Diagnosing Cloud Microphysical Process Information from Remote Sensing Measurements—A Feasibility Study Using Aircraft Data. Part I: Tropical Anvils Measured during TC4

Gerald Mace University of Utah, Salt Lake City, Utah

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Sally Benson University of Utah, Salt Lake City, Utah

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Abstract

The authors investigate whether radar remote sensing of a certain class of ice clouds allows for characterization of the precipitation rates and aggregation processes. The NASA DC-8 collected the measurements in tropical anvils during July and August 2007 as part of the Tropical Composition, Cloud and Climate Coupling (TC4) experiment. Measured hydrometeor size distributions are used to estimate precipitation rates (P) and to solve the hydrodynamical collection equation. These distributions are also used to estimate radar reflectivity factors (Z) and Doppler velocities (Vd) at W, Ka, and Ku bands. Optimal estimation techniques are then used to estimate the uncertainty in retrieving P and aggregation rates (A) from combinations of Z and Vd. It is found that diagnosing information about A requires significant averaging and that a dual-frequency combination of W and Ka bands seems to provide the most information for the ice clouds sampled during TC4. Furthermore, the addition of Vd with expected uncertainty contributes little to the microphysical retrieval of either P or A. It is also shown that accounting for uncertainty in ice microphysical bulk density dominates the retrieval uncertainty in both P and A causing, for instance, the instantaneous uncertainty in retrieved P to increase from ~30% to ~200%.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Gerald “Jay” Mace, jay.mace@utah.edu

Abstract

The authors investigate whether radar remote sensing of a certain class of ice clouds allows for characterization of the precipitation rates and aggregation processes. The NASA DC-8 collected the measurements in tropical anvils during July and August 2007 as part of the Tropical Composition, Cloud and Climate Coupling (TC4) experiment. Measured hydrometeor size distributions are used to estimate precipitation rates (P) and to solve the hydrodynamical collection equation. These distributions are also used to estimate radar reflectivity factors (Z) and Doppler velocities (Vd) at W, Ka, and Ku bands. Optimal estimation techniques are then used to estimate the uncertainty in retrieving P and aggregation rates (A) from combinations of Z and Vd. It is found that diagnosing information about A requires significant averaging and that a dual-frequency combination of W and Ka bands seems to provide the most information for the ice clouds sampled during TC4. Furthermore, the addition of Vd with expected uncertainty contributes little to the microphysical retrieval of either P or A. It is also shown that accounting for uncertainty in ice microphysical bulk density dominates the retrieval uncertainty in both P and A causing, for instance, the instantaneous uncertainty in retrieved P to increase from ~30% to ~200%.

Denotes content that is immediately available upon publication as open access.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Gerald “Jay” Mace, jay.mace@utah.edu

1. Introduction

Earth’s climate system is tightly coupled to the atmospheric branch of the hydrological cycle (Stephens 2005). Because the processes that define the atmospheric water cycle are linked to motions that exist over a continuum of spatial and temporal scales, our ability to predict changes to the climate system depend fundamentally on our understanding of the hydrologic cycle (Stevens and Bony 2013). Improvements in representing moist processes in models have been difficult to realize, however. Knutti et al. (2013) show that the current generation of climate models in phase 5 of the Coupled Model Intercomparison Project (CMIP5) show only incremental improvements in reducing biases in temperature and precipitation fields relative to observations. Klein et al. (2013) show that cloud properties also demonstrate only incremental improvements relative to previous IPCC assessments and that the current generation of models continues to show significant differences relative to observations in both cloud microphysical and macrophysical properties. See also Jiang et al. (2012) who make similar points using A-Train data as a point of reference.

Many of the biases noted by Klein et al. and Knutti et al. can be traced to errors in the atmospheric water cycle and are ultimately derived from uncertainties in physical processes that, in the real atmosphere, operate at spatial and temporal scales where water droplets and ice crystals evolve in turbulent vertical motions to become precipitation. Indeed it is the time rates of change of hydrometeors through the continuum of particle sizes within air motions that define the key microphysical processes. As examples of the sensitivity of the atmosphere to these processes, consider the papers of Adams-Selin et al. (2013), Saleeby and Cotton (2008), and Morrison et al. (2015) that all show large sensitivity in predictions of surface rain rates and convective to stratiform area ratios for physically reasonable changes in various aspects of microphysical parameters that control process rates in cloud-resolving models. Given the emerging importance of process parameterizations in cloud-resolving models, it is incumbent on the observational community to provide rigorous physical constraints on these processes to the extent that such constraints can be provided. Understanding whether that goal is attainable is the specific objective of this study. Here, we explore whether single- or dual-frequency Doppler radar at vertical incidence could theoretically provide information regarding the aggregation process and precipitation rates in tropical anvils using data from the Tropical Composition, Cloud and Climate Coupling Experiment (TC4) airborne field campaign (Toon et al. 2010).

