs;…” Röttger and Larsen [1990, p. 242] write: “The terms Fresnel scatter and Fresnel reflection have been introduced because the horizontal correlation distance of the discontinuities is longer than the radar wavelength but of the order of the Fresnel zone…” A similar view is documented in Gage [1990, p. 551]. This disagrees with our result that the critical length scale Lc is equal to De/4 and not equal to f. In the transition region between the near field and the far field, f and De/4 are of comparable magnitude, which may be the reason of the misinterpretation. In the “outer far field,” however, where f is large compared to De/4, “asymptotic specularity” is reached at much smaller L than previously thought. (With asymptotic specularity we mean independence of the degree of aspect sensitivity as function of L if L is larger than Lc.) For VHF CDR observations in the mesosphere, f is of order 1 km and much larger than De/4, which for a VHF radar is of order 20 m.

7. Summary and Conclusions

[84] In this paper, the existing Fresnel-approximated theory of radio-wave backscatter observed with CDRs [Doviak and Zrnić, 1984, 1993] has been extended in two main directions. First, local sampling has been introduced, such that the local velocity and refractive-index statistics are allowed to vary randomly within the radar's sampling volume and during the dwell time. Second, a first-principle formulation of all moments of the instantaneous Doppler spectrum has been derived from a local version of the moments theorem in combination with the random Taylor hypothesis. Although it is beyond the scope of this paper to give a comprehensive discussion of the implications, a number of conclusions can be drawn:

[85] 1. The Fraunhofer approximation, which neglects effects of the curvature of the wave fronts within the sampling volume [Tatarskii, 1961], and the Fresnel approximation, which takes the curvature of the wave fronts within the sampling volume into account [Doviak and Zrnić, 1984, 1993], lead to the same radar equation if the refractive-index perturbations are statistically isotropic at the Bragg wavenumber.

[86] 2. If the local refractive-index spectrum is strongly Bragg- anisotropic, and if the laminae's planes are nearly perpendicular to the beam axis, then the echo intensity tends to become aspect sensitive, as is commonly observed in the lower VHF regime at near-zenith beam directions. If the echo intensity is not aspect sensitive, however, then the refractive-index spectrum is either Bragg-isotropic at least somewhere within the sampling volume, or the variability of Bragg-anisotropic laminae orientations is large compared to the radar's beam width, or both. Therefore, absence of aspect sensitivity at VHF and UHF frequencies does not necessarily imply Bragg-isotropy, in contrast to what appears to be widely believed.

[87] 3. For asymptotic specularity to be reached, the refractive-index correlation length perpendicular to the beam axis must be larger than one quarter of the effective antenna diameter but not necessarily larger than the Fresnel-zone radius. (Here we have assumed a Gaussian beam and a refractive-index spectrum that is Gaussian in the transverse directions and white in the radial direction.) This result is in agreement with Gurvich and Kon [1992] but in contrast to what appears to be widely believed.

[88] 4. The refined theory gives a unifying description of three mechanisms that cause systematic differences between Doppler velocities and radial wind velocities: (1) a correlation between Bragg-isotropic radar reflectivity and radial wind velocity [Nastrom and VanZandt, 1994]; (2) a correlation between the orientation of Bragg-anisotropic scatterers and the radial wind velocity [Muschinski, 1996b; Yamamoto et al., 2003]; and (3) a nonzero Bragg wave vector component of the spatial quadrature spectrum of refractive-index and radial wind velocity [Tatarskii and Muschinski, 2001]. Whether the refined theory is useful to explain the puzzling downward bias of a few tens of centimeters per second, which is often observed with UHF CDRs in the atmospheric boundary layer [e.g., Angevine, 1997; Beyrich et al., 1998; Lothon et al., 2002], remains to be seen.

[89] 5. If the correlation velocity [Tatarskii and Muschinski, 2001] is negligible and the refractive-index perturbations Bragg-isotropic, then the instantaneous Doppler spectra may be interpreted as histograms of power- and reflectivity-weighted radial wind velocities. This is not necessarily true for incoherently averaged (i.e., time-averaged) Doppler spectra.

[90] 6. The local sampling theory presented here allows CDR signals to be expressed in terms of local turbulence statistics that may vary randomly in space and time. In this regard, there is a conceptual similarity between the refined CDR theory and the large-eddy simulation (LES), which has become the key technique to computationally simulate high-Reynolds number turbulent flows [e.g., Lilly, 1967; Leonard, 1974; Schmidt and Schumann, 1989; Muschinski, 1996a; Lesieur and Métais, 1996; Stevens and Lenschow, 2001; Muschinski and Lenschow, 2001; Wyngaard et al., 2001].

