1. Introduction

During the last few decades, radio acoustic sounding systems (RASSs) have developed as a reliable tool for remotely measuring vertical profiles of virtual temperature. RASSs retrieve virtual temperature profiles by using Doppler radar to measure the propagation speed of a sound wave, which is a function of temperature. While early systems were based on radars operating at wavelengths of a few meters (e.g., Marshall et al. 1972; Frankel et al. 1977), more recently there has been an emphasis on UHF wind profilers observing the atmospheric boundary layer (e.g., May et al. 1990; Carter et al. 1995; Angevine et al. 1998).

The theory behind RASS has been treated extensively in a number of studies. Earlier work studied the focusing effect of the spherical acoustic wave front and the size and position dependence of the focused spot on the mean horizontal wind (Clifford et al. 1978; Bhatnagar and Peterson 1979). Later work considered the effect of wind and temperature gradients on the spatial structure of the radar backscatter from the sound wave (hereafter referred to as the RASS echo) (Masuda 1988; Lataitis 1992). In these studies, the ability of a RASS to capture the focused spot was considered to be the limiting factor. More recent work has recognized that the broadening of the spot resulting from the decoherence of the acoustic wave front caused by atmospheric turbulence would allow sensitive radars to significantly extend their height coverage in windy conditions (e.g., Lataitis and Clifford 1996). This effect is more pronounced for shorter-wavelength RASSs, since the decohering effect on the acoustic wave is more acute.

A number of studies suggest corrections to enhance the accuracy of the RASS temperature data. These include corrections to biases caused by the vertical wind component (May 1989), horizontal wind and turbulence (Peters and Angevine 1996), and biases due to gradients in the reflectivity profile (Angevine et al. 1994; Görsdorf and Lehmann 2000).

Although use of RASSs for mean vertical profiles of virtual temperature has become commonplace, few experimental studies of the spatial structure of the RASS echo exist. For example, Masuda (1988) and May et al. (1996) use the middle and upper (MU) radar, a 103-m-diameter 46.5-MHz wind profiler, for this purpose. Similarly, few studies have addressed the potential to retrieve turbulent fluctuations in temperature. Most probably, the lack of appropriate hardware has limited such studies for UHF systems.

During the past decade, the Microwave Remote Sensing Laboratory at the University of Massachusetts has developed the turbulent eddy profiler (TEP), a 915-MHz volume-imaging Doppler radar for atmospheric boundary layer (ABL) research (Hopcraft 1997; Mead et al. 1998; Pollard et al. 2000). TEP is designed to study the local three-dimensional structure of boundary layer clear-air radar echoes. Recently, TEP has been augmented with an acoustic source and modified to implement RASS. This implementation offers new possibilities to study the spatial structure of the RASS signal: the spatial distribution of the intensity of the RASS echo on the receive array can be investigated, as well as the volumetric distribution of the RASS-induced radar backscatter. It also opens up the possibility to retrieve information about the three-dimensional structure of the temperature field at scales of tens of meters. For the latter purpose, observable temperature fluctuations must exceed the instrument sensitivity. Although these fluctuations are usually small, large eddy simulations predict horizontal temperature variations up to 0.7 K for moderately convective conditions (Khanna and Brasseur 1998).

In this paper we present observations of the spatial structure of RASS echoes under developing convective conditions using volume-imaging radar. This structure is interpreted both through individual realizations of the echoes and through time averages. We explore the possibility of retrieving turbulent statistical information of the virtual temperature field by computing profiles of the structure function parameter, C²_Tυ. The paper is organized as follows. Section 2 reviews RASS theory and concepts. Section 3 describes the experimental setup including a brief description of the TEP-based RASS system and the signal processing used to retrieve temperature. Time–height cross sections of radar echo and RASS are interpreted. Section 4 studies the spatial structure of the RASS echo and the statistical fidelity of near-instantaneous measurements of T_υ are considered by examining structure functions and their vertical profiles. A summary and conclusions are presented in section 6.

2. Radio acoustic sounding

RASS employs a coherent radar to measure backscattered radiation from fluctuations in atmospheric density induced by an acoustic source to determine the local speed of sound. The speed of sound, c_a, is related to virtual temperature, T_υ, to a first approximation by

View Expanded

where R is the universal gas constant, M is the molecular weight of dry air, and γ = C_p/C_υ is the ratio of specific heats at constant pressure and constant volume (Görsdorf and Lehmann 2000). In deriving RASS temperatures it is common to use

View Expanded

which corresponds to M = 28.96 g mol⁻¹ and γ = 1.4, although these values are not constant in the real atmosphere.