By process we refer to the physical mechanisms that cause hydrometeor distributions to change in time. Since typically remote sensors and in situ aircraft can collect only single looks at populations of hydrometeors, inferring time derivatives from these measurements is not possible using finite differences. The process we consider here is that due to hydrometeors collecting other hydrometeors through differential sedimentation rates as described by the hydrodynamical collection equation (e.g., Verlinde et al. 1990):
e1
where indicates the time rate of change of the mixing ratio (r) of species n due to collection of species m. The quantity D is the hydrometeor diameter for species n or m as indicated by the subscripts, N(D) is the number concentration of species n and m, m(Dm) is the mass of species m at size D, υ is the terminal fall speed, ρ0 is air density, and Emn is the collection efficiency or the probability that a collision between two ice crystals will result in a permanent fusion of them. Field et al. (2006) considered the self-collection mode where species m and n refer to the same distribution of ice hydrometeors. They evaluated a version of Eq. (1) using in situ data collected during spirals in thunderstorm anvils over Florida and found that N(D) became more heavily weighted to the largest particles with time from which they inferred aggregation. McFarquhar et al. (2007) found similar signatures of aggregation using spirals that extended to the melting layer in convective environments.

To infer with Eq. (1) using aircraft in situ data we first fit the optical array probe N(D) with modified gamma functions (Hammonds et al. 2014; Posselt and Mace 2014) and we approximate V(D) using the approach described by Mitchell and Heymsfield (2005). When applying Eq. (1) to ice clouds, the mass- and area-dimensional (M-D and A-D) properties of the ice particles and Emn must be assumed. Hashino and Tripoli (2011) and Szyrmer and Zawadzki (2014a,b), among others, identify these empirical properties of ice particles as major sources of uncertainty. For instance, Field et al. (2006) find that Emn ranges between 0.1 and 1 and that it may be a function of particle size. Hashino and Tripoli (2011) fix Emn in their study at 0.4. In developing the statistics for the analysis described later, we assume that Emn is uniformly distributed between 0.1 and 0.5 to account for the uncertainty in this quantity. With these assumptions, Eq. (1) is solved numerically using Romberg quadrature using the quantities derived from TC4 aircraft measurements.

Past studies use aircraft and remote sensing data to infer aggregation and accretion rates in ice- and liquid-phase clouds. [Note that hereinafter we use the term aggregation (A) when referring to ice-phase processes and accretion to refer to liquid-phase processes.] Using ground-based W-band radar, Westbrook et al. (2007) found that radar reflectivity (Z) increases exponentially from the top of an ice cloud layer to its base and that this vertical rate of increase can be used to infer aggregation. In essence, Westbrook et al. found that condensed mass is approximately conserved in the vertical dimension and particle growth due to aggregation results in an increase in Z with depth from the layer top. Essentially mass migrates to larger sizes through aggregation as the particles sediment causing Z to increase with depth in the layer. Such a mass-conserving particle growth process suggests that ice water path and layer-integrated Z may be uncorrelated. This violates the assumptions implied by many retrieval algorithms (e.g., Protat et al. 2007; Liu and Illingworth 2000; Mace et al. 2002; Delanoe and Hogan 2008; Deng et al. 2010).

What we find intriguing about Eq. (1) is that no time derivative appears on the right-hand side even though the result is a time rate of change. The time dependency emerges from the differential fall speeds of the hydrometeors implying that it would not be necessary to make multiple measurements of a particular volume with remote sensors to infer a time-dependent process rate. It is, however, necessary that the microphysical characteristics of N(D) be inferred accurately.

