1. Introduction

The lidar return backscattered from clouds is a powerful tool in the remote sensing of clouds (e.g., Platt 1979; Sassen et al. 1989; Winker et al. 1996). Quantitative interpretation of the backscattered lidar signal in terms of useful properties of clouds, such as their optical extinction, cloud particle phase, and size, is difficult because of the variable extinction-to-backscatter ratio for cloud particles (the so-called lidar ratio), severe attenuation of the backscattered signal by the cloud, and the effects of multiple scattering. Before attempting such interpretations it is vital to ensure that the observed backscattered signal itself is well calibrated. A rough calibration can be accomplished from an assessment of the characteristics of the lidar such as the transmitted power, the optics, and the receiver sensitivity, but absolute calibration is usually achieved using the known Rayleigh molecular backscatter signal from the atmosphere (e.g., Fernald et al. 1972; Platt 1973). To detect molecular backscatter, wavelengths of about 700 nm or less are used and, to avoid any contamination from aerosol returns, the calibration is carried out with the molecular signal from the stratosphere and upper troposphere.

In this paper we propose a method of automatic self-calibration that relies on measuring the total path-integrated backscatter in thick stratocumulus cloud that completely extinguishes the lidar signal. The integrated backscatter is equal to the reciprocal of twice the lidar ratio (Platt 1979), and for the spherical water droplets in stratocumulus, at a lidar wavelength of 905 nm, the lidar ratio is known and has a constant value (Pinnick et al. 1983; Derr 1980). Accordingly, the new calibration technique is to scale the received backscatter signal until the integrated backscatter agrees with the theoretical lidar ratio. This method is applicable for lidars that are set up to measure clouds, and the advantage of this system is that accurate calibration is achieved every time there is thick stratocumulus cloud: a relatively common occurrence. In addition, the technique works for lidars with wavelengths up to 1.1 μm and for lidar ceilometers with shorter wavelengths that are not sensitive enough to detect the molecular return. However, it may not be appropriate for lidars that are optimized for aerosol backscattering because the strong lidar return from low liquid water clouds may saturate the detectors. Multiple scattering introduces some uncertainty, unless the lidar has a very narrow beam, but can be accounted for by using suitable models (e.g., Eloranta 1998). The technique has certain parallels with the self-calibration of precipitation radars with polarization diversity (Goddard et al. 1994), whereby every time there is reasonably heavy rain the redundancy of the two parameters, differential reflectivity (Z_DR) and specific differential phase shift (K_DP) can be used to calibrate the reflectivity (Z) to within 0.5 dB (10%).

In section 2, the theoretical basis for the technique and the treatment of multiple scattering is discussed. Application to real data is presented in section 3, and validation of the technique is shown in section 4. Observations of supercooled clouds are presented in section 5, and examples of the variability of the mean lidar ratios for ice clouds are displayed in section 6.

2. Outline of the method

If we now consider the total integrated backscatter, B, that is the value of observed backscatter β_OBS integrated along the path until the signal is totally attenuated at z = ∞, we have

View Expanded

where we have assumed that S and η do not vary along the path. This is essentially the inverse of the technique described by Spinhirne et al. (1989), who used a calibrated lidar to estimate the value of η in liquid water clouds, and by Platt (1979) to infer the mean S values for cirrus.

The principle of the calibration technique is to measure B, the path-integrated observed backscatter for a totally attenuating cloud that has a known value of S and η, and then to scale the calibration factor for β_OBS until B = 1/2ηS. Ice has a variable value of S and therefore is not suitable for this technique, but, as shown below, the value of S in stratocumulus is constant, so lidar returns from optically thick stratocumulus can be used to calibrate the lidar backscatter.

The assumption of a constant value of η with height may not be strictly valid but Kunkel and Weinmann (1976) and Platt (1981) both indicate that η tends toward a constant value for optically thick water clouds. We will show that η can be estimated reliably for thick stratocumulus and that the expected variation in η is small.

