Introduction

This work reexamines the inference of millimeter-wave and optical attenuation in—and reflection from—fogs and/or hazes in the light of relatively new and interesting data (Zak 1994) on the distribution of drop sizes in four advection fogs in the first few hundred meters above the ground. The wavelengths of interest lie in the 35–95-GHz radar regime (2–8 mm) and in the infrared to visible regime (0.4–12 μm). Knowledge of reflectivity and attenuation is of special importance for evaluating the visibility in situations in which aircraft are involved. The theoretical background linking electromagnetic (EM) to meteorological parameters seems to be well understood in general (Doviak and Zrnic 1993), and there are abundant empirical relationships (Atlas 1964) that relate reflectivity, attenuation, and liquid water content to each other and to relevant parameters of the medium and measuring instruments. The ability to measure drop sizes reasonably accurately has also made it possible to discriminate between some of the various empirical relationships and to provide a theoretical basis. However, the wavelengths in question approach and become as small as (if not smaller than) the drop sizes, resulting in some of the customarily used relationships becoming invalid because they are based on the Rayleigh-regime approximations (drop size ≪ wavelength). The data analyzed in this work provide an opportunity to predict what radars and optical instruments might observe.

Several findings can be summarized as follows. First, we found that the distribution of drops with diameter D, namely, n₄(D, z) [in m⁻⁴], can be fitted with a gamma distribution α(z)D⁴ exp(−D/D_o) in which the parameter D_o appears to be a weak function of location and almost not at all of altitude z. Second, we have found, at least in the advection fogs, that the Rayleigh-regime reflectivity coefficient Z(z) is closer to being a linear function of the liquid water content M(z) rather than a quadratic one, as predicted by Atlas (1964) for clouds. Third, we have found that the ratio between the reflectivity and the attenuation coefficient to M(z) is a function of wavelength that varies weakly from location to location for the infrared-to-visible wavelength regimes.

Brief summary of the experiment

The drop size data upon which this work is based (Zak 1994) were obtained during flight tests involving descents along a 3° glide slope in which onboard instrumentation measured the number of drops in “bins” of 3-μm spreads in drop diameter up to a diameter of 47 μm using a Particle Systems, Inc., forward scattering spectrometer probe (FSSP-100) mounted on a wing (Dye and Baumgartner 1984; additional information available from Web site raf.atd.ucar.edu/∼darrell/htmFiles/fssp100.html and raf.atd.ucar.edu/∼darrell/htmFiles/260X.html). These descents were part of a Federal Aviation Administration program to test the capability of millimeter-wave radar and infrared cameras to provide a runway image in the cockpit of aircraft attempting landings in fog. Larger drops were noted in 20- and 300-μm bins by means of an optical array probe (OAP 200X or 200Y) mounted on the other wing (Baumgardner and Korolev 1997), but there is uncertainty as to what extent the noted effect upon drop size distribution of these larger particles will really be in fog. The flights occurred at four distinct advection fog sites: Vandenberg AFB; California (VAN); Worcester, Massachusetts (WOR); Arcata, California (ARC); and Santa Maria, California (SM). The VAN and ARC sites are directly adjacent to the Pacific Ocean. Measurements at VAN on 27 August 1992 and at ARC on 28 August 1992 were on similar ocean-produced fogs moving inland at surface wind speeds of less than 3 m s⁻¹. Some drizzle was reported. The SM site is 18 km farther inland. The SM measurements of 23 September 1992 dealt with a fog that was nearing a dissipation stage more than the others, but there was no rain. The WOR site is 73 km west-southwest (WSW) of Boston’s Logan Airport, and measurements there on 26 September 1992 concerned remains of Tropical Storm Danielle moving northeast at a speed of 4–5 m s⁻¹, leading to fog and rain locally with drizzle during some of the airplane approaches during fog conditions.

Basic formulation

The following very brief summary of the relationships between EM and meteorological parameters should suffice as a background. The radar backscatter cross section per unit volume of spherical scatterers, Q (in m² m⁻³), expresses the ratio of reflected plane-wave flux to incident spherical electromagnetic power flux in terms of an equivalent area per unit volume:

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where σ_bs(λ, D, ε) is the backscatter power attenuation coefficient (Bohren and Huffman 1983) for a single particle, as described by the Mie theory (Mie 1908) for a spherical particle, and ∫^z dζγ(ζ) is the one-way path integral of the attenuation coefficient γ(ζ) between the power source and the unit volume. The radar/laser wavelength and particle permittivity are given, respectively, by λ and ε. Formulation in terms of the Mie theory is increasingly important as radar wavelengths move into the millimeter regime. The attenuation coefficient at altitude z is given (van de Hulst 1957) by