Consider a simpler representation of a collection process that would be represented by Eq. (1) as parameterized by Khairoutdinov and Kogan (2000) for warm liquid clouds. These parameterizations of autoconversion and accretion are widely used in models (e.g., Morrison and Gettelman 2008) and have been critically evaluated (e.g., Wood 2005) and found to be reasonable approximations of Eq. (1):
eq1
e2

The temporal rate of creation of precipitation (droplet mode with subscript n) through self-collection by cloud droplets (subscript m) depends directly on the amount of cloud water present (rm) and also indirectly on the number of cloud droplets (Nm). Similarly, accretion of cloud water by precipitation describes the rate of creation of precipitation mixing ratio or in Eq. (2). These regression relationships show that a set of measurements must together be capable of simultaneously characterizing information regarding not only the hydrometeors doing the collecting but also the small and more numerous hydrometeors that are being collected. The requirement to infer the smaller-size hydrometeor properties in the presence of larger hydrometeors is what makes inferring process-level information from remote sensors such a challenge, as we discuss next.

2. Methodology

Our objective is to characterize the uncertainties in A and precipitation rates (P) in extended tropical anvils using nadir-pointing Doppler radar measurements. The term A is defined by Eq. (1), and
e3
The challenge we address in this paper is to account for the uncertainties that would cause error in retrievals of A and P using datasets from past, current, and hopefully future orbiting nadir-pointing radars. While passive remote sensing data such as microwave brightness temperatures, infrared emission, and visible reflectance always provide useful constraints, we neglect their contribution in this study and focus only on combinations of radar measurables.

Since we can derive a reasonable, if approximate and uncertain, estimate of A and P by integrating Eqs. (1) and (3), we use N(D) measured by aircraft with assumed M-D as input to a simple radar forward model, building on the methodology in Hammonds et al. (2014). This allows us to explore what set of radar measurables provide enough information to retrieve A and P and to determine the sensitivities of A and P to assumptions. We treat this aspect of the problem using Bayesian optimal estimation (OE) techniques. We anticipate that measurements could be of single- or double-frequency W-, Ka-, and/or Ku-band Z with or without Doppler velocity (Vd).

a. The optimal estimation algorithm

As discussed in Rodgers (2000) and elsewhere (Wood et al. 2014; Mace et al. 2016), Bayesian OE allows us to derive a most likely state [x or the quantities we desire to know, x = (P, A)] and the covariance of that state given measurements (y) and uncertainty in those measurements via a covariance matrix . Forward models F(x) link x and y and use simplifying assumptions whose uncertainties are normally represented as a covariance matrix. We also utilize prior knowledge of the state x, given y as a state vector xa and its covariance
and is a variance or covariance as defined by the subscripts. Linking these various components of the problem are sensitivity matrices or Jacobians that record the first-order sensitivities of y to x, that is, ∂y/∂x denoted as and that list the sensitivities of y to assumptions . Inherent in the use of first-order derivatives to describe the sensitivity is an assumption of linearity equivalent to a first-order Taylor expansion. Evaluating and comparing uncertainties in A and P using OE can be reduced to quantifying the error covariances matrix defined as
e4
For the purpose of explanation, we assume a measurement system that has two frequencies (W and Ka bands) each recording Z and Vd. To refer to this combination, we use the superscript ZVaw where Ka-band Z and Vd are denoted Za and Va and w and a refer to radar frequencies. So,
e5
In words, we use the higher-frequency W-band measurements as a difference from the lower-frequency Ka-band measurements or, that is, . The various parts of the problem can now be described in more detail for the ZVaw measurement combination:
e6
The terms in are estimated by perturbing the parameters of the gamma-fitted N(D), calculating P, A, Z, and V from perturbed particle size distributions (PSDs) and then finding the slope of a linear regression to the perturbed quantities.