The method makes use of the enhancement in electromagnetic backscatter resulting from the matching of the acoustic wavelength, λ_a, to half of the radar wavelength providing a Bragg resonant scattering structure. Since the speed of sound is unknown a priori and since it varies with height, a range of acoustic frequencies must be used in practice in order to satisfy the Bragg resonance condition for the range of expected speeds of sound. Diverse modes of acoustic excitation have been considered in the past. In Angevine et al. (1993) it was found that a frequency modulated acoustic signal of constant amplitude gave the best results. If the power spectrum of the acoustic signal is approximately flat over the range of frequencies employed and it is assumed that the acoustic wave fronts are aligned with the propagation direction, the Doppler spectrum of the received RASS signal has a single maximum at the acoustic frequency:

View Expanded

where λ_a is the acoustic wavelength and λ_EM is the electromagnetic wavelength. Since the acoustic wave propagates in a moving medium, the sound wave travels at the vector sum of the sound velocity and the velocity of the underlying parcel of air. Hence, to retrieve the true speed of sound, the component of air velocity in the direction of travel of the acoustic wave, u_r, must be subtracted from the Doppler velocity measured by the radar; thus,

View Expanded

Several factors limit the performance of a RASS. The spherical acoustic wave fronts tend to concentrate or retrofocus the backscattered signal on a small spot (Clifford et al. 1978; Bhatnagar and Peterson 1979; May et al. 1996). The analysis is further complicated if horizontal winds and temperature gradients are present (Masuda 1988; Lataitis 1992). To obtain backscatter, assuming a quasi-specular reflection of the electromagnetic wave on the acoustic wave front, this acoustic wave front must propagate in a direction parallel to the radius vector with its origin at the center of the radar transmitter and the receiver array. This results in backscatter regions whose locations depend on the wind and temperature profile.

Turbulence further distorts the acoustic wave front, dispersing a portion of the backscattered signal away from the focused spot, or broadening it (Bhatnagar and Peterson 1979; Lataitis and Clifford 1996). This allows sensitive clear-air radar systems to detect the RASS echo even when the focused spot lies away from the receiving antenna, which is often the case. Under these conditions, the intensity of the RASS echo strongly depends on a path-weighted value of the acoustic refractive index structure parameter (Lataitis and Clifford 1996):

View Expanded

where C²_na is the acoustic refrative index structure function parameter and R is the range to the radar. Thus, the RASS measurement is inherently conditioned to a certain level of turbulence within the intervening air column when the detected echo lies outside the focused spot.

3. Experimental setup

4. Spatial structure of the RASS echo

The combination of RASS with a volume-imaging radar suggests that, if the horizontal fluctuations of the virtual temperature over scales within the radar's field of view are sufficiently large, then these might be imaged. However, it is sensible to first observe the spatial structure of the RASS echo. This structure may be studied through observation of the diffraction pattern, that is, the backscattered electromagnetic field incident on the receive array, of the RASS echo, or by examining beamformed images of the radar reflectivity of the insonified air.

Figure 6 shows an example of the time-averaged intensity of the diffraction pattern of the RASS echo from 250-m height incident on the hexagonal array. For comparison, the right panel shows the intensity of the diffraction pattern corresponding to the atmospheric clear-air echo from the same height. The radiometric confidence in these intensity estimates is limited by Rayleigh fading, and is improved by averaging independent samples. The confidence interval is calculated assuming that the statistics of the averaged intensity follow a χ²(2N) distribution, where N is the number of independent samples that are calculated from the dwell time and the spectral width of the signal (Ulaby et al. 1986). It is worth noting that this confidence interval estimate implicitly assumes that the signal is stationary. In both cases the intensity patterns are 5-min averages, resulting in an estimated 90% confidence interval of less than 1 dB for each location. The average atmospheric echo intensity pattern is nearly uniform as one might expect. However, the spatial distribution of the RASS echo shows a well-defined structure. Given the light wind conditions, it is appealing to interpret the structure as the focused spot, centered slightly off the array and which is consistent with the geometry of the setup. This spot is blurred due to the wandering of the spot as the wind changes in strength and direction.

Figure 7 shows a sequence of images displaying the evolution of the near-instantaneous intensity of the diffraction patterns and the corresponding beamformed RASS echo intensity patterns. The images correspond to 2.5-s averages spaced approximately 1 min apart. Although the number of independent samples averaged is smaller, the 90% radiometric confidence interval is approximately 2.5 dB, which implies that these images are not dominated by fading but by the actual structure of the RASS echo. Given the 12° acoustic beamwidth, the scale of the structures in the pattern is expected to be approximately 1 m, which is consistent with the structures observed in the images. The first two images in the sequence suggest the presence of the focused spot in the observed diffraction pattern. The dominant feature in the associated beamformed patterns is the acoustic beam pattern. In the following frames the regions of high RASS-induced reflectivity is less well defined and the intensity of the echo is lower. In the corresponding diffraction patterns the spot appears to have wandered off of the receive array, causing the lower measured intensity.

