# Use of the Doppler spectral with to improve the estimation of the convective boundary layer height from UHF wind Profiler observations

Bok-Haeng Heo, Sandra Jacoby-Koaly, Kyung-Eak Kim, Bernard Campistron, Bruno Bénech, Eun-Sil Jung

► To cite this version:

Bok-Haeng Heo, Sandra Jacoby-Koaly, Kyung-Eak Kim, Bernard Campistron, Bruno Bénech, et al. Use of the Doppler spectral with to improve the estimation of the convective boundary layer height from UHF wind Profiler observations. Journal of Atmospheric and Oceanic Technology, 2003, 20, pp.408-424. 10.1175/1520-0426(2003)0202.0.CO;2. hal-00136466

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NOTES AND CORRESPONDENCE

Use of the Doppler Spectral Width to Improve the Estimation of the Convective
Boundary Layer Height from UHF Wind Profiler Observations

BOK-HAENG HEO

Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu, South Korea

SANDRA JACOBY-KOALY

Laboratoire d'Aérologie (CNRS), Observatoire Midi-Pyrénées, Université Paul Sabatier, Toulouse, France

KYUNG-EAK KIM

Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu, South Korea

BERNARD CAMPISTRON AND BRUNO BENECH

Laboratoire d'Aérologie (CNRS), Observatoire Midi-Pyrénées, Université Paul Sabatier, Toulouse, France

EUN-SIL JUNG

Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu, South Korea

25 September 2000 and 2 August 2002

ABSTRACT

Enhancement of the air refractive index structure parameter $C_n^2$ often occurs at the top of the convective boundary layer (CBL), where the absolute values of the vertical gradients of virtual potential temperature and mixing ratio have a peak. This well-known behavior of the $C_n^2$ profiles is often used to locate the height of the mixed layer $Z_i$ from UHF wind profiler observations. In the present study, $Z_i$ determination with the $C_n^2$-based technique was investigated for a case of clear-air CBL and a case of cloud-topped CBL. In certain circumstances, such as multifold $C_n^2$ peaks or poorly defined peaks, these techniques fail to correctly retrieve CBL height. In order to improve $Z_i$ determination, a new method based on the conjoint use of $C_n^2$ and Doppler spectral width profiles is proposed and discussed.

# 1. Introduction

The height of the atmospheric boundary layer $Z_i$ governs the vertical mixing of atmospheric pollutant and has been used as an important parameter in air pollution monitoring and boundary layer studies. It is also an important scaling length for the normalization of boundary layer parameters such as fluxes and variance including vertical gradients of wind, potential temperature, and moisture. This characteristic level is required as a basic input parameter in numerical weather and

climate forecasting models of the boundary layers (Stull 1988; Beyrich and Weill 1993).

The daytime continental convective boundary layer (CBL) considered in the present study is generally capped by a temperature inversion layer that arises in the morning in response to solar heating and the subsequent vertical mixing. It defines the interface between a well-mixed layer and the free atmosphere and prevents the vertical exchange of pollutants toward the free atmosphere. This interfacial layer called the entrainment zone is characterized by a significant vertical gradient of virtual potential temperature and a minimum heat flux. According to Stull (1988), the CBL height is the top of mixed layer, often defined as the average base of the overlying stable layer, and responds to surface forcings with a timescale of about an hour or less. Different definitions of CBL height have been proposed based on

<footnote>Corresponding author address: Dr. Kyung-Eak Kim, Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu, 702-701, South Korea.
E-mail: kimke@knu.ac.kr</footnote>
<figure><img src="image_2.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_3.png" /></figure>

FIG. 1. Vertical profiles of the virtual potential temperature ($\theta_v$), mixing ratio ($q$), virtual potential temperature gradient ($d\theta_v/dz$), and mean vertical gradient of potential refractivity ($M^2$) from rawinsonde observations at (a) 0803 and (b) 1059 LST, and $C_n^2$ profiles obtained from the UHF profiler observation on 19 Jun 1998 at Viabon. Parameter $Z_i$ denotes the CBL height determined by the gradient method of Sullivan et al. (1998) from rawinsonde observations. Parameters $Z_{peak1}$ (or $Z_{peak}$) and $Z_{peak2}$ represent the primary and secondary peaks in $C_n^2$ profiles of the UHF profiler, respectively.

various physical quantities. Sullivan et al. (1998) defines $Z_i$ as the height where the vertical gradient of potential temperature has its maximum value. This level corresponds to the middle of the inversion layer. It has also been defined as the height where the buoyancy flux reaches its most negative value (Deardorff et al. 1980; Wyngaard and LeMone 1980). In that case, the CBL

height is defined as the base of the temperature inversion. During daytime, low and shallow stratus clouds can exist in or at the CBL top and often occupy a significant portion of the CBL. However, there is no consensus as to how to define CBL height in the presence of clouds. In that case, two characteristic levels are identified. The first one is the cloud base corresponding to
<figure><img src="image_4.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_5.png" /></figure>

FIG. 2. As in Fig. 1, except for (a) 0802, (b) 1119, and (c) 1403 LST on 29 Jun 1998. Vertical solid bars with the B and S characters in the leftmost panel represent a cloud layer and broken and scattered clouds, respectively.

the height of dry mixing layer and the second one is the cloud top generally linked with the inversion layer. The difficulty in determining cloudy boundary layer height is due to the fact that a new turbulent source appears in the cloud layer linked with radiative and condensation/evaporation processes. This turbulent source can be strong at the cloud top and on the cloud edge when the cloud layer is fragmented. We will consider the case in which the cloud top corresponds to the $Z_i$ level.