b. Uncertainty estimation

The measurement error covariance matrix is typically taken to include instrumental uncertainties and forward model uncertainty (Austin and Stephens 2001) or
e7
where is the instrumental uncertainties:
e8
The quantity incorporates assumptions that cause uncertainty equivalent to measurement error as written in Eq. (7). We allow for forward model error (FME) that is due to uncertainties in the ice crystal microphysical properties that we represent using the mass-dimensional power law following Hammonds et al. (2014) where the uncertainty in M-D influences the radar backscatter cross section. Recent work has highlighted the limitations of using spheroidal models for deriving backscatter cross sections in precipitating ice clouds (Leinonen et al. 2013) and also the potential advantage of using triple-frequency Doppler measurements (Kneifel et al. 2015). However, these issues become relevant in volumes where the low-frequency dBZ values are in excess of +10 and N(D) is dominated by sizes in excess of several millimeters. We found those characteristics to be mostly absent in the in situ data collected in TC4. M-D also causes uncertainty in Vd because of the backscatter cross-section weighting of Vd but also because of how the M-D influences terminal velocity (V) following Mitchell and Heymsfield (2005) who show that V depends on the mass to cross-sectional area ratio. We fix the cross-sectional area A-D relationship using the power law A = aADbA, where aA = 0.23 and bA = 1.88 from Mitchell’s (1996) Table 1 for aggregates of side planes. The FME covariance matrix is the second term on the right in Eq. (7) and we represent and , respectively, as follows:
e9
e10
The is calculated in a way that is similar to that of . However, the covariances of am and bm needed for are largely unknown. While studies exist that have attempted to calculate these quantities from sporadic field data (Mitchell 1996; Heymsfield et al. 2004; Schmitt and Heymsfield 2010; A. J. Heymsfield et al. 2010; Szyrmer and Zawadzki 2010; Wood et al. 2014), the covariances of the M-D power law in a particular class of clouds are unknown. We examine two versions of the M-D power law that would result in low and high bulk density ice crystals. The low bulk density version of the M-D power law has (cgs units) am = 0.0039 and bm = 1.9, reported by Szyrmer and Zawadzki (2010), and is likely on the low end of what should be considered realistic for these types of clouds (A. J. Heymsfield et al. 2010). The high bulk density M-D power law is more appropriate to heavily rimed ice or graupel and has am = 0.037 and bm = 2.65 (cgs units). The high-density ice assumption is presented for comparison and is likely not typical of ice crystals in extended anvils. While the variances and covariances in am and bm are largely unknown, studies (Mitchell 1996; Heymsfield et al. 2004; Schmitt and Heymsfield 2010; Szyrmer and Zawadzki 2010; Wood et al. 2014) suggest that values of 100% and 20% are, respectively, reasonable for the variances. The values of am and bm are strongly correlated (we assume a value of 0.8) as shown in Wood et al. (2014). This correlation can be derived from the data in Mitchell (1996, his Table 1).

In addition to the ZVaw algorithm, we also examine a combination that includes just W-band Z and Vd, which we denote with the superscript ZVw, a combination that includes W- and Ka-band Z with no Vd, Zaw, and a combination that includes Ka- and Ku-band Z, Zua. The versions of Eqs. (5), (6), (8), and (9) that vary with measurement combination are listed in the appendix.

c. Approach to the prior and information content

To determine, we build up a large database of information from measurements collected by in situ aircraft that describes the microphysical properties of interest (Fig. 1 shows example statistics for TC4). These quantities derived from in situ data form the basis of our prior statistical understanding that is used to populate the and matrices as a function of environmental condition. Sourcing our database and searching for all measured volumes with a certain set of observable characteristics, we then derive a statistical description of the microphysical properties of interest, process rates, etc. The variances and covariances of these desired properties derived from our in situ database is how we populate the matrix.

Fig. 1.
Fig. 1.

Frequency distributions of various observed and derived quantities from the TC4 dataset that we analyzed. Included here are ~24 000 5-s averages. Shown are (a) temperature, (b) pressure, (c) ice water content, (d) number concentration, (e) effective radius, (f) derived Ku-band dBZe, (g) derived Ku-band V, (h) precipitation rate, (i) aggregation rate derived using Eq. (1), and (j) the correlation between the precipitation rate and the aggregation rate in the ice cloud data collected during TC4.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

With this understanding, our objective can be restated more explicitly. Can we demonstrate that retrieval algorithms, when applied to some combination of measurements, actually add information beyond climatological statistics that we already know from our airborne databases when we fully account for uncertainties in the algorithms due to assumptions? It is entirely possible that our uncertainties [i.e., the second term inside the parentheses in Eq. (4)] due to our lack of knowledge in and the sensitivities of our forward modeled measurements to our assumptions () are large enough that simply using a microphysical statistical description derived from climatologies generated from in situ datasets is as informative as the retrieval results themselves. Defaulting to climatological or statistical results is, as a matter of fact, precisely what Bayesian algorithms are designed to do in circumstances where uncertainties dominate (Rodgers 2000).