The method is illustrated in Fig. 1, where the theoretical profile for β_OBS(z) is plotted for two clouds, both with a lidar ratio of 18.8 sr, but one with a liquid water content increasingly linearly with height at a rate of a = 1 g m⁻³ km⁻¹ and a constant drop concentration, N, of 2000 cm⁻³ (assuming the droplets are monodispersed with their size increasing with height), and a second where a = 0.25 g m⁻³ km⁻¹ and N = 50 cm⁻³. The area to the left of the two curves is the same; the first cloud is denser with a higher peak value of β_OBS but is then rapidly attenuated, whereas the second has a lower peak value of β_OBS but is then less attenuated.

The value of the lidar ratio in stratocumulus clouds can be derived theoretically because the cloud droplets are spherical and have a well-defined refractive index. This is not the case for ice particles, which can have variable shapes and air/ice densities. The values of σ and β for single spherical droplets, expressed as an efficiency relative to their geometric cross-sectional area, as a function of their diameter, D, are shown in Fig. 2 for the size range 0.1 μm to 1 mm. The curves have been calculated using Mie theory (Mie 1908) for a wavelength of 905 nm at a temperature of 25°C, where the refractive index of water is 1.327 + i0.672 × 10⁻⁶ (Hale and Querry 1973). It is assumed that the slight temperature dependence of the refractive index of water at visible and near-IR wavelengths (Kou et al. 1993) can be neglected and that one value will suffice for the temperature range −40° to 25°C. The variation of the refractive index with wavelength can be found in various sources (e.g., Hale and Querry 1973; Kou et al. 1993; Querry et al. 1991).

The value of σ tends to twice the cross-sectional area for large droplets, but the backscatter, β, oscillates wildly when the diameter is comparable to the light wavelength; this can be visualized as a result of the interference between reflections from the front and back surface of the spherical droplets. The lidar ratio, S, which is then simply the ratio of σ to β, also oscillates. In real clouds there is always a distribution of sizes of cloud droplets, and the fluctuations in S will be smoothed out.

We assume that the cloud droplet size distribution can be described by a normalized gamma function of the form (e.g., Bringi and Chandrasekar 2001)

View Expanded

where N_W is the drop number concentration normalized so that the liquid water content is independent of μ, D₀ is the median equivolumetric diameter, and μ represents the shape of the distribution. For a value of μ = 0, (7) reduces to the familiar inverse-exponential distribution, but as μ increases the droplet size spectrum becomes more and more monodispersed.

The variation of S with D₀ at a lidar wavelength of 905 nm for gamma distributions of droplets is shown in Fig. 3. The values of μ were chosen to correspond to the range expected for cloud droplet spectra in thick stratocumulus, which, according to Miles et al. (2000), is 2–10. Over the droplet diameter range 10–50 μm the value of S is almost constant, with a mean of 18.8 sr and a range of ±1 sr, where the range is determined from the extreme values of S. A similar value was found by Pinnick et al. (1983), who calculated S to be 18.2 sr at 1.06 μm. Typical values of D₀ lie between 8 and 20 μm for stratocumulus with significant liquid water content (Miles et al. 2000); lower median diameters were found for clouds that were either too thin or had too low a liquid water content to fully attenuate the lidar and therefore are not suitable for calibration. For this size range, and for a range of μ between 2 and 10, the value of S is 18.8 ± 0.8 sr where the range, again, is calculated from the extreme values, and, since the range of median diameters for droplet size distributions obtained from observations coincide well with the theoretical region in which the lidar ratio is constant, calibration of the lidar is therefore theoretically achievable with an accuracy better than 5%.

Figure 3 also shows that once the diameter exceeds 50 μm the value of S starts to fall. Fox and Illingworth (1997) showed that drizzle is ubiquitous in marine stratocumulus deeper than 200 m. These drizzle droplets dominated the radar reflectivity (proportional to the sixth moment of the drop spectrum) but usually made a negligible contribution to the liquid water content (third moment) and so should also have a negligible effect on σ and β (second moment) within cloud. However, occasions on which there is strong drizzle falling below cloud with significant backscatter should not be used for calibration purposes because the value of S will be below 18.8 sr. The backscatter from the drizzle drops contributes significantly to the integrated backscatter, B, but the corresponding attenuation is much lower (reduced S) and renders the assumption in (6) invalid.