The entrainment in the interfacial layer can create a strong vertical gradient in the temperature and humidity profiles, which causes a maximum in the profile of the refractive index parameter $C_n^2$ resulting in an enhancement of sodar and ultrahigh frequency (UHF) profiler returns. A maximum value in backscattered intensity profiles can be also found in clouds because of the enhancement of reflectivity by strong turbulent mixing within the cloud and entrainment mixing near cloud boundaries (Angevine et al. 1994). Furthermore, the ra-
<figure><img src="image_6.png" /><figcaption>Fig. 2. (Continued)</figcaption></figure>

dar backscattered intensity profiles can also present a maximum at the top of a residual layer, linked to the CBL developed on the previous day (Dye et al. 1995). Numerous studies based on numerical modeling and/or experimental data have shown that the vertical profiles of the refractive index structure parameter $C_n^2$ have a pronounced maximum near the base of the capping inversion layer or where the humidity gradient is large (Burk 1980; Wyngaard and LeMone 1980; Fairall 1991; Angevine et al. 1994; Muschinski et al. 1999). Benech et al. (1997) also showed with UHF observations that

the $C_n^2$ peak occurred between the boundaries of temperature inversion layers and that the dissipation rate of turbulent kinetic energy significantly decreased near the base of the inversion. Grimsdell and Angevine (1998) showed that when a cloud layer of significant depth is present at the CBL top, the $C_n^2$ peak is poorly defined or displaced to the height of the cloud top or cloud edges.

Recently, different $C_n^2$-based techniques used to determine CBL height from the UHF profiler have been investigated. White (1993) estimated CBL height using

<figure><img src="image_7.png" /><figcaption>FIG. 3. Vertical profiles of the UHF backscatter intensity and CBL heights (asterisks) estimated by the maximum backscatter intensity method at (a) 0800 and (b) 1100 LST on 19 Jun, and (c) 0801, (d) 1121, and (e) 1401 LST on 29 Jun 1998. Parameters $Z_i$ denotes the CBL height determined by the gradient method of Sullivan et al. (1998) from rawinsonde observations.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>FIG. 4. As in Fig. 3, except for the median filtering method of Angevine et al. (1994). Solid lines represent the $C_n^2$ profiles of the UHF profiler over the given periods.</figcaption></figure>

range-corrected signal-to-noise ratio (SNR) data observed in cloud-free boundary layers. Angevine et al. (1994) determined a representative CBL height from the median value of the heights at which SNR peaks occurred during a certain period. However, these methods based on maximum backscattered intensity are only applicable for estimating CBL height in cloud-free boundary layers. Dye et al. (1995) used a median profile algorithm to discard the $C_n^2$ maximum occurring inside scattered fair weather cloud, and pointed out that the estimation of CBL height using their algorithm was nearly inefficient in the case where persistent convective clouds existed above the profiler.

Actually, although UHF radar is a more recent tool used in the boundary layer studies and monitoring, remote sensing of the boundary layer and in particular the determination of $Z_i$ have a long history. The work reported by Kaimal et al. (1982) is a representative example showing CBL depth estimation with a K-band, X-band, and S-band radar, a lidar, and a sodar during the PHOENIX experiment. The remotely determined $Z_i$ level, based on reflectivity profiles, agreed with the CBL depth deduced from different in situ sensors within about ±10%. In an earlier study, Campistron (1975) showed the capability of a millimetric radar for the detection of the CBL top and other stable layers located in the free atmosphere.

In the present paper, the behavior of $C_n^2$ peaks, which is often used to estimate CBL height, is examined, and a new method is suggested that can be used to improve the CBL height estimate from UHF wind profiler observations by making joint use of $C_n^2$ peaks and vertical air velocity variance $\sigma_w^2$. The capability of the present method to estimate the CBL height is analyzed comparing the results obtained by other methods for each of the two cases, a clear day and a cloudy day.

## 2. Overview of the TRAC experiment: Data collection and processing

### a. TRAC experiment

The data used in the present study were collected during the Turbulence, Radar, Aircraft, Cells (TRAC) experiment, the description of which is detailed by Campistron et al. (1999). The experiment was designed to investigate coherent structures in the boundary layer, and to quantify turbulent and coherent transport, surface heat flux and entrainment, and interactions between the boundary layer and clouds. An important goal of the experiment was to provide a relevant database for numerical simulation of the boundary layer. It was carried out from 15 June to 5 July 1998 over a rural flat area of about 60 km × 60 km located in the middle part of France, several hundred kilometres away from the influence of high mountain ranges and maritime zones. Important instrumental facilities were deployed to reach the scientific objectives. Among them, a C-band Doppler radar, two UHF wind profilers, two powerful sodars, two instrumented aircrafts, and five surface meteorological stations equipped to measure turbulent heat fluxes and other more conventional atmospheric parameters including radiative fluxes were used. Rawinsondes were launched five times a day (0200, 0500, 0800, 1100, and 1400 LST) from the main site of the experiment (48.21°N, 1.68°E) with a typical vertical resolution of about 30 m.