Figure 8 shows two examples of the volumetric signature of the RASS echo, displayed as a stack of horizontal slices of the reflectivity at each range gate. The left image is similar to what one would expect intuitively: the region with significant backscatter corresponds approximately to the portion of the TEP volume illuminated by the narrower acoustic beam. In the right-hand image, the backscatter region is smaller and appears to be divided into two regions. Ray-tracing simulations described in Masuda (1988) have shown similar split regions appearing if the vertical gradient of the wind velocity exceeds the vertical gradient of the speed of sound. Presumably, other mechanisms could produce similar structures. The time interval between these images is only 90 s, showing the rapid temporal variability of the spatial signature of the RASS echo. Simple two-dimensional ray-tracing calculations using RASS-retrieved temperature profiles and radar-retrieved wind profiles show considerable temporal variability (not shown in this paper) that is consistent with the observations. However, more complex three-dimensional calculations are needed to understand this variability in detail.

Figures 7 and 8 also suggest that, while a large receiving aperture may help to capture a possible focused spot, the resulting narrower beam may point in a direction of low RASS backscatter. In a beamforming system the latter is avoided since the beam may be steered into regions of high backscatter. For example, for the time period studied, comparing the intensity corresponding to TEP's vertical beam to the intensity maxima among all available beams, an average improvement of approximately 9 dB is found. This allows a beamforming system to retrieve vertical temperature profiles at a faster rate and with extended height coverage compared to a single-beam system. However, it is also clear from the beamformed intensity images that the spatial variability of the RASS signal may preclude the ability to retrieve instantaneous volumetric snapshots of the complete temperature field. Nevertheless, such variability does not preclude the computation of spatial statistics of temperature.

5. Structure function parameter, C²_Tυ

Temperature fields obtained with TEP allow for an explicit computation of the structure function and the structure function parameter, C²_Tυ, using pairs of samples spaced horizontally (see Fig. 9). The structure function is calculated by averaging squares of temperature differences, (T_i,j − T_k,l)², for pairs of points equally spaced (e.g., the pairs {T_i,j, T_k,l} and {T_i,j, T_m,n} in the figure) and over time. The structure function parameter is calculated using (9).

The error in the computation of the structure function is the result of the error in the measurement of two temperatures, T₁ = T_υ(r₁) and T₂ = T_υ(r₂), where r₁ and r₂ are two locations that, in the cases examined, correspond to the same height and are separated horizontally. If the estimated temperatures are T̂₁ = T₁ + ε₁ and T̂₂ = T₂ + ε₂, with ε₁ and ε₂ the measurements errors, the error in the estimation of (T₁ − T₂)² is given by

View Expanded

Assuming that the errors are zero-mean random variables, the expected value of ε_d is

View Expanded

which will be greater than zero, introducing a bias in the measurement. The ρ₁₂ term accounts for the possible correlation of the errors in the temperature estimates. The variance of the error depends on the higher moments of the probability density functions of ε₁ and ε₂. In order to estimate the temperature differences, T₁ − T₂, the measurement error of the individual temperature estimates needs to be small compared to these random temperature differences. In this case the variance of ε_d can be approximated by

View Expanded

It is reasonable to assume that for closely spaced beams, the sign of ρ₁₂ is positive. Thus common-mode measurement errors (e.g., in accounting for the vertical wind) will tend to cancel. In the worst case, they are independent, which we assume henceforth.

The variance of T̂_υ is dictated by uncertainty in the velocity estimates of the RASS return and the atmospheric echo. The virtual temperature is calculated by rearranging (2):

View Expanded

To characterize the error for the temperature estimation, standard error propagation has been applied to (13), which results in

View Expanded

The accuracy of the Doppler velocity estimate is limited both by the presence of additive noise and the finite spectral width of the atmospheric radar signal. An approximate expression for the variance of the estimate of the radial velocity, or the apparent velocity of sound using the pulse-pair algorithm, is given by Doviak and Zrnić (1993):

View Expanded

where σ_υ is the Doppler spectral width, M is the number of pulse pairs averaged, T_s is the time interval between samples, ρ is the correlation function of the noise-free signal, and SNR is the signal-to-noise ratio after coherent integration at the receiver. In the limit of high signal to noise and narrow spectral width compared to the Nyquist interval this expression becomes

View Expanded

With the TEP radar settings this gives var(υ̂) = 0.014σ_υ. To be valid, (15) assumes a large number of independent samples. When this not the case, computer simulations may be used to obtain the variance of the velocity estimation error. Figure 10 shows curves of the standard deviation of the estimation error versus signal-to-noise ratio for various values of the spectral width obtained using (15) and through numerical simulation. The simulated data were generated following Galati and Pavan (1995). Analytical and numerical results agree well except for the lowest spectral width, in which case the number of independent samples tends to be small, violating the condition for validity of (15).

