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Oriented particles can exhibit different polarization properties than randomly oriented particles. These properties cannot be resolved by conventional polarization lidar systems and are capable of corrupting the interpretation of depolarization ratio measurements. Additionally, the typical characteristics of backscatter phase matrices from atmospheric oriented particles are not well established. The National Center for Atmospheric Research High Spectral Resolution Lidar was outfitted in spring of 2012 to measure the backscatter phase matrix, allowing it to fully characterize the polarization properties of oriented particles. The lidar data analyzed here considers operation at 4°, 22° and 32° off zenith in Boulder, CO, USA (40.0°N,105.2°W). The HSRL has primarily observed oriented ice crystal signatures at lidar tilt angles near 32° off zenith which corresponds to an expected peak in backscatter from horizontally oriented plates. The maximum occurrence frequency of oriented ice crystals is measured at 5 km, where 2% of clouds produced significant oriented ice signatures by exhibiting diattenuation in their scattering matrices. The HSRL also observed oriented particle characteristics of rain at all three tilt angles. Oriented signatures in rain are common at all three tilt angles. As many as 70% of all rain observations made at 22° off zenith exhibited oriented signatures. The oriented rain signatures exhibit significant linear diattenuation and retardance.
© 2014 Optical Society of America
1. Introduction
Conventional polarization lidar instruments perform two polarization measurements [1–4]. One detection channel corresponds to the original transmitted polarization state (the parallel channel), and the second channel corresponds to the polarization orthogonal to the original transmitted state (the perpendicular channel). The polarization properties of the scattering volume are subsequently described as the depolarization ratio written
where N⊥ is the background subtracted signal on the perpendicular channel and N‖ is the background subtracted signal on the parallel channel.It should be observed, however, that the depolarization ratio is not necessarily a complete description of the polarization properties of the scattering volume. For a theoretically complete description, we generally use the backscatter phase matrix, a sixteen element Mueller matrix capable of representing all polarization effects exhibited by scattering volumes.
In most cases, polarization lidar data is interpreted by assuming the scattering volume consists of randomly oriented particles. A volume consisting of randomly oriented particles is macroscopically isotropic which means the backscatter phase matrix is rotationally invariant [5]. This property combined with reciprocity gives a backscatter phase matrix of randomly oriented particles with three degrees of freedom and the form [5, 6]:
The scalar β is the volume backscatter coefficient and d is the depolarization of the volume. The f14 element is the result of circular diattenuation (also called circular dichroism) in the individual particle scattering properties. This term is commonly ignored and is zero if the particles are axially symmetric (therefore each particle has f14 = 0) or their mirror particles exist in equal numbers (a mirror particle would have an f14 element opposite the original particle and therefore cancels the term in the ensemble matrix) [7]. Measurements of this term reported in [5] as well as this work suggest this element is typically zero in atmospheric scattering processes. This means that the most commonly assumed atmospheric backscatter phase matrix has two degrees of freedom.The relationship between the depolarization ratio δ and d depends on the ellipticity of the polarization lidar state (generally linear or circular polarizations), but assuming f14 is zero, their relationship is monotonic, invertible and fully defined. Because the medium is macroscopically isotropic, direction of incidence is omitted from the backscatter phase matrix argument and only the scattering angle is needed to describe the scattering geometry (π for backscattering).