### b. UHF profiler

The UHF wind profiler used is the DEGREWIND PCL1300 profiler, designed and manufactured by the Etablissement Degreane. Located at the rawinsonding site, it was operated continuously during the entire cam-
<figure><img src="image_9.png" /><figcaption>FIG. 5. As in Fig. 3, except for the median filtering method of Dye et al. (1995). Circles indicate the $C_n^2$ intensity of the UHF profiler at each gate over an hour. Solid lines indicate profiles of the $C_n^2$ median value taken at each gate.</figcaption></figure>

paign in order to observe mean and turbulent clear air conditions in the lower atmospheric layer. The main characteristics of this Doppler radar were a 1238-MHz transmitted frequency, with a 4-kW peak pulse power, a 25-kHz pulse repetition frequency, and a 150-m pulse length. In order to get the three components of the wind, the profiler alternatively used five beam positions—one vertical and four oblique—with a one-way half-power aperture of 8.5°. The oblique beams, with an off-zenith angle of 17°, were disposed every 90° in azimuth. Signals of time series were acquired with range gates evenly spaced along the radial at 75-m intervals, starting at 65 m AGL. A running coherent integration over 125 points on the times series was performed. In addition, 20 suc-

cessive Doppler spectra, which were obtained from a 128-point discrete Fourier transform applied on the decimated series (1 point out of 125 points was used), weighted with an Hanning window, were incoherently averaged. As a result, the dwell time on each beam was 13 s, the spectral resolution 0.1 m s⁻¹, and the Nyquist velocity ±6.1 m s⁻¹.

The spectra, usually contaminated by noise and non-meteorological echoes, were carefully edited in order to extract the first three moments of the atmospheric backscattered peak. In the first step, the mean noise level was determined using the objective technique proposed by Hildebrand and Sekhon (1974). The zero velocity and the two adjacent spectral lines, usually contami-

<figure><img src="image_10.png" /><figcaption>FIG. 6. As in Fig. 3, except for the new method using $C_n^2$ (solid profile) and $\sigma_w$ (dotted profile) data. The straight solid lines and $Z_0$ represent lines of linear regression of the heat flux from the vertical air velocity variance against the height and zero heat flux height estimated by extrapolation of the linear line, respectively. The values for $\sigma_w$ indicate the uncertainties on the $Z_0$ intercepts.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>FIG. 7. Time-height cross section of the refractive index structure parameter $C_n^2$ and CBL heights (asterisks) estimated by (a) the maximum backscatter intensity method (b) the median filtering methods of Angevine et al. (1994) and (c) Dye et al. (1995), and (d) the new method using $C_n^2$ and $\sigma_w$ for 19 Jun 1998 at Viabon. Triangles indicate the launch of time soundings during the day.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 7. (Continued)</figcaption></figure>

nated by ground clutter, were discarded and replaced by interpolated values. In addition, the statistical averaging technique developed by Merritt (1995) was applied to remove strong coherent-like echoes (bird, aircraft, etc.).

Usually, distinct Doppler peaks emerge above the noise level. Thus, in the second step of data processing, the selection of the meteorological peak is done through a consensus based on vertical, temporal, and spectral con-
<figure><img src="image_13.png" /><figcaption>FIG. 8. Time-height cross section of $\sigma^2/z$ with the CBL heights (asterisks) estimated by the new method using $C_n^2$ and $\sigma_w$ for 19 Jun 1998 at Viabon.</figcaption></figure>

tinuity as well as thresholding. In fact, the consensus gives a Gaussian weight to each spectral line remaining after thresholding, the value of which depends on the continuity tests. The first three moments are computed with the weighted contiguous spectral lines selected during the last 30 min. Tests have shown that this consensus technique, when removing or smoothing out spurious echoes, performs a temporal average over the considered period of the three moments of the meteorological peak. The use of a consensus period of 5 min or less results in more fluctuating values and sometimes fails to remove unwanted spectral lines. However, when averaging in time, these high-temporal definition moments, the values obtained with the 30-min consensus are recovered. This in particular shows that the spectral width discussed in this paper is not subject to a perceptible broadening factor due to the use of a long-duration-based consensus. Since all the raw Doppler spectra are recorded, it is then possible during offline processing to optimize the parameterization of the consensus to obtain the best results. For the data presented here, we used only a five-beam cycle acquired every 5 min. Additionally, we used only the reflectivity provided by the vertical beam. Tests have shown that reflectivity values acquired on the vertical or oblique beam are in fairly good agreement (with a difference less than 2 dB).

Quality control of the UHF profiler data was made during a 1-yr validation campaign with the use of rainsondings, sodar, and sonic anemometers (Dessens et al. 1997). On the other hand, the ability of UHF radar to detect rain of even weak intensity was used, in com-

parison with disdrometer measurements at the ground, to calibrate the instrument in reflectivity and to assess vertical velocity and spectral width retrieval (Campiton et al. 1997).

### c. $C_n^2$ relation

In a turbulent field with a locally homogeneous and isotropic inertial subrange, radar reflectivity due to backscattering from refractive index inhomogeneities is directly proportional to the refractivity structure constant $C_n^2$ (Tatarskii 1961; Ottersten 1969b). According to Tatarskii (1971), $C_n^2$ is related to the turbulent field characteristics and to the vertical variation of thermodynamic quantities of the mean field through the following relationship given by Doviak and Zrnic (1993, p. 466):

$$C_n^2 = a^2 \varepsilon^{-1/3} K_{\phi} 10^{-12} M^2, \quad (1)$$

where $a^2$ is a dimensionless constant ranging from 3.2 to 4.0, $\varepsilon$ is the dissipation rate of turbulent kinetic energy, $K_{\phi}$ is the eddy diffusion coefficient of the potential refractivity index, and $M$ is the mean vertical gradient of the generalized potential refractive index. All the quantities in (1) are in Systeme International (SI) units.