[91] 7. Perhaps the most important feature of this theoretical development, which in its general form has been presented in sections 2 and 4, is that it offers a rigorous and systematic access to the zeroth as well as the higher-order moments of the Doppler spectrum (in the single-signal case) and of the Doppler cross-spectrum (in the two-signal case) of observed and simulated CDR signals under a wide variety of meteorological conditions in the atmospheric boundary layer and the free atmosphere. This has been achieved by combining the random Taylor hypothesis, the Fresnel approximation, and a local version of the moments theorem. All local turbulence statistics are allowed to vary randomly in time and space, without the need to assume isotropy, homogeneity, or stationarity of the refractive-index and velocity turbulence within the sampling volume and during the dwell time. Moreover, the general theory makes no assumptions on the specific form of the instrument functions. The theory can be used to analytically explore and optimize the design of multifrequency and multireceiver CDRs as well as the setup of computer simulations based on modern computational fluid dynamics techniques.

Appendix A: Moments Theorem

[92] The moments theorem allows the mth moment of a cross-spectrum ϕ12(ω) to be expressed in terms of the mth derivative of the corresponding cross-covariance function C12(m)(τ) at zero lag:

equation image

A proof is given in the following.

[93] By definition,

equation image

Consider the Fourier expansion of C12(τ),

equation image

Expanding the exponential into a Taylor series,

equation image

and taking the mth derivative at τ = 0 leads to

equation image

With

equation image

we obtain

equation image

Therefore,

equation image

q.e.d.

Appendix B: Fourier-Stieltjes Representations

B1. Statistical Orthogonality

[94] Consider the Fourier-Stieltjes representations of two complex, random quantities, p(t) and q(t):

equation image

and

equation image

Now, assuming statistical stationarity, we define a cross-covariance function:

equation image

Taking the conjugate complex of p(t + τ) instead of p(t + τ) is an arbitrary but commonly accepted convention. Taking

equation image

as the defining equation for the cross-spectrum ϕpq(ω), we find

equation image

(This looks artificially complicated. But we will see in a moment why we write Cpq(τ) in this peculiar way.)

[95] On the other hand, we find from the Fourier-Stieltjes representations:

equation image

Comparing with (B5) leads to the statistical orthogonality relationship

equation image

B2. Correlation Theorem

[96] Consider the expression

equation image

where G(r) is a window function and R(r) is a covariance function. Write G(r) and R(r) in terms of their Fourier transforms, H(k) and Φ(k), where H(k) is a spectral transfer function and Φ(k) is a spectrum:

equation image

and (according to the Wiener-Khinthine theorem)

equation image

Therefore

equation image

where we have used the identity

equation image

Comparing the two expressions for C leads to the correlation theorem:

equation image

Note the minus sign in Φ(−k).

Acknowledgments

[97] This work would have been impossible without innumerable discussions with colleagues in the various disciplines that are connected by an interest in CDR. I am grateful to Steve Frasier and Paco Lopez-Dekker (both Univ. Mass.) for many interesting discussions and for providing me access to their beautiful and inspiring TEP and FMCW radar measurements; to Rod Frehlich (CIRES), Peter Sullivan (NCAR), Dave Fritts, Joe Werne (both at Colorado Research Associates), and John Wyngaard (Pennsylvania State Univ.) for many discussions about boundary-layer turbulence, LES, DNS, and intermittency; to my colleagues at CIRES and the NOAA Environmental Technology Laboratory: Bob Banta, Alan Brewer, Phil Chilson, Chris Fairall, Graham Feingold, Reg Hill, Rich Lataitis, Dan Law, Ken Moran, Vladimir Ostashev, Barry Rye, Valerian Tatarskii, Bob Weber, and Alan White for countless conversations on turbulence and atmospheric remote sensing; to Phil Chilson, Bob Palmer (Univ. Nebraska), and Tian-You Yu (Univ. Oklahoma) for a decade of joint efforts to get the operational community excited about the potential of frequency-domain interferometry (FDI) and range imaging (RIM); and to Peter Czechowsky, Jürgen Klostermeyer, Rüdiger Rüster, and Gerhard Schmidt from the SOUSY group, who introduced me to the world of radar remote sensing in the early 1990s. Thanks are due to Koki Chau (Jicamarca Observatory, Peru), Rod Frehlich (Univ. Colorado), Ken Gage (NOAA Aeronomy Lab.), Reg Hill, Volker Lehmann (German Weather Service, Lindenberg, Germany), Hubert Luce (Univ. de Toulon et du Var, France), Alex Praskovsky and Eleanor Praskovskaya (both NCAR), Valerian Tatarskii, Ulrich Schumann (DLR Oberpfaffenhofen, Germany), Gerd Teschke (Univ. Bremen, Germany), and John Wyngaard for many helpful comments on earlier versions of the manuscript. A particularly thorough and valuable review was given by Dick Doviak (NOAA National Severe Storms Lab.), who served as one of the Radio Science reviewers. I am particularly indebted to Ben Balsley (CIRES), Steve Clifford (CIRES), Dick Doviak, Earl Gossard (CIRES, now retired), Mike Hardesty (NOAA-ETL), Don Lenschow (NCAR), M. J. Post (CIRES), and Dick Strauch (formerly NOAA-ETL, now NOAA Forecast Systems Lab.) for many years of support, encouragement, and collaboration. This study was supported by the U.S. Army Research Office under grant 40136-EV (program management Dr. Walter Bach).

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