While there is often good argument for the assumption of random orientation, there are known cases where particles in the atmosphere exhibit preferential orientation. In the most general interpretation, the backscatter phase matrix has ten degrees of freedom. Making no assumptions about the orientation or symmetries but assuming reciprocity applies, the scattering matrix is given by [7]
where R(φ) is a Mueller rotation matrix of angle φ. The superscript (φ) denotes that the matrix elements are from the matrix with ten unique elements. The rotation term allows that the matrix has some linear coordinate basis given by the angle φ, that would set one or more matrix element to zero (thus φ represents one of the ten degrees of freedom in the matrix). When we cannot assume randomly oriented particles, the scattering medium is not macroscopically isotropic so the backscatter phase matrix is a function of the direction of incidence, k⃗i.Rotating the matrix F(φ)(k⃗i, − k⃗i) by the angle −φ allows us to obtain the matrix in its base coordinates F(k⃗i, − k⃗i). Note the rotation angle φ is dictated by the angle between the instrument’s polarization coordinate basis and the s-polarization of the orientation plane (most often assumed to be the horizontal plane in atmospheric scattering). In this work we assume that the particles under interrogation are not optically active. Beyond the presence of strong magnetic fields in the ionosphere or lightning there seems to be little argument for this polarization effect in optical atmospheric scattering. From these assumptions we obtain the form of the scattering matrix in its base coordinates as [5]
where there are now six degrees of freedom (we report seven matrix elements, but reciprocity reduces the number of degrees of freedom to six [8]).When the scattering volume is described by Eq. (4), the depolarization ratio is not sufficient to accurately determine the polarization properties of the scattering volume, and the measured value of the depolarization ratio depends on the degree-of-polarization (DOP), linear polarization angle and ellipticity of the polarization state used to interrogate the volume [9]. For polarized incident light (DOP = 1), the depolarization ratio is generally given by
where ψ is the linear angle and χ is the ellipticity angle of the lidar’s polarization state. For linear depolarization lidar χ = 0 or and for circular depolarization lidar . The value of ψ is generally not specified in linear depolarization lidar. Linear depolarization ratio measurements of oriented particles depend on four scattering matrix terms (β, f12, f22 and f33) while circular depolarization ratios depend on three terms (β, f14 and f44). Observations reported here and by [5] indicate that f14 is typically zero, thereby reducing the dependency of circular depolarization to two matrix elements [10].Figure 1 considers two cases of oriented scatterers and the possible linear depolarization ratios (χ = 0) as a function of the lidar’s linear polarization angle, ψ. The red case uses scattering matrix measurements of oriented ice crystals and the blue case is calculated using scattering matrix measurements of oriented rain drops. The oriented ice crystals analyzed for this plot produce linear depolarization ratios between approximately 0.15 and 0.25 depending on the lidar’s linear polarization state. The depolarization ratio of this instance of rain may be anywhere between 0.1 and 0.6 depending on the linear polarization state, ψ, used to interrogate the volume. Thus, the presence of oriented particles can contribute significant ambiguity to depolarization ratio measurements.
Fig. 1 Example of measured depolarization ratio as a function of linear polarization angle for rain (blue) and oriented ice crystals (red). The matrices used for this simulation were obtained from an observation of oriented ice crystals on June 24, 2012 and rain on July 9, 2012 over Boulder, CO, USA (40.0°N,105.2°W).
It should be noted that the presence of oriented particles is necessary but not sufficient to give ambiguous depolarization ratio measurements. To observe the matrix described in Eq. (4), three conditions must be met:
- A subpopulation of the particles must have a non-uniform orientation distribution.
- The particles’ orientation distribution must not exhibit rotational symmetry about the observing instrument’s line of sight.
- The backscatter from the oriented population cannot be significantly diluted by randomly oriented populations [11].
It has been noted in [12, 13] that horizontally oriented plates have strong backscatter cross sections at lidar tilt angles near 32.5°. It is most likely that conditions (2.) and (3.) are satisfied by oriented plates when the lidar is tilted at this angle. Oriented columns are expected to exhibit a similar backscatter peak for lidar tilt angles near 57.5° but practical limitations prevent us from operating the lidar at such a large tilt angle.
In the spring of 2012, the National Center for Atmospheric Research (NCAR) High Spectral Resolution Lidar (HSRL) was modified to measure the full backscatter phase matrix of oriented particles [14]. The instrument uses temporally varying transmitted and measured polarization states at a wavelength of 532 nm to reconstruct the ten element backscatter phase matrix described by Eq. (3). After obtaining the ten element matrix, it is reduced to the form in Eq. (4) by applying a linear rotation that best reflects the expected block diagonal form in Eq. (4). Because the HSRL performs a polarimetrically complete measurement, it cannot exclusively operate in the eigen states of of mirrors, beamsplitters, etc (a technique common to many conventional polarization lidar). All independent polarization terms are accounted for including partial polarization of the laser, mirror retardance/diatenuation and imperfect polarization analyzers. The details of the measurement as well as a complete list of all 30 calibration terms are provided in [14].
Observations of full scattering matrices reported in [5] suggest that preferential azimuthal orientation over western Siberia appears to the be the rule, more than the exception. More than 90% of observed ice clouds had non-zero f12 scattering matrix elements (f12 = 0 when the volume is composed of randomly oriented particles). The question as to whether this orientation behavior is common in other regions remains uninvestigated.