Following Ottersten (1969a), $M$ can be formulated as follows:
$$M = -77.6 \times 10^{-6} \frac{P}{T} \left( \frac{d \ln \theta}{dz} + \frac{15 600 q}{T} \frac{d \ln \theta}{dz} - \frac{7800}{T} \frac{dq}{dz} \right), \quad (2)$$

where P, T, q, θ, and z are the atmospheric pressure (hPa), temperature (K), mixing ratio (kg kg⁻¹), potential temperature (K), and height (m), respectively.

The CBL top is usually characterized by a strong increase in the potential temperature as well as a strong decrease in the mixing ratio. Thus, these two factors that add positively in (2) enhance the radar signal provided, according to (1), that sufficient turbulent mixing exists to create refractivity index irregularities at half the radar wavelength. This is the principal scale of the inertial subrange participating in the radar signal (Tatarskii 1961).

In a well-mixed CBL, the vertical profiles of potential temperature and mixing ratio are, on average, nearly constant with height (Stull 1988). Despite the strong turbulence, we might expect weak radar echoes from refractive index irregularities in the CBL. Grimsdell and Angevine (1998) have suggested that, in clear-air mixed layers, part of the radar return might be contaminated by hard targets such as insects.

### d. Doppler spectra width

In the present work, we make use of the variance of the Doppler spectra $\sigma_w^2$ acquired on the vertical beam. According to Sloss and Atlas (1968), $\sigma_w^2$ has two principal atmospheric contributors. They are the turbulent velocity fluctuations at scales smaller than the dimension of the radar resolution volume and a broadening factor due to the finite width of the beam. These two contributions are independent of each other and their variances are additive in $\sigma_w^2$. The value of the variance $\sigma_b^2$ induced by the beam-broadening effect is given by these authors by the following relationship:

$$\sigma_b^2 = \theta_b^2 (16 \ln 2)^{-1} V^2, \quad (3)$$

where V is the horizontal wind speed and $\theta_b$ is the one-way half-power angular aperture of the beam assumed to have a Gaussian shape. In the following, we use $\sigma_w^2$ values corrected for the beam-broadening effect in order to retain only the turbulent information.

### 3. Comparison between CBL height estimated from rawinsonde data and UHF $C_n^2$ profiles

In this section a comparison is made for two characteristic days of TRAC between the vertical profiles of $C_n^2$ provided by the UHF profiler and the vertical profiles obtained with the rawinsondings of thermodynamical parameters related to $C_n^2$ in Eq. (2).

#### a. Definition of the CBL top

As discussed previously in this paper, we consider the CBL top as the height corresponding to the thermal inversion layer (maximum thermal gradient). This definition is totally compatible with both dry and cloudy CBLs where the cloud layer is related to the surface layer, as generally observed in continental conditions. In that case, the CBL height $Z_i$ was determined from rawinsonde data using the gradient method of Sullivan et al. (1998).

#### b. Analysis of two case studies

For this study, we selected a clear-air CBL (19 June 1998) and a cloud-topped CBL (29 June 1998). They are representative of the main difficulties encountered in $Z_i$ determination using radar. For these two days, the vertical profiles of $\theta_v$, q, virtual potential temperature gradient ($d\theta_v/dz$), and $M^2$ deduced from rawinsonde observations in the early morning, near midday, and in the midafternoon (only for the cloudy day), are presented in Figs. 1 and 2, respectively. The vertical derivatives were computed using a centered difference at each radiosonde level. A running average over 150 m was applied on the $M^2$ profiles in order to reduce computing noise. Associated with these thermodynamical parameters, $C_n^2$ vertical profiles of the UHF profiler obtained shortly around the launch time of the soundings are presented. The rapid decrease of the $C_n^2$ value, which can be observed below 250 m, is an artifact produced by a range variable attenuation voluntarily introduced to avoid receiver damage from strong returns and transmitter leakage. This artifact does not permit the accurate observation of the CBL properties under a height of 250 m, which corresponds to the early morning state.

#### 1) CASE OF 19 JUNE 1998

On 19 June 1998, the observations were made under fair weather conditions, with a weak synoptic wind of less than 7 m s⁻¹ and a nearly cloud free CBL. Figure 1a displays the vertical profiles obtained at about 0800 LST. The profiles of $\theta_v$ and its vertical derivative $\partial\theta_v/\partial z$ show two distinct strong stable layers located between 0.3 and 0.7 km and between 1.1 and 1.6 km. These layers are also marked by strong vertical variations of q and correspond to two major peaks in the $M^2$ profile, a result expected from Eq. (2) considering the vertical evolution of $\theta_v$ and q.