This error analysis assumes that the radar signal is governed by Rayleigh statistics (i.e., that it can be modeled as the sum of many uncorrelated random scatterers). Due to the presence of a (coherent) specular component in the RASS echo, errors in the RASS sound speed estimate may be less than the errors in the clear-air Doppler estimate; thus, the total velocity error estimate may be somewhat pessimistic.

In order to control this error, especially since RASS echoes are elusive in nature, data points may be qualified such that only points exceeding a certain SNR are used. Since the temperature is obtained using two velocity measurements, both the RASS echo and the atmospheric radar echo should exceed a threshold. Figure 11 shows examples for different heights of joint histograms of the atmospheric echo SNR and RASS SNR. The figure also shows lines of constant uncertainty in the temperature measurement, obtained using (15) and (14). This error grows very rapidly for low SNR and plateaus for higher SNR at a level set by the spectral width of the echo. If a low SNR threshold is selected, the estimate of C²_Tυ is dominated by the temperature uncertainty due to noise. However, selecting a high threshold reduces the number of available data points, which increases the statistical uncertainty. Also, a high threshold may bias the estimate of C²_Tυ if its actual value and the signal-to-noise levels are correlated. The results presented were obtained by selecting an SNR threshold that minimized, a posteriori, the uncertainty of the estimate of C²_Tυ.

The left panel in Fig. 12 shows the structure function, D²_Tυ(r), calculated for two independent 20-min periods, one starting at 1230 LST and the other at 1615 LST. In both cases, the structure function appears to follow the r^2/3 behavior for small spacings expected for isotropic turbulence. For larger spacings, the points fall below the line. Rather than an indication of anisotropy, it is likely that the acoustic beam is not large enough to insonify fully the volumes centered at these locations (see Fig. 7). For those points, the measurement does not correspond to the center of the pixel, but to a position that will, in general, be closer to the center of the image.

The right panel in Fig. 12 shows profiles of C²_Tυ corresponding to the same periods. The first period is marked by a deepening convective boundary layer (CBL). The profile appears to show a characteristic z^−4/3 behavior, with a sharp increase near the BL top, as expected for a convective boundary layer (Wyngaard and LeMone 1980; Fairall 1987). The second profile, corresponding to the late afternoon, shows consistently lower values of C²_Tυ.

The left panel in Fig. 13 shows the first profile in Fig. 12 normalized using an estimated BL depth of 500 m and a temperature scale, , of 0.12 K, with the velocity scale w∗ defined as . The measured surface heat flux, , was 0.18 K m s⁻¹, calculated using sonic anemometer data obtained at a height of 6 m. For reference, the right panel shows a profile of C²_n for the same period, obtained from the radar reflectivity. The C²_n profile shows little evidence of a z^−4/3 profile, which may be a consequence of the dominant influence of humidity fluctuations.

Although the shape of the normalized profile of C²_Tυ appears in agreement with theoretical expectations and other experimental data, the values below the top of the BL are about an order of magnitude larger than expected for a convective boundary layer in the steady state based on the available surface measurements (Wyngaard and LeMone 1980). It is not yet clear why this is so. The discrepancy may be due to entrainment effects in this deepening boundary layer or to errors in estimating the surface flux due to the effects of heterogeneous complex terrain. The conditional nature of the estimates of C²_Tυ based on the signal-to-noise ratio may limit observability to the larger fluctuations.

6. Summary

This paper has presented an implementation of a RASS using a UHF volume-imaging radar and analysis of temperature measurement uncertainties due to a finite signal-to-noise ratio and to the finite spectral width of the radar signal. The temporal dynamics of profiles of the RASS-derived virtual potential temperature are found to be consistent with radar echo characteristics below, through, and above the capping inversion.

The spatial structure of the RASS echo has been studied considering both the diffraction pattern on the receiving array and the beamformed intensity patterns observed during a period of low horizontal winds. In the former case, evidence of a focused spot of a size consistent with RASS theory is found. Beamformed images of the RASS reflectivity show a complex structure that varies considerably in space and time. Such variability hinders retrieval of “filled” virtual temperature fields at short time scales.

For spatial separations within the insonified volume, structure functions of virtual temperature show an r^2/3 dependence typical of isotropic turbulence, while vertical profiles of the virtual temperature structure parameter, C²_Tυ, show the expected z^−4/3 behavior in the mixed layer. Thus, although continuous fields of turbulent temperature appear to be somewhat elusive, the statistics of near-instantaneous temperatures are indicative of turbulent fluctuations.

In the future, the current system will be improved by adding extra, spatially separated, acoustic sources (as in most operational RASS systems). This should certainly improve the spatial coverage of the virtual temperature field. However, this may complicate the interpretation of the spatial structure of the resulting RASS-induced radar echo. Careful selection and synchronization of the transmitted acoustic wave forms may avoid this problem and result in overall richer datasets.