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for a glide slope at elevation angle θ with single-particle forward scattering amplitude f_fs(λ, D, ε). The liquid water content of a unit volume of droplets at altitude z is given by,

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where ρ_w is the specific mass of water, for example, in g m⁻³. Equations (1)–(3) are well known and are cited in many places. In the Rayleigh regime of very small particles (D ≪ λ), it follows (van de Hulst 1957; Kerr 1965) that

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where σ_bs(λ, D, ε) = |f_bs(λ, D, ε)|² so that (1) becomes

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in what is known as the Rayleigh regime of scattering. Here, Z(z) is known as the reflectivity factor, and it is purely a function of the drop size distribution. Alternative expressions for Z(z) specifically in the Rayleigh regime and for M(z) in general are,

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where n₃(z) is the particle density (in m⁻³), and 〈υⁿ(z)〉 is the nth moment of the particle volume πD³/6 [i.e., the sum of all particle volumes per unit volume weighted by n₄(D, z) and divided by n₃(z)]. Similar formulas can be found, among others, in Paluch et al. (1996). From (6), it follows that

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This can express a linear relationship between M and Z, provided 〈υ²(z)〉/〈υ(z)〉 is not a function of altitude. For the above-mentioned gamma distribution model, n₄(D, z) = α(z)D⁴ exp(−D/D_o) as well as for many other commonly used analytical models (Deepak 1982), it is easily seen that 〈υ^m(z)〉 is not a function of altitude z, so that the coefficient of Z(z)/M(z) is not altitude dependent (as long as D_o also is not a function of z). However, nonlinear relationships between M and Z can result when the measured quantities underlying the calculations are averaged differently (see below).

The particle drop size data

The data (Zak 1994) consist largely of number densities for particles with diameters D in 3-μm “bins” such that d − 1.5 < (D in μm) < d + 1.5 in a bin, with d = 3.5, 6.5, . . . , 45.5 μm. The particle diameters thus lie between 2 and 47 μm. The data were gathered every 10 m of altitude from about 400 m altitude down. Two problems were encountered: 1) 2 to 10 runs were conducted for each fog so that the sampling varied highly and 2) the instrumentation was not able to register densities below 10⁴ m⁻³ per bin, so that false zeros were registered where there may have been lower densities. Figure 1 shows cumulative densities n₃(z) obtained at each altitude and site from the raw data by averaging over the runs. Not shown but implicit are fairly large standard deviations of up to 50% or so around these averages (which reflect changing weather conditions from run to run). The small-scale irregularities in these data also are partly due to the false zeros registered for low density values.

The liquid water content M(z), as defined by (3), is determined by the particle drop size distribution. It is obvious from use of (8) in 〈υⁿ(z)〉 that both 〈υ(z)〉 and 〈υ²(z)〉 are independent of altitude and that therefore the ratio of Z(z) to M(z) is independent of altitude z when both of these quantities are computed with the model (8). To test whether this is really the case for the actual data at these sites, we have calculated 〈υ²(z)〉/〈υ(z)〉 using the averaged n₄(D, z) data at each site rather than the model (8). Figure 4 shows equivalent graphs of 10 log₁₀[〈D⁶(z)〉/〈D³(z)〉] versus z for each of the advection fog sites. This ratio at WOR is 35.9 ± 1.28 dB in the regions of significant measurements; the standard deviations are smaller for VAN and ARC and only slightly larger for SM. It seems reasonable to expect little variation of Z/M with altitude.

The relationship between Z(z) and M(z)

The constancy with altitude of 〈υ²(z)〉/〈υ(z)〉 suggests [see (7)] that the reflectivity factor Z(z) is a linear function of liquid water content M(z) for these data at low frequencies. The quadratic relationship (Atlas 1964) for uniform size droplets in clouds would not be valid here. That different power laws are measured seems often to be a function of what is averaged. If, for instance, the data for both quantities are averaged at a fixed altitude, z = 150 m, over the four sites, then the relationship (Z in mm⁶ m⁻³) ≈ 0.0251 (M in g m⁻³)^1.583 is found from the model. Figure 5 shows an evaluation of ln[Z(z)] versus ln[M(z)] for the VAN site at significant altitudes 40 ⩽ (z in m) ⩽ 150, and for the ARC site at 60 < (z in m) < 240, for the actual averaged n₄(D, z) data. The overbar indicates averaging at fixed altitude over a number of runs. Regression analysis yields

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Only these two sites had enough data runs so that averages of the n₄(D, z) values would overcome the low-threshold limitations sufficiently. In checking the linear relationship for these same data we find for the ratio Z(z)/M(z):

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It is difficult to discriminate between (9a) and (9b) for these data as the errors in Z/M, Z/M^1.282 and Z/M^1.106 are similar.¹ Furthermore, radar or optical measurements take into account the nonzero drop concentrations that the VAN, etc., measurements were not able to see. For these reasons. it is not straightforward to predict a unique Z versus M relationship that holds for EM reflectivity under the conditions of these measurements.