The vertical profile of $C_n^2$ exhibits two distinct peaks with nearly the same magnitude, centered at 0.4 and 0.8 km, respectively. The level of the lower $C_n^2$ peak coincides with the height of the lower $M^2$ peak. The other $C_n^2$ peak is close to a minor peak of the $M^2$ profile and is in the upper part of a quasi-neutral layer extending from 0.7 to 1 km. This is a typical case in which a $C_n^2$-based technique might fail in selecting the peak cor-
<figure><img src="image_14.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>FIG. 9. As in Fig. 7, except for 29 Jun 1998.</figcaption></figure>

responding to the $Z_i$ level unless more information is taken into account.

Considering that these observations were made in the early morning and taking into account the temporal con-

tinuity of the UHF observations and rawinsonding measurements, we found that the most probable CBL top is associated with the lower stable layer. Based on the gradient method, the height of the $Z_i$ level was located
<figure><img src="image_16.png" /><figcaption>Fig. 9. (Continued)</figcaption></figure>

at 0.5 km, that is, 100 m above the corresponding $C_n^2$ peak. The previously mentioned neutral layer corresponds to the residual layer and the upper-level inversion extending up to 1.6 km might be attributed to the remaining CBL inversion of the previous day or to a

synoptic feature. This inversion is not associated with a pronounced wind shear (not shown), which is able to generate dynamic turbulence, and is outside the influence of the turbulent mixing of the CBL. This is an instructive example showing that despite strong tem-
<figure><img src="image_17.png" /><figcaption>FIG. 10. As in Fig. 8, except for 29 Jun 1998.</figcaption></figure>

perature and humidity gradients but in the absence of sufficient mixing, very slight enhancement in the radar signal occurs.

At about 1100 LST on the same day, the vertical profiles presented in Fig. 1b show that the estimations of the CBL height are simpler than in the early morning. We are left with a $C_n^2$ profile with a single prominent maximum located at a height of about 1.15 km. This level is about 50 m below the heights of $M^2$ and $\partial\theta_v/\partial z$ maxima, near the base of a temperature inversion extending from 1.1 to 1.3 km. Beneath this layer, the CBL appears well mixed with nearly constant $\theta_v$ and q profiles. The gradient method gives a $Z_l$ level at 1.2 km, in good concordance with the $C_n^2$ peak.

## 2) CASE OF 29 JUNE 1998

On 29 June 1998, a humid but weak (less than 6 m s⁻¹) westerly flow prevailed in the CBL. Figures 2a to 2c present vertical profiles obtained from radiosondes and UHF radar at about 0802, 1119, and 1403 LST, respectively. This fair weather day differs from the first case study because of the partly cloudy sky present from early morning to late afternoon with cloud fraction varying from 60% to 80%, according to visual reports made at the UHF site. Radiosonding offers only a snapshot in time and space of the atmosphere. Thus, in order to get a more complete description of the cloud layer, we used measurements collected by one of the two research aircraft that performed continuous flights over the study area between 0916 and 1207 LST.

This flight was divided into eight 45-km constant lev-

els and course legs between 100 and 1800 m above ground and centered on the vertical of the UHF. Four legs were performed in a north-south-oriented vertical plane and the other in an east-west-oriented vertical plane. Following Grimsdell and Angevine (1998), the cloud-base height was taken to be the lifting condensation level deduced from the aircraft data series of pressure, temperature, and mixing ratio sampled along the flight every 4 m. The results for the considered flight period show a mean cloud-base height of 800 m and an increase in the cloud base of about 150 m toward both the north-south and the west-east directions over the 45-km legs. In addition, on average, there was a temporal increase of about 120 m during the 3-h flight. At the beginning of the flight, during the ascent, the aircraft crossed the cumulus layer and visual report indicated a cloud-base and cloud-top height of 900 and 1400 m, respectively. The same observations made at the end of the flight situated the cloud-base and cloud-top level at about 950 and 1400 m, respectively.

The early morning vertical profiles at 0802 LST in Fig. 2a display similar characteristics to those of the previously discussed day (Fig. 1a). With the same reasoning, we identify a residual layer including clouds, between about 1.0 and 1.3 km, with a nearly constant virtual potential temperature with height, capped by a stable layer extending, from 1.3 to 1.45 km. In this inversion layer, the vertical increase of the potential temperature and the vertical decrease of the mixing ratio are much more acute than in the case of the cloud-free day. The most probable CBL top in Fig. 2a is found at about 0.9 km, at the base of a weakly stable layer where
some thin clouds could be identified. This level corre-
sponds to the height of one of the two main peaks, with
nearly the same intensity, of the radar–C²<sub>n</sub> profile. The
second C²<sub>n</sub> peak at a height of 1.3 km is linked to the
base of the strong inversion aloft but is displaced 150
m below the well-defined M² peak positioned in the
upper part of this stable layer. There is a third distinct
but less prominent peak C²<sub>n</sub> near a height of 1.9 km,
located at the base of an increase of the virtual potential
temperature and at the same level of the M² maximum,
which can be linked with the cloud layer disconnected
from the CBL.

Around midday (Fig. 2b) and midafternoon (Fig. 2c)
only one prominent C²ₙ peak existed at about 1.4 km,
coincident with a M² maximum at the base of a strong
inversion layer. Without ambiguity, this level can be
defined as the CBL top in view of the well-mixed po-
tential temperature and mixing ratio profiles beneath.
Secondary relatively weak peaks of C²ₙ are apparent at
a height of 2.3 km (Fig. 2b) and at 2.0 km (Fig. 2c),
and are not correlated with an evident singularity in the
radiosonde profiles. However, at a height of 1.4 km, the
C²ₙ peak in Fig. 2c was produced by mixing at the cloud
top.