The results presented in [5] represent the only polarimetrically complete observations of backscatter polarization properties from oriented particles found in the atmosphere. As a result, the general impact of oriented particles on depolarization lidar data remains largely unquantified. Further, optimization and development of new polarimetic lidar techniques will inevitably require some a priori information about the scattering matrices common in the atmosphere. The intent of this work is to provide some additional insight into observed polarization properties of oriented particles in the atmosphere, though these observations are limited to brief periods at one geographical location (Boulder, Colorado, USA, 40.0°N,105.2°W).
2. Observations
We filter for oriented particles in HSRL measurements by resolving non-zero off diagonal terms, f12 and f34, of the scattering matrix. The f12 element is the linear diattenuation of the scattering matrix and describes the particles’ preference for scattering one linear polarization over its orthogonal state [9]. The f34 element is most closely associated with retardance which generally transforms the ellipticity of the backscattered radiation. The scattering matrix is recorded as the set of ten scattering matrix elements, F(φ)(k⃗i, − k⃗i) in Eq. (3), to account for linear rotation [14]. This matrix is then reduced to the seven element form shown in Eq. (4) by applying the necessary Mueller matrix rotations. The reduced matrix data is analyzed for statistically significant deviation from zero in the f12 and f34 terms by awarding an orientation index value based on the computed signal-to-noise. The orientation index is bounded between zero and one by using the definitions
and where erf is the error function or Gauss error function and σjk is the standard deviation of the jk element of the scattering matrix. These functions provide a bounded scale for assessing the likelihood that a scattering volume has statistically significant oriented scattering terms.The NCAR HSRL has operated at tilt angles of 4° (as close to vertical as possible with this instrument), 22° and 32°. The observational periods analyzed in this work and their durations are shown in Table 1. All observations were made in Boulder, Colorado, USA (40.0°N,105.2°W). Observations during summer and late fall 2012 at 22° and 32° tilt are analyzed, while analysis of near zenith operation (4°) is limited to late summer 2013. The selection of these data sets for analysis is based on the lidar’s operational status, data set duration, and quality of data and calibrations. The number of observation days at 4° off zenith are substantially less than the specified duration. This is because the lidar had several days in late August and early September where the transmit laser was not locked to the iodine absorption line in the molecular receiver channel, so retrieval of the backscatter ratio was not possible.
Table 1. HSRL Data Sets Analyzed
Figure 2 shows a clear case of oriented ice observation on December 18, 2012. The top plot shows the particle linear depolarization. The plots have been filtered to highlight cloud scattering by removing signals that have a backscatter ratio less than 2, where backscatter ratio is computed as the total backscatter divided by the molecular backscatter. The depolarization ratio is calculated based on the measured f44 element which reflects the HSRL’s original configuration that measured circular depolarization. The results are reported as an equivalent linear depolarization ratio. The bottom plot shows the resulting maximum orientation index, max(O12, O34). In this case the oriented particles are exclusively identified by non-zero f12 elements. For oriented ice crystals, the f34 element is not typically large enough for the HSRL to resolve a non-zero value, which is consistent with findings in [5].
Fig. 2 A case of oriented ice crystal observations from December 18, 2012, Boulder, CO, USA (40.0°N,105.2°W) where the lidar is tilted 32° off zenith. The top plot shows the equivalent particle linear depolarization ratio (calculated using the f44 element). The bottom panel shows the orientation index used as a metric for identifying regions containing oriented particles. Water or mixed phase clouds are precipitating ice virga. Oriented particles are observed in this virga at 2 and 3 km, just below the liquid cloud base.
Oriented ice crystals are observed just below or in very close proximity to liquid water clouds (determined by their low depolarization). This observation, like most oriented ice crystal observations seen by HSRL during this field test, appears to be a case of ice virga precipitating out of liquid water or mixed phase clouds, which is qualitatively similar to observations of horizontally oriented plates reported in [15].
Figure 3 shows observation of oriented rain drops which occur when drag forces cause the otherwise round drops to flatten along the vertical direction [16, 17]. In this case, it is a combination of f34 and f12 that allows the oriented drops to be identified.