A convenient construct for formalizing this concept is the information content I, which is the ratio of to or . This quantity was defined initially in Shannon and Weaver (1949) and explored by L’Ecuyer et al. (2006), Cooper et al. (2007), Wood et al. (2014), and Mace et al. (2016). We use it here, also, to determine the degree to which a particular retrieval result provides information beyond what is contained in the prior information. It may be that the term in Eq. (4) is larger than such that I is equal to 0 in individual sample volumes. If our assumptions are unbiased and normally distributed, we can average similar volumes to reduce the variance in the result as the inverse of the number of independent samples. The primary assumptions of most concern would be those regarding the ice crystal microphysics M-D and A-D relationships, the aggregation efficiency, and the instrument error. It is common in the remote sensing literature to consider the statistics of results where many similar independent events are averaged. In the results presented below we consider the number (Navg) of independent volumes that would be required to result in an overall uncertainty of 5%.

While Navg is essentially just a scaling factor, it provides an intuitive understanding for how many independent samples are needed to actually result in useful statistics. Beyond that, the concept of averaging naturally raises questions as to what type of averaging would be done. Statistically, one would want independent samples of like volumes and certainly not simple spatial averaging. Similarity could be defined in a number of ways, but practically, histograms of retrieval results derived from volumes that occur with similar characteristics, such as temperatures and optical depths, would allow for extraction of meaningful information from remote sensing data. Again, many remote sensing studies evaluate results using this approach (e.g., Henderson et al. 2013; Lebsock and L’Ecuyer 2011, their Fig. 6; L’Ecuyer et al. 2008; Suzuki et al. 2011, their Fig. 5; Mace et al. 2016). We are quantifying the uncertainty in those averages here.

Another issue of concern is the requirement for unbiased and normally distributed assumptions. If this requirement is violated, then the average converges on a biased mean and/or the variance of the result is incorrect. Christensen et al. (2013) explore this issue for the retrieval of warm low cloud properties derived from composites of CloudSat retrievals. It is unlikely at present if we could confidently say what the characteristics of the M-D statistics actually are in nature so we cannot confidently say that averaging would result in unbiased results. However, we are confident that over time this will be remedied as additional data are collected in ice clouds (i.e., Wood et al. 2014; Szyrmer and Zawadzki 2010).

3. Results

As described by Toon et al. (2010), TC4 took place primarily in the extreme tropical eastern Pacific Ocean during the summer of 2007. The NASA DC-8 collected the data we use here, and much of the data is summarized in Fig. 1. In addition, the NASA ER-2 also participated in TC4 and often flew tightly coordinated patterns with the in situ aircraft. The NASA WB57 also participated in TC4 but we do not use any of the WB57 data in this study. While the TC4 dataset is quite varied, we use a subset of the data that corresponds to ice-phase samples collected between 215 and 240 K corresponding to pressures from 180 to 300 hPa. The 5-s average sample volumes we analyze were collected in layers that had recently been or were at the time of sampling associated with deep convection and would be described as extended stratiform anvils. Many of the layers were actively raining at the surface. The water contents were high and particles sizes were large relative to those of typical tropical cirrus (Berry and Mace 2014). Hydrometeor number concentrations tended to peak in the tens per liter range. The precipitation rates averaged 10 mm day−1, and radar reflectivities calculated from the PSDs were in the −10-dBZe range on average. So, while these clouds had much more condensate than typical tropical cirrus, their precipitation rates and radar reflectivities put them below the detection threshold of the Tropical Rainfall Measuring Mission (TRMM) and the dual-frequency radar on the Global Precipitation Measurement (GPM) mission (Hou et al. 2014).

a. Case study

It is useful to examine a particular case that was observed on 17 July 2007. The DC-8 and ER-2 sampled anvil outflows that were detraining toward the southwest from an active convective line that formed just after sunrise approximately 100 km offshore. The two aircraft established a pattern that kept them in close coordination while sampling the freshly produced anvils (Fig. 2). We focus on a short period when the aircraft sampled a particularly dense part of the anvil at approximately 1528 UTC marked by the arrow in Fig. 2 and the transition between the red and green segments of that flight track. Figure 3 shows an X-band radar reflectivity height–time cross section measured by the ER-2 Doppler radar system (EDOP; G. M. Heymsfield et al. 2010). The linear reflectivity feature at 9 km is the radar reflection of the DC-8 in the EDOP beam. Figure 4 shows that in the period leading up to the penetration of the fresh anvil plume, the particle spectra were quite narrow, ranging between 300 um and 1 mm. At about 1528, the anvil plume was penetrated and the particle spectra changed dramatically with both larger ice crystals extending out to 3 mm and a large increase in smaller ice crystals between 100 and 300 um. This increase in both larger and smaller ice crystals caused the calculated A to increase substantially.