3) CONCLUDING REMARKS

In summary, the analysis of these two case studies
shows ambiguity in the determination of the Zᵢ from
C²ₙ peaks, mainly in the early development of the CBL
when a residual layer from the previous day persists. In
the presence of clouds, the C²ₙ peak is always very close
to the CBL depth defined at the level of the capping
inversion layer. However, due to the cloud-base vari-
ability deduced from aircraft measurement, it is not pos-
sible to conclude that the cloud base was also positioned
at the CBL top. However, the cloud top seems to be
associated with a secondary radar reflectivity maximum.
Finally, distinct peaks of the mean vertical gradient of
the generalized potential refractive index are apparently
not always linked to the local maximum of observed
C²ₙ. As expected from Eq. (1), to understand radar return
intensity it is necessary to take into account the degree
of small-scale turbulent mixing.

4. Estimation of the CBL height with C²ₙ-based techniques

Two main algorithms to estimate the CBL height from
UHF profiler data have been suggested by several re-
searchers, as quoted in the introductory section. These
algorithms are based on the entrainment-induced max-
imum of backscatter intensity, such as maximum value
of SNR or C², and are compared and tested by esti-
mating the CBL height for two cases—a clear day and
a cloudy day.

a. Maximum backscattered intensity method

This method, proposed by White (1993), simply as-
signs the CBL height as the level of C²ₙ or SNR max-
imum in an instantaneous vertical profile. Figure 3
shows vertical profiles of UHF backscatter intensity and
the CBL heights estimated by the maximum backscat-
tered intensity method at 0800 and 1100 LST on 19
June 1998, and at 0801, 1121, and 1401 LST on 29
June 1998. Except for Fig. 3a, the CBL heights deduced
from the C²ₙ peaks agreed with the rawinsonde-estimated
CBL heights. In the case of Fig. 3a, the C²ₙ profile had
a double peak structure where the primary and second-
ary peaks occurred from the residual layer and the CBL
height, respectively.

The maximum backscattered intensity method takes
the primary C²ₙ peak as the CBL height. This method,
therefore, has a tendency to erroneously estimate a C²ₙ
peak due to the enhancement of reflectivity by a cloud
layer or a stable residual layer when the C²ₙ profile had
a double peak structure as in Fig. 3a. However, the
method can provide a good time resolution in the es-
timation of CBL height since it estimates the height from
the C²ₙ maximum of a single profile.

b. Median filtering method

This method is based on an increase in backscatter
intensity in the same way as the maximum backscattered
intensity method. However, a median filter is used to
remove C²ₙ peaks from the enhancement of reflectivity
by clouds, stable residual layer, precipitation, and bio-
logical targets such as insects and birds.

Angevine et al. (1994) suggested two algorithms to
find CBL height. In the first, a C²ₙ peak in each profile
is selected and then the median value of the heights at
which the peaks occur during the considered period is
determined to be the CBL height. The other is that after
taking the median value of the C²ₙ values at each range
gate during the considered period, the height that has
the peak value in the median C²ₙ profile is determined
as the CBL height. Dye et al. (1995) included threshold
logic to remove the enhancement of reflectivity by non-
meteorological targets such as migrating birds, aircraft,
and so on and computed the median C²ₙ value at each
range gate over a period. They then selected the height
of the maximum C²ₙ in the median profile as the CBL
height for that period. The threshold removes all values
of C²ₙ greater than 10⁻¹¹ m⁻²/³.

Vertical profiles of UHF backscattered intensity and
the CBL heights determined from the first median fil-
tering method of Angevine et al. (1994) are shown in
Fig. 4. The solid lines in each panel represent the C²
profiles during the observation. Except for Fig. 4a, the
CBL heights estimated using this method agreed with
the rawinsonde-estimated CBL heights. In the case of
Fig. 4a, the heights of the primary peaks from the re-
sidual layer were constant, while the heights of the sec-
ond peaks from the CBL height were raised with a lapse in time. This method, therefore, erroneously estimated the height at which the primary peaks consistently occurred as the CBL height.

Vertical profiles of UHF-backscattered intensity and the CBL heights determined from the median filtering method of Dye et al. (1995) for the same case as Figs. 3 and 4 are shown in Fig. 5. In the case of Fig. 5a, with both elevated residual layer and CBL height, this method determined the lower $C_n^2$ peak as the CBL height. However, in the case of Fig. 5c, where cloud layers were distinct from the CBL height, the upper $C_n^2$ peak due to cloud layer was selected as the CBL height during the considered period. According to these results, this method has a tendency to estimate heights at which the $C_n^2$ values and their fluctuations with time were as large as the CBL height. In the case of a rapidly growing CBL in the morning or a very fluctuating cloud top, this method may produce inaccurate estimates of CBL height.

The median filtering methods of Angevine et al. (1994) and Dye et al. (1995) performed best when scattered fair weather cumulus clouds were present or vertical profiles of backscatter intensity had a single peak structure, but performed worst when deep clouds were persistently present over the UHF radar or vertical profiles of backscattered intensity had a double peak structure. These algorithms find a representative CBL height from several profiles of backscattered intensity observed during an hour. Consequently, these methods provide CBL height estimations of lower time resolution than the maximum backscattered intensity method, which can estimate its height from a single profile (in about 10 min).