The calculations in Fig. 3 are for a wavelength of 905 nm and scale approximately with lidar wavelength. The calibration technique will not work for a 10-μm lidar because the range of D₀ with constant S does not correspond to the size range found in water clouds. For shorter wavelengths, the ranges of D₀ with constant S are 5–25 and 3–17 μm at 532 and 355 nm, respectively, so should also encompass the range of median diameters expected in stratocumulus clouds. The values of S at 532 and 355 nm are 18.6 ± 1 and 18.9 ± 0.4 sr, respectively, for the same range of parameters (D₀ between 8 and 20 μm, μ between 2 and 10).

We now assess the contribution to the lidar backscatter return due to multiple scattering and the impact that this has on the calibration technique. Multiple scattering occurs when the photons scattered by a cloud particle undergo subsequent scattering events and are collected by the receiver (e.g., Eloranta and Shipley 1982). Multiple scattering is reduced by minimizing the receiver field of view (FOV). For a given divergence and FOV the amount of multiple scattering increases with range as the beam diameter increases. As particle size increases the forward-scattering peak becomes more pronounced and scattered photons are more likely to stay within the lidar beam. Increasing the particle concentration shortens the mean photon path between collisions and also leads to increased multiple scattering. All these effects are included in Eloranta's (1998) multiple-scattering model.

Eloranta's model was used to calculate the increased value of B, the integrated backscatter, for typical water clouds as a function of their height for a specific lidar setup. The required lidar parameters are the (half angle) beam divergence and receiver FOV, which, for the Vaisala CT75K lidar ceilometer used in this study, are 0.75 and 0.66 mrad, respectively. The required cloud parameters, extinction coefficient and effective diameter, are chosen to cover the maximum spread of the range of typical stratocumulus given by Miles et al. (2000).

The technique requires that the lidar signal be fully attenuated, and extinction coefficients of 15 and 20 km⁻¹ were chosen to describe the range that also fitted this criterion. More important for multiple scattering is the range of droplet sizes that might be encountered in both continental and marine air masses. The mean droplet effective diameters selected to cover the extremes of the observed range were 8 and 20 μm. The smallest mean droplet effective diameter noted by Miles et al. for clouds that were thick enough to fully attenuate the lidar was about 8 μm, similar to the small mean droplet effective diameter measured in some supercooled cloud layers (Hogan et al. 2003a). Larger mean droplet effective diameters were also noted, typically in marine clouds, with values that were closer to 20 μm. Clouds with mean droplet effective diameters larger than this are also likely to contain significant amounts of drizzle, the presence of which can be detected below cloud base.

The error in the calibration is, therefore, the uncertainty in the range of η that might be encountered. For the Vaisala CT75K lidar ceilometer η is 0.83 ± 0.09 at 1 km and 0.73 ± 0.06 at 4 km. The observed values of the apparent lidar ratio, ηS, should therefore lie within 14.5–17 sr at 1 km and 13–14.5 sr at 4 km, and the uncertainty in calibration for this lidar is about 10%.

For a lidar with a much narrower divergence and receiver FOV, η approaches 1 and the uncertainty in η is reduced accordingly. Significant molecular backscatter is present for lidars with wavelengths < 600 nm, and the contribution to the integrated backscatter should be accounted for.

In this derivation we have implicitly assumed that the lidar system parameters are known accurately, that the beam profile is Gaussian, and that there is full overlap of the lidar beam and telescope FOV. It is presumed that the major source of uncertainty in estimating the multiple-scattering factor is the spread in observed cloud droplet size distributions. The multiple-scattering factors must be recalculated for each lidar system because the lidar wavelength and telescope FOV, particularly, can have a significant impact on their value. If the lidar system parameters are not known accurately then it will be necessary to include this extra uncertainty when utilizing Eloranta's (1998) model to determine the range of possible multiple-scattering factors.

3. Observations of integrated backscatter in water clouds

The calibration technique was applied to a zenith-pointing Vaisala CT75K ceilometer consisting of an InGaAs diode laser operating at 905 nm with a divergence of 0.66 mrad and a field of view of 0.75 mrad (both half angle). It is a fully automated system that produces averaged profiles every 30 s with a range resolution of 30 m and is located at Chilbolton in southern England (51.1445°N, 1.4370°W).