Fig. 3 A case of oriented rain observations from July 17, 2012, Boulder, CO, USA (40.0°N,105.2°W) where the lidar is tilted 22° off zenith. The top plot shows the equivalent particle linear depolarization ratio (calculated using the f44 element). The bottom panel shows the orientation index used as a metric for identifying regions containing oriented particles. Large rain drops flatten as they fall and have strong oriented particle polarization signatures.
During the summer of 2012 several forest fires were burning in Colorado and their smoke frequently appears in the lidar profiles. The smoke can have a backscatter ratio exceeding 2 (the criteria previously used to filter for clouds) so we have added an additional criteria to the processed data sets shown below. A scattering volume is only counted as a cloud if the backscatter ratio is greater than 2 and
where BSR is the measured backscatter ratio and f44 is the scattering matrix element from Eq. (4) which is bounded between values of −1 and 1. This additional criteria avoids including smoke in orientation frequency statistics.Figure 4 shows the total time of cloud observations for each lidar tilt angle as a function of altitude. Figure 5 shows the frequency at which those clouds produced statistically significant oriented scattering matrices as a function of altitude. The observation frequency is determined by the time of oriented particle observation (time where O12 > 0.4 or O34 > 0.4) divided by the total time of cloud observations (where backscatter ratio is greater than 2 and satisfying Eq. (8)). We selected an orientation index threshold of 0.4 to focus this analysis on clear cases of oriented particles. All data is analyzed in 40 second integrations at 30 m range resolution. The observations made in summer are shown as solid lines, while the late fall/winter observations are dashed lines. The frequent oriented particle signatures seen below 3 km during the summer are caused by rain.
Fig. 5 Fraction of cloud observations where clouds have oriented scattering matrices. The range resolution is 0.03 km. There are no oriented scattering matrix observations in November and December 2012 when the lidar is tilted at 22°.
Observations of oriented ice crystals conforming to Eq. (4) are quite rare, with a maximum frequency just under 2% of all cloud observations between 5 and 6 km when the lidar was directed 32° off zenith. Those observations are easily identified in several data sets as clear f12 signatures in ice falling out of mixed phase clouds (similar to Fig. 2). In a small number of cases, oriented ice crystal signatures are seen at other tilt angles or higher altitudes. These are typically associated with the tops of convective storms but do not have a clear consistent structure that makes them stand out as a real atmospheric effect. It is difficult to determine if the signatures seen in convective storms represent false positives or actual preferential orientation of ice crystals at the cloud tops.
In late fall 2012, no oriented particles were identified when the lidar was tilted at 22° off zenith. Cirrus clouds were frequently present in this data and no convective storms were observed during this time. There were no convective storms when the lidar operated at 32° off zenith during December 2012, but there were two days where oriented ice crystals were clearly identified (one is the example shown in Fig. 2). The conditions under which oriented ice crystals were observed in the winter were similar to those in the summer (ice precipitating out of liquid or mixed phase clouds), but the ice crystals were observed at lower altitudes.
Figure 6 shows a histogram of measured f44 vs. f33 for all cloud observations when the lidar was tilted 32° off zenith in June/July 2012. The green line shows the expected relationship
for randomly oriented particles. Note from Eq. (2) that f22 = − f33 and there is no distinction between F and F(φ) when the volume consists of randomly oriented particles. Figure 6 shows that most clouds conform to the relationship in Eq. (9). This indicates that the HSRL matrix observations are consistent with fundamental scattering theory, and therefore reliable measures of the scattering matrix. There is a sizable population that does not fall on the green line. That spur above the green line corresponds to rain observations. The oriented ice crystal observations with significant nonzero f12 elements are still scattered about green dashed line. Thus, testing the relationship described by Eq. (9) does not appear to be a good method for identifying scattering effects of oriented ice crystals.Fig. 6 Histogram of f44 vs f33 of all cloud observations from June/July 2012 when the lidar operated at 32° off zenith. The green line shows the expected relationship for randomly oriented particles given by Eq. (9). The color bar is log10 of the number of cloud events (40 second integration at 30 m resolution) recorded in each bin.
A unique feature of high spectral resolution lidar is its ability to separately measure molecular and total (aerosol and molecular) backscatter [18]. This allows for a straight forward computation of atmospheric extinction with relatively few assumptions. The extinction properties of oriented ice crystals did not appear to be significantly different from spatially and temporally close randomly oriented ice. For example, the virga in Fig. 2 demonstrated no resolvable difference in extinction between instances with high and low orientation index.