## 5. Estimation of CBL height using $C_n^2$ and $\sigma_w^2$ profiles

The fourth method introduced here is a proposed improvement to the existing methods by making use of the UHF Doppler spectral width. The maximum backscattered intensity method and median filtering method were found to be incomplete in their estimations of CBL height when the backscattered intensity profile had a double peak, and the primary peak in each profile occurred in cloud layers or at the top of a residual layer rather than at the top of the mixed layer. In this paper, an algorithm is proposed to distinguish the peak caused by CBL height from other peaks due to either a cloud layer or a residual layer. This is done by using the variance of vertical air velocity measured by a UHF profiler.

In a dry, convectively well-mixed layer, heat flux decreases linearly with height from the ground to the top of CBL, where it reaches a minimum. The heat flux profile is linked with the standard deviation of the vertical velocity by the relation (Weill et al. 1980)

$$ \frac{\sigma_w^3}{z} \approx \alpha^{3/2} \frac{g}{\theta} (w' \theta_v'), \quad (4) $$

where $\sigma_w$, $z$, and $\alpha$ represent the standard deviation of the turbulent vertical velocity, height, and a universal constant ($\alpha = 1.4$), respectively. The term $(g/\theta)(w'\theta_v')$ represents a local buoyancy production, where $g$, $\theta$, $\theta_v$, and $w'$ are the gravitational acceleration, potential temperature, virtual potential temperature, and vertical velocity perturbation, respectively. Implicitly, this relation implies that the profile of vertical velocity standard deviation is linked with the CBL sensible heat flux profile. In a well-mixed layer, the heat flux decreases linearly with height. Consequently, $\sigma_w^3/z$ in Eq. (4) decreases linearly from the top of the surface layer to the CBL top. To find CBL height, the present method employs a comparison of the $C_n^2$ peak heights with the height corresponding to the “zero flux level” (the level where $\sigma_n^3/z$ becomes null), which is deduced from the vertical profile of the variance of the vertical velocity (see Fig. 6). The nearest peak of $C_n^2$ peaks to the zero flux level is taken as the CBL height. This level implies that there is a continuity of turbulent motion between the ground and the selected level. In cloudy conditions, this condition is only correct within the mixed layer up to the cloud base.

The new method follows three steps to find the reference height using the $\sigma_w^3/z$ profile, and then to determine the CBL height from the $C_n^2$ peaks.

First, the primary and secondary peaks were selected in each $C_n^2$ profile, and the height of a peak close to the surface was chosen as our first guess of the CBL height $Z_1$. The selection of the lowest peak ensures that the data below it are inside the mixed layer.

Second, a linear regression line of Eq. (4) was obtained by least square fitting between the preceding estimation of $Z_1$ and $0.2Z_1$. The lower bound ($0.2Z_1$) secures the only data in the mixed layer for the least square fitting. Therefore, the values of $\sigma_w^3/z$ in the surface layer are not used in the least square fitting. This restriction is required because the relation (4) is correct only in the mixed layer and the UHF profiler data may not be satisfactory below 200 m.

Third, the zero heat flux height, $Z_0$ was estimated by an extrapolation of the regression line as shown in Fig. 6. Therefore, the height of the $C_n^2$ peak nearest to $Z_0$ was taken as the CBL height.

Figure 6 shows the vertical profiles of UHF-backscattered intensity and CBL height determined using $C_n^2$ and $\sigma_w$ data of UHF profilers for the same case as in Fig. 4. The height of the zero heat flux in Fig. 6 was found to be near the radiosonde-estimated CBL heights; the new method estimated the CBL heights well even though the profile had double peaks. The uncertainties in the $Z_0$ intercepts in Figs. 6a–e are 0.028, 0.026, 0.049, 0.027, and 0.067 km, respectively, and the distances between the two $C_n^2$ peaks in Figs. 6a and 6c are 0.420 km and 0.300 km, respectively. As in the cases of Figs.
6a and 6c, the errors of the zero intercepts are much smaller than the distances between the two peaks. Thus, the peak closest to the intercept is well identified as the CBL height.

**6. Estimation of diurnal variation of CBL height**

Four methods, as previously mentioned, were com-
pared and analyzed to examine their ability to estimate
the diurnal variation of CBL height during a clear day
and a cloudy day. In addition, the time resolution of the
analyzed CBL heights was compared using the diurnal
variations of the CBL heights obtained by the four dif-
ferent methods.

The CBL heights were estimated from the vertical profiles of UHF backscattered intensity on a clear day using four different methods. The diurnal variations of CBL height are shown in Fig. 7 for 19 June 1998. According to the strongest peak in the time–height cross section of $C_n^2$, around 0700 LST, the CBL formed and started growing. Its height grew rapidly from 0.3 km at 0700 LST to 1.1 km at 1100 LST and reached nearly 1.3 km around 1200 LST. It ranged between approximately 1.2 and 1.3 km during the afternoon. After 1700 LST, corresponding to sunset, the 1.3-km inversion level corresponded to the top of the residual layer. A stable layer was formed near the ground that grew throughout the night. Our comments on the mixed layer are correct only up to 1700 LST, where the surface heat fluxes are always in relation to the CBL top.