A typical time series of vertical profiles of lidar backscatter observed from the ground by the Vaisala CT75K lidar ceilometer is displayed in Fig. 4, together with the apparent lidar ratio values derived from the integrated lidar backscatter, B. Intermittent thick stratocumulus is present at an altitude of 1.5 km. At this range, the theoretical value of the multiple-scattering factor, η, is 0.79, which corresponds to ηS = 15 sr. From the figure it is clear that there are long periods when the lidar signal is completely extinguished and the lidar can be calibrated by computing the mean value of ηS during the periods 1810–1850 and 1930–2020 UTC and scaling it to match the mean theoretical value. The instantaneous values of ηS are in the range 14.5–16.5 sr during these two time periods, and the standard deviation of ηS is 1.01, which indicates that the calibration is potentially consistent to within ±7% if there is no variation in multiple scattering. There is still an uncertainty of about 10% in the absolute calibration because of the range of values that the multiple-scattering factor can have.

During most periods of stratocumulus the value of B is constant, but as indicated in Fig. 4 there are occasional excursions from this value. The most obvious occur when the cloud is not thick enough to totally attenuate the lidar signal (between 1900 and 1930 UTC and after 2200 UTC in Fig. 4), which leads to B being too low and the derived value of ηS being too high and rapidly varying. The value of ηS decreases when strong drizzle is present below cloud base (Fig. 4, between 2130 and 2145 UTC) and can be identified where β values are much higher (at least 10⁻⁵ sr⁻¹ m⁻¹) than the background aerosol backscatter below cloud base. The lidar drizzle signature is accompanied by large values of radar reflectivity, and in both cases coincident 94-GHz cloud radar data provide an excellent means of confirmation.

A convincing demonstration of the treatment of multiple scattering in the calibration procedure is provided in Fig. 5, in which the variation with height of ηS, derived from B, is given by theoretical lines for clouds with extinction coefficients of 15 and 20 km⁻¹ (for an optical depth of 1 at distances of 66 and 50 m, respectively) and monodispersed droplet diameters of 8 and 20 μm. These values have been chosen to represent the range of cloud parameters discussed by Miles et al. (2000). Superposed on Fig. 5 are the observations of the apparent lidar ratio over a 4-month period expressed as a normalized probability density at each 30-m gate, where the height refers to the cloud-base height. Only observations of thick highly attenuating cloud that satisfied the following conditions (similar to supercooled layer identification in Hogan et al. (2003b)] are included.

1. The peak value of β should be more than 10⁻⁴ sr⁻¹ m⁻¹ and be at least a factor of 20 times greater than the value of β 300 m above.

2. Drizzle and rain events are excluded.

3. Strong background aerosol events are excluded.

4. Validation

The technique was validated by comparison with the UV lidar at Chilbolton, which operates at a wavelength of 354.7 nm and has been calibrated using the method described by Fernald et al. (1972). This instrument experiences strong molecular scattering and has a lidar beam divergence and telescope FOV of 0.05 and 0.2 mrad, respectively (both half angle). Full overlap is achieved at about 1.5 km.

To validate the technique it is necessary to calculate the amount of multiple scattering expected for this instrument and to account for the atmospheric transmittance at the shorter wavelength. The two-way atmospheric transmittance has been calculated using mean values for the atmospheric number density taken from the U.S. Standard Atmosphere, 1976 (COESA 1976).

The UV lidar was calibrated using averaged profiles from a clear air period earlier in the day at around 1135 UTC, shown in Fig. 6, where the observed and theoretical attenuated molecular backscatter agree well above 1.5 km, and with earlier calibrations. In the boundary layer (below 1.5 km) some aerosol is present and, as expected, the observed profile deviates somewhat from the attenuated molecular backscatter profile.