Altitude integrated histograms of randomly oriented clouds and rain, oriented rain and oriented ice crystals are shown in Fig. 7. The rain observations exhibit a higher extinction at 32° off zenith than 22° or 4°. This is unexpected, since flattened rain drops should have a larger projected area when the lidar is directed vertically. It is possible this is the result of different rain size distributions between the observational periods. Because we could not identify significant oriented populations at tilt angles other than 32°, it is not clear whether or not oriented ice crystals significantly change extinction as a function of lidar tilt angle.
Fig. 7 Altitude integrated histograms for extinction of randomly oriented clouds and rain (top), oriented rain (middle) and oriented ice crystals (bottom). The histograms are separated according to lidar tilt angle with summer observations at 32° (blue), 22° (green) and 4° (red) off zenith.
Histograms of the oriented particle matrix elements as a function of altitude were created for three of the five data sets considered here. The November/December 2012 data was omitted due to the relatively few cases of oriented particles observed. The histograms for June/July 2012 at 32°, July/August 2012 at 22° and August/September 2013 at 4° are shown in Figs. 8, 9 and 10 respectively. In interpreting these plots, oriented ice crystals are attributed to occurrences above 3 km and rain is below 3 km. Also the data presented in these plots has been filtered for only those cases where the particles are determined to be oriented based on an orientation index greater than 0.4. Note that f14 is not reported as this term shows no statistically significant deviation from zero in any of the analyzed observations.
Fig. 8 Histograms of measured matrix elements for scattering matrices conforming to Eq. (4) (instances of high orientation index). Observations here are from June and July 2012 when the lidar was tilted 32° off zenith. Oriented ice crystals are seen at approximately 5 km, while oriented rain is below 3 km. The color bar is the same for all plots and is log10 of the fraction of total observations where rain or clouds were observed to have the specified element value.
Fig. 9 Histograms of measured matrix elements for scattering matrices conforming to Eq. (4) (instances of high orientation index). Observations here are from July and August 2012 when the lidar was tilted 22° off zenith. Oriented rain below 3 km is the only major contributor to oriented particle signatures. The color bar is the same for all plots and is log10 of the fraction of total observations where rain or clouds were observed to have the specified element value.
Fig. 10 Histograms of measured matrix elements for scattering matrices conforming to Eq. (4) (instances of high orientation index). Observations here are from August and September 2013 when the lidar was tilted 4° off zenith. Oriented rain below 3 km is the only major contributor to oriented particle signatures. The color bar is the same for all plots and is log10 of the fraction of total observations where rain or clouds were observed to have the specified element value.
As mentioned previously, oriented ice crystals only appear to produce clear, statistically significant oriented scattering matrices when the lidar is tilted near 32° off zenith. It’s notable that the ice crystals tend to form at the same altitudes. This is likely because plate formation at sizes known to orient (approximate diameter larger than 100μm [19]) occur in a relatively narrow temperature range [20–22]. Analysis of CALIPSO data by [23, 24] found that observations of oriented plates were consistently bounded between −30°C and −10°C. In [24] it is reported that oriented ice crystal observations by CALIPSO were typically near an altitude of 5 km at Boulder’s latitude of 40°N.
Oriented ice crystals tend to be most easily identified by their non-zero f12 elements, or linear diattenuation. The f34 element associated with retardance is generally not resolvable with ice crystals. Additionally, the diagonal matrix terms tend to closely approximate the relationship in Eq. (9). As part of that relationship f22 = − f33 which is more or less reflected in Fig. 8. Oriented scattering matrices are only consistently and clearly observed from ice crystals when the lidar is tilted at 32° off zenith. While some instances of oriented ice crystals appear at other tilt angles, it is not clear if they are the result of poor filtering or actual instances of oriented particles.
In contrast to oriented ice crystals, rain appears to be best described by the scattering matrix in Eq. (4) for all three lidar tilt angles. Even at 4°, rainfall still produces a significant (though reduced) population of oriented particles. Rain produces both significant linear diattenuation (f12) and retardance (f34) though there does not appear to be an obvious functional relationship between the two parameters. Also, oriented rain does not typically conform to the relationship in Eq. (9), a fact clearly visible in Fig. 6. It is notable that in most cases |f22| > |f33|. This is probably because the depolarizing characteristics of the rain are relatively small, where the smaller |f33| is caused by retardance.