The diurnal variations of CBL heights derived from
the maximum backscattered intensity method shown in
Fig. 7a agree well with the rawinsonde-estimated CBL
heights and also provided reasonable structure and evo-
lution of the CBL during the growth of the CBL height.
During the late afternoon between 1600 and 1750 LST,
however, this method overestimated the CBL height
when partial clouds were persistently present at 2.1 km
and higher. Figures 7b and 7c show the diurnal varia-
tions of the CBL heights determined by the median
filtering method of Angevine et al. (1994) and Dye et
al. (1995), respectively. The results are similar to that
of the CBL heights estimated by the maximum back-
scattered intensity method. The diurnal variations of the
CBL heights estimated by the new method using $C_n^2$ and
$\sigma_w$ data are shown in Fig. 7d. The results of the CBL
height derived from the new method agree well with
not only the rawinsonde-estimated CBL heights but also
provide also a reasonable structure and evolution of the
CBL height without overestimation in the presence of
partly persistent clouds. This method also gives a good
time resolution for CBL height detection. Figure 8 clear-
ly shows that the parameter $\sigma_w^3/z$ is at its maximum
during daytime near ground level and decreases from
the ground to the top of the boundary layer in relation
to the turbulence inside the CBL. The maximum of
$\sigma_w^3/z$ is well linked with the maximum of radiative en-
ergy, which arrives at ground level. During the period

between 1800 and 1900 LST, the singularities of $\sigma_w^3/z$
was observed by the UHF profiler.

The diurnal variations of the CBL heights on a cloudy
day estimated by four different methods are shown in
Fig. 9. Considering both the variation of the strongest
peak in the time–height plots of $C_n^2$ and the rawinsonde-
estimated CBL heights, the CBL height grew rapidly
from 0.5 km at 0700 LST to 1.2 km at 0830 LST. Ad-
ditionally, the elevated CBL height coincided with the
top of the cloud layer. After 0830 LST, it was controlled
by the cloud layer and fluctuated between approximately
1.2 and 1.4 km. Sharp variations in both $\theta_v$ and $q$ at the
top of the CBL and significant turbulence within the
cloud layers resulted in the reflectivity enhancement be-
tween 1.2 and 1.4 km and a narrow width of the $C_n^2$
contour of more than –12.5 dB. The sharp peaks due
to reflectivity enhancement aided the estimation of the
CBL height in all four method, and allowed the attain-
ment of a reasonable structure and evolution of the CBL
except for the case in which the median filtering method
of Dye et al. (1995) overestimated the CBL height
around 0800 LST. In this case, the new method esti-
mated the CBL heights well and provided a reasonable
structure and evolution of the CBL without overesti-
mation in the presence of partly persistent clouds. Figure
10 shows, as in Fig. 8, that the parameter $\sigma_w^3/z$ represents
the evolution of the turbulence inside the CBL between
0700 and 1700 LST. During this day, the turbulence
evolution is very different from the previous case with
a clear sky and seems to be modulated by the cloud
cover. As in the previous case, this parameter clearly
shows that the turbulent structure is well linked with
the CBL top deduced from $C_n^2$.

**7. Summary and conclusions**

In the present study, the applicability of $C_n^2$ peaks in
the estimation of CBL height was investigated and a
new method to objectively estimate the CBL height us-
ing $C_n^2$ and $\sigma_w$ data of UHF profilers was developed.
This method was compared with other methods for two
cases, one on a clear day and the other on a cloudy day.

The peaks in each $C_n^2$ profile often occurred not only
at the top of the CBL but also at the residual layer.
When there was no significant solar heating and vertical
mixing in the morning, the height of the strongest ver-
tical gradient of $\theta_v$ was not consistent with that of a
strong vertical gradient of $q$. When the vertical gradients
of $\theta_v$ and $q$ in the entrainment zone were very strong
due to strong vertical mixing in the afternoon, the $C_n^2$
peak corresponding to the CBL height was in good
agreement with the rawinsonde-estimated CBL height.
If there were cloud layers or a residual layer over the
top of the CBL height, the vertical profile of backscat-
tered intensity had a double peak structure. In particular,
when the CBL height was consistent with the height of
the cloud layer, vertical profiles of both $d\theta_v/dz$ and
$C_n^2$ had a sharp peak near the top of the cloud layer
because of the strong vertical gradient of $\theta_v$ due to the entrainment mixing near cloud boundaries and at the entrainment zone.

The maximum backscattered intensity method and median filtering method correctly estimated the CBL height when the $C_n^2$ profile had a single peak, but these methods erroneously estimated the CBL height when there was a residual layer or a cloud layer over the CBL. The new method distinguished the peak due to the CBL height from the peak due to a cloud layer or a residual layer by using both $C_n^2$ and $\sigma_w$ data. It was then able to correctly estimate the CBL height. With regard to diurnal variations of the CBL height, the new method provided more stable and reliable estimations of CBL heights in real time than the other methods even when the vertical profiles of backscattered intensity had two peaks around CBL height due to a residual layer or cloud layer.

The new method developed in this study can estimate CBL height using $C_n^2$ and $\sigma_w$ data as well as provide information about the evolution of CBL height over the entire day. The analysis was based on two case studies and more observations are indeed needed to confirm the reliability of the proposed method of $Z_i$ determination.

*Acknowledgments.* The TRAC experiment was mainly funded by the Institut National des Sciences de l'Univers (INSU), Météo-France and Electricité De France (EDF). Two authors (Kim and Heo) gratefully appreciate the financial support of the Korea Science and Engineering Foundation Grant KOSEF 985-0400-004-2.

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