Figure 6 displays a calibrated attenuated UV elastic backscatter profile and a calibrated attenuated CT75K ceilometer profile with a supercooled cloud at 2.7 km for comparison. The UV backscatter profile has been corrected for the transmittance at 354.7 nm, which is T = 0.72 at 2.7 km for a standard atmosphere (at 905 nm T ≈ 1). The multiple-scattering factors for liquid water clouds at 2.7 km were calculated to be η = 0.86 ± 0.06 for the UV lidar and η = 0.74 ± 0.4 for the CT75K ceilometer, where the range of uncertainty was estimated by using the cloud droplet sizes at the extremes of the range found by Miles et al. (2000) as inputs to Eloranta's (1998) multiple-scattering model. The integral of the UV elastic backscatter from the cloud layer, ηS, in the absence of any other attenuating medium, should fall in the range 15.1–17.4 sr. Likewise, the value of ηS for the 905-nm CT75K ceilometer should fall in the region 13.1–14.7 sr.

A value of ηS = 17.2 sr is obtained from the profile in Fig. 6 that lies within the theoretical envelope and toward the extreme, which corresponds to the theoretical multiple-scattering factor calculated for the smallest droplet size found by Miles et al. (2000). Therefore, the likely explanation is that the observed cloud consists of small cloud droplets, which is reasonable since the observed cloud is a supercooled layer, which often consist of small cloud droplets (Heymsfield et al. 1991; Hogan et al. 2003a). This assumption is corroborated by the value of ηS found for the CT75K ceilometer, 14.5 sr, which is also toward the extreme, which corresponds to the smallest droplet size. It also possible that there may be some attenuation due to boundary layer aerosol, shown to be present in Fig. 6 below about 1.5 km, which would increase the value of ηS slightly. However, consistency between the values of ηS found in the cloud layer at 2.7 km, and agreement between the two independent calibration methods, provides validation of the technique.

5. Supercooled and mixed-phase clouds

Once the calibration procedure has been carried out it is possible to estimate the value of the lidar ratio for other types of clouds that are thick enough to totally extinguish the lidar beam. A frequent occurrence over Chilbolton is layers of clouds occurring at temperatures below freezing that highly reflect the lidar signal but are not accompanied by any increase in radar reflectivity (Hogan et al. 2003b). This suggestion was supported by in situ aircraft measurements (Hogan et al. 2003a), which confirmed the presence of supercooled droplets. Because the ice particles are so much larger than the liquid droplets, they are associated with a much higher radar reflectivity than liquid water clouds, so the suggestion has been made that these layer clouds, which highly reflect the lidar light but not the radar beam, are predominantly supercooled liquid water droplets. A 2-yr study (Hogan et al. 2003b) showed that they can be found at temperatures as low as −40°C and are present 27% of the time that any cloud between −5° and −10°C is present, falling to 6% of the time that any cloud between −25° and −30°C is present.

Further evidence that these clouds are supercooled layers was also presented by Hogan et al. (2003a), who, using polarimetric radar observations, found high Z_DR signatures beneath such layers, indicating the growth of pristine crystals; however, there is always the possibility that such clouds are composed of enormous numbers of small ice crystals so that they have a high extinction and attenuation of the lidar signal and also a low radar reflectivity.

The effective droplet radii in these layers, measured by in situ aircraft data, ranged from 2 to 5 μm (Hogan et al. 2003a) and correspond to a D₀ range of 6–12 μm. For most of this size range, the lidar ratio, S, is still relatively constant but, as D₀ decreases to below 8 μm, the value of S will begin to rise at 905 nm for broader droplet size distributions (Fig. 3). At shorter wavelengths, however, S remains relatively constant for these size ranges (S is constant down to D₀ = 5 μm at 532 nm and D₀ = 3 μm at 355 nm).

An example of the lidar profile and the value of ηS inferred from the integrated backscatter through such a layer between 2130 and 2300 UTC is depicted in Fig. 7. Isotherms taken from the 0–6-h forecast output of the Met Office Unified Model indicate that the temperature of this layer is between −20° and −15°C, which is in good agreement with temperature profiles obtained from sondes released nearby at Larkhill at 1100 UTC 17 November 2000 (solid line) and 0700 UTC a day later. The value of ηS for most of this period and for the layer present between 1930 and 2045 UTC remains constant and comparable to that expected for warm stratocumulus. Departures from this value occur when the cloud is not thick enough to fully attenuate the lidar return (after 2230 UTC).

