## 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 10^{5} 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 _{12}^{(m)}(*t*) ≡ 〈*I**_{1}(*t*)*I*_{2}^{(m)}(*t*)〉 are developed, where *I*_{1}(*t*) and *I*_{2}(*t*) are two phase-coherently measured, complex CDR signals, where *I*_{2}^{(m)}(*t*) ≡ ∂^{m}*I*_{2}(*t*)/∂*t*^{m} is the *m*th time derivative of *I*_{2}(*t*), and where *t* is time. (Here, *I*_{1} and *I*_{2} 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 _{12}^{(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):

where the local cross-covariance function

is a random function of its arguments, *G*_{12}(**x**, **r**) is a deterministic instrument function, and 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 (*I*_{1}(*t*) ≡ *I*_{2}(*t*)) sampling function *G*_{11}(**x**, **r**) for a monostatic radar. It is shown that _{11}^{(m)}(*t*) can be written as a weighted **x**-space integral over contributions _{nn}^{(m)}(**x**, −_{B}(**x**), *t*), where _{B}(**x**) is the local Bragg wave vector, and _{nn}^{(m)}(**x**, **k**, *t*) is the local spectrum associated with _{nn}^{(m)}(**x**, **r**, *t*).

[11] In section 4, explicit models for _{nn}^{(1)}(**x**, **k**, *t*) and _{nn}^{(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.