At 4° off zenith, the oriented signatures of rain are significantly weaker and less frequent than the other two tilt angles but still common enough that it is never practical to assume the matrix in Eq. (2) when observing polarization properties of rain.
It is not entirely clear why rain still produces an oriented scattering matrix at 4°. One possibility is that 4° tilt is still sufficient for rain to satisfy condition (2.). Alternately it is possible that winds are shifting the mean canting angle of the rain.
Before reduction to its seven element form, the linear diattenuation elements have the relationship
A cursory analysis of the angle φ in rain indicates that the orienting effect at 4° is consistently in the same direction. This suggests that the oriented scattering matrix of rain is probably due to the 4° tilt of the lidar. This analysis does not completely rule out wind, as it is possible the wind direction during rain fall was always the same over the August/September 2013 observational period. However radar observations have indicated the mean canting angle of rain is not expected to deviate by more than 1–2° [25], which suggests that the lidar tilt angle is the more significant term.3. Conclusion
The presence of oriented particles has the potential to render conventional polarization lidar measurements ambiguous. Only with information about the scattering matrix elements common in the atmosphere, is it possible to assess the magnitude of such depolarization ratio ambiguities. Further, such scattering matrix observations allow us to determine common cases where oriented particles significantly contribute to the backscatter light.
The NCAR HSRL has observed the full backscatter matrix of clouds and aerosols over Boulder, Colorado, USA (40.0°N,105.2°W) at three tilt angles. These observations have been analyzed to report information about the types of scattering matrices that are typical for atmospheric observations.
We have described how the presence of oriented particles in a scattering volume is not sufficient to produce an oriented scattering matrix. Two additional criteria are required, a break in rotational symmetry along the lidar line-of-sight, and sufficient backscatter from the oriented population to be observable over a randomly oriented sub population. Observations by the NCAR HSRL have shown clear cases where ice crystals and rain drops satisfy all three criteria to produce a statistically significant oriented backscatter phase matrix. In circumstances where the oriented particle backscatter is not observable (does not satisfy all three criteria), there is no issue in using conventional polarization lidar to interrogate the volume.
In this work we have shown observed statistics on scattering matrix elements for lidar operation at 4°, 22°, and 32° off zenith. Oriented ice crystals are most visible where the lidar is tilted 32° and are identified by significant linear diattenuation signatures (f12). Their backscatter matrix diagonals, however, typically conform to the relationship expected for randomly oriented particles, and they do not appear to exhibit significant retarding effects (f34). The instances of oriented ice crystals that were clearly visible were those of ice virga precipitating out of water or mixed phase clouds, which is similar to observations of oriented plates reported in [15]. These precipitating oriented ice crystals were observed in both summer and late fall, though the ice crystals formed at lower altitudes in the late fall. We found that clouds produce scattering matrices conforming to Eq. (4) in less than 2% of cloud observations made by HSRL. It should be noted that this analysis focused on cases of clearly oriented particles and instances of smaller oriented populations are not likely to be counted as “oriented” in this analysis. Our intent is not to definitively determine where particles orient, but rather, where their scattering properties cannot be approximated by Eq. (2).
To our knowledge, the NCAR HSRL is the first lidar instrument to fully observe the optical polarization properties of rain. The backscatter matrices of rain in the data analyzed here commonly have the form of oriented particles, even at the smallest lidar tilt angle of 4°. The highly ambiguous depolarization ratio expected from these scatterers is perhaps somewhat mitigated by the fact that lidar is not commonly used to interrogate rain (though not all together absent from polarization lidar studies [1]). It should be emphasized that one should never assume rain can be accurately described using the linear depolarization ratio. Circular depolarization ratio measurements can be used, but they should be qualified to make clear that the scattering matrix has the form in Eq. (4). Additionally, our observations of rain suggest a potential use of more advanced lidar polarimetry in characterization of precipitation. Thus, more complete polarization lidar may be a useful tool in characterizing raindrop evaporation, coalescence and liquid water content.
Acknowledgments
The National Center for Atmospheric Research is sponsored by the National Science Foundation.
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