At infrared to visible wavelengths, the second relationship in (5) is no longer valid. In that case, one needs to work with

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where Q(z) is given by (1). Using the model (8) and Mie theory for the scattering amplitudes in (1), and then substituting the result for zero attenuation into (10), we obtain a linear relationship between Z(z) and M(z), the ratio of which is a function of wavelength λ (but not of z), as shown in Fig. 6. The data in Fig. 6, sampled at frequencies, each of which is a factor 1.02 larger than the previous one, appear to show most of the salient features, and the relatively small standard deviation curve indicates that one average curve is useful for all four sites at infrared to visible wavelengths. The apparent universality of this curve is largely determined by σ_bs(λ, D, ε), as can be understood from (1). A relationship between the zero-attenuation cross section η(z) ≡ Q_o(z) and M(z) follows from this curve and from

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The ratio Q_o/M is shown as a function of wavelength in the infrared to visible regime in Fig. 7a for all four sites, and the average and standard deviation are depicted for 11 representative frequencies in Fig. 7b. To obtain the cross section Q(z) with the exponential attenuation factor in (5), the relationship between γ(z) and M(z) is required. Using (2), the ratio γ(z)/M(z) is shown as a continuous function of wavelength in the infrared to visible regime in Fig. 8a, for all four sites, and the average and standard deviation are depicted in Fig. 8b. Figure 8b also shows a sixth-order polynomial fit to the average γ(z)/M(z) versus λ curve.

From these observations, it seems possible to deduce attenuation and reflection coefficients in the infrared to visible regime from knowledge of the liquid water content M(z). Both Q_o(z)/M(z) and γ(z)/M(z) then appear to be predictable at these frequencies from single curves of each ratio versus wavelength λ, from which we also can construct Q(z) as well as the attenuation ∫ dz γ(z).

Prediction of M(z) required for reflectivity and attenuation

Here, we will describe briefly a possible method for predicting infrared and visible attenuation that an airplane preparing to land in fogs (of the type discussed) would encounter. The method rests on the assumption that a millimeter-wave radar is available at the landing site to probe the fog. We assume that the radar can measure accurately, at millimeter wavelength, the radar cross section of a known object (e.g., a sphere of known radius rising by means of a balloon through the fog). Assume that the radar cross section of the object in free space would be Q_obj. The actually measured cross section of the object, with known radar cross section Q_obj, rising in a fog would be

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if θ is the elevation angle, and with γ(z) given by the Rayleigh-regime formula,

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From these formulas, it follows that

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from which we obtain, by inversion, the liquid water content M(z). The above curves of γ/M and Q/M then predict attenuation and reflectivity at infrared to visible frequencies.

Final comments

We have noted that several of the sites measured a number of drops with diameters larger than 50 μm. For example, the VAN and ARC data shows some runs in which drops with diameters up to 270 μm (partially if not wholly due to drizzle) were noted. An analysis shows that these sparsely distributed large drops do not significantly affect our estimates of M(z), nor of γ(z), but they do significantly change the estimates of Z(z), given as a sum or integral of D⁶n₄(D, z), due to the strong effect of the sixth power of diameter D. However, the particle backscatter amplitudes, f_bs(λ, D, z), at infrared to visible wavelengths are much lower than would be predicted from (4) at the same wavelengths. Nevertheless, an exploratory analysis of the effect of these large particles does indicate an appreciable effect upon Q(z)—of the order of an extra 10 dB at λ = 3 μm for VAN. A gamma distribution of n₄(D, z) does not seem feasible for the large-drop tails, as these do not decrease sufficiently rapidly with increasing D. Nevertheless, modeling similar to the above can be done with inclusion of the larger particles. The above analysis has not included the effect of these drops because 1) not all sites exhibited appreciable numbers of large drops, 2) there are greater inaccuracies in the large-particle measurements by FSSP (Paluch et al. 1996), and 3) advection fogs normally do not have drops much larger than D = 50 μm; the larger drops may have been due to drizzle. The effect of large drops upon attenuation is expected to be considerably smaller, for example, because that quantity depends on an integral of approximately D³n₄(D, z)dD, as opposed to a D⁶ factor. In fact, a check shows the effect to be less than 10%.