6. Ice clouds

We now consider the lidar return from thick ice clouds leading to extinction of the lidar signal after a distance of 1–2 km or so. Examples of the lidar profiles and inferred values of ηS are provided in Figs. 8 and 9. The accompanying 94-GHz cloud radar reflectivity plots confirm that the clouds were generally much deeper. In Fig. 8, values of ηS are seen to vary between 5 and 30 sr over a period of 40 min between 2010 and 2050 UTC. Before and after this time the lidar signal also indicates the presence of supercooled cloud layers. This is confirmed by the steady value of ηS of 15.5 sr, with lower values when ice is falling below. The mixed-phase nature of this layer is also apparent as the radar reflectivities steadily increase toward 2230 UTC as the value of ηS falls below 10 sr, indicating the growth of much larger ice crystals. Values of ηS are also seen to vary rapidly in Fig. 9, by as much as 10 sr and on time scales as short as 5 min, before 1800 UTC. After 2000 UTC the fluctuations in the value of ηS are much smaller. A thin, strongly backscattering layer is present between 4 and 5 km during much of this period, which effectively extinguishes the lidar fully. This layer coincides with the drop in radar reflectivity seen above this height and is indicative of a supercooled layer.

The multiple-scattering factor for ice crystals can vary considerably because it depends on both the ice crystal size distribution and the ice crystal morphology. Since it is difficult to determine η in ice, it is not possible to provide precise estimates of S.

One method of measuring the mean S of ice from nonextinguishing ice cloud using backscatter lidar is reported by Grund and Eloranta (1990), in which the molecular Rayleigh lidar backscattering on the far side of the cloud is used to estimate the attenuation due to the cloud and to provide an attenuation-corrected value of β. A mean value of the lidar ratio can then be derived. Grund and Eloranta (1990) reported values of S ranging from 15 to 50 sr for this method.

A method of directly measuring the value S in ice using the Raman inelastic backscatter is reported by Ansmann et al. (1992). Typical values of S at a wavelength of 332 nm were between 5 and 15 sr. This technique was capable of measuring the vertical profile of S, which was shown to be not necessarily constant, and a substantial variation of S of 2–20 sr can be observed within one vertical profile.

The full range of S for cirrus observed by Ansmann et al. (1992) of 2–30 sr compares favorably with the range of ηS values that our observations indicate, 5– 40 sr. Of course our values of ηS are integrated over the extinction path and are limited to thick severely attenuating clouds, where the Raman method might not work because of the loss of weak Raman signals.

7. Conclusions

We have described a technique for absolute calibration of cloud lidars at visible and near-IR wavelengths that are not able to detect molecular backscatter. This technique has the potential to calibrate lidars to better than 5% if the design of the lidar ensures that multiple scattering is small, and to within 10% for a lidar that experiences significant multiple scattering, if suitable conditions of stratocumulus are available.

This technique is appropriate for any lidar setup optimized for observing cloud backscatter where the multiple-scattering factor can be estimated accurately, provided the range in the estimated multiple-scattering factor, arising from the potential variability in the cloud droplet size distribution, is not too large. This should include lidars on board high-altitude aircraft with narrow FOVs and also lidars in space. However, it may be necessary to use a Monte Carlo approach to model the multiple-scattering effect reliably at longer ranges, where the multiple-scattering return may have been significantly delayed relative to the single-scattered return at the same height because it has traveled over a significantly longer distance. This technique may not be appropriate for lidars that are optimized for aerosol backscattering.

The effects of multiple scattering are minimized if the laser divergence and telescope FOV are small but can be accounted for if the lidar's optical attributes are known. Mie theory is used to calculate the single-scatter and extinction characteristics of stratocumulus and, using Eloranta's (1998) model for multiple scattering, the amount of enhanced backscatter can be predicted for the expected range in stratocumulus optical depths and cloud droplet sizes. The design of the Vaisala CT75K lidar ceilometer is such that significant multiple scattering does occur, and calibration has been demonstrated to within 10% after accounting for multiple scattering.

It has been shown that this technique can be used to confirm the presence of supercooled water layers as they have essentially the same characteristics as warm stratocumulus clouds. The mean ηS of ice clouds can also be inferred using this technique as described by Platt (1979), and it is shown that they have a large range of variability with ranges of 5–30 sr being measured in one ice cloud over a 40-min period.