Fig. 2.
Fig. 2.

Visible GOES image that overlays a short segment of the flight track of the DC-8 on 17 Jul 2007. The arrow marks the location of the case study discussed in the text. The image was obtained through the courtesy of Dr. P. Minnis and L. Nguyen, NASA Langley Research Center.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

Fig. 3.
Fig. 3.

The X-band radar reflectivity cross section collected by the EDOP instrument on the ER-2 between 1526 and 1532 UTC 17 Jul 2007 along the flight track depicted in Fig. 2. The period we analyze is between 1528 and 1529 UTC. The linear feature at 9 km is the reflection of the NASA DC-8 in the EDOP radar beam.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

Fig. 4.
Fig. 4.

Particle size spectra measured by the (a) PIP and (b) 2DS probes on the NASA DC-8 during the period represented in the radar cross section in Fig. 3.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

Two particular N(D)s created by combining the two-dimensional stereo (2DS; Lawson et al. 2006) and the Droplet Measurement Technologies precipitation imaging probe (PIP) data are shown in Fig. 5. The specific properties of these two resolution volumes are listed in Table 1. The EDOP cross section (Fig. 3) shows that the feature was relatively shallow, extending at the time of measurement down to ~6 km. The measured X-band radar reflectivities near the sample volumes of the DC-8 were in the +10 to +15 dBZe range, agreeing reasonably well with the calculated Ku-band values using the low bulk density assumption. We note that the change in N(D) from 1528:00 to 1528:35 resulted in a 10-dB increase in dBZe, while Vd increased by only ~10%.

Fig. 5.
Fig. 5.

Particle size distributions measured by the 2DS and PIP on the NASA DC-8 at the point shown by the arrow in Fig. 2 at (a) 1528:00 and (b) 1528:35 UTC.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

Table 1.

Derived properties of the PSDs in Fig. 5 being used in the case study.

Table 1.

For the PSDs in Fig. 5 we calculate and Navg for various combinations of synthetically derived measurements Ze and Vd from several frequency combinations (W, Ka, and Ku bands). To test the sensitivity of the algorithms to uncertainties in Vd, the standard deviation of Vd (σV) is incremented from 2 to 20 cm s−1. Results are listed in Table 2 for situations in which we neglect the forward model error (FME = 0) and in which we allow the FME to influence and Navg (FME = 1). Results are presented for the low bulk density ice crystal assumption. Overall, we find that neglecting FME returns uncertainties in P that are in line with expectations from multiple-frequency and multiple-parameter retrieval algorithms such as those applied to dual-radar-frequency GPM data using Ka and Ku bands (e.g., Iguchi et al. 2010). We also find that none of the algorithms applied to either volume can return or, in other words, have I > 0 for A even when FME = 0. In other words, to retrieve A, even under the best of circumstances, some amount of averaging is required. In Table 2, we present Navg, or the number of like samples that would be required to bring the A uncertainty to 5%.

Table 2.

The I, and Navg results for the PSDs in Fig. 5 from the case study. FME = 0 denotes that forward model error was neglected, and FME = 1 indicates that forward model error was allowed to modify the measurement uncertainty. The notation is as described in the text.

Table 2.

Another overall finding evident from Table 2 is that accounting for FME in these volumes significantly influences and Navg. Even for P, I = 0 when FME is fully accounted for and several hundred samples are required to bring the uncertainty in P to 5%. Similarly for A, Navg increases by a factor of ~20 when we fully account for the uncertainty in backscatter cross section and Doppler velocity due to uncertainty in the microphysical properties of the ice crystals as represented in the M-D power law.

A more detailed look at Table 2 shows that including Vd in the retrieval algorithms tends to lower the uncertainties in both quantities, yet the most benefit from Vd is found for very small σV. While no Doppler radar has yet flown in space, the engineering specification for the uncertainty in Vd for the W-band radar to be flown in the upcoming joint European Space Agency–Japan Aerospace Exploration Agency Earth Clouds, Aerosols and Radiation Explorer (EarthCARE) mission (Illingworth et al. 2015) is on the order of 1 m s−1. A requirement for the NASA Advanced Composition Explorer (ACE) mission that is planning a dual-frequency Doppler radar at Ka and W bands is ~20 cm s−1 (S. Tanelli 2015, personal communication). We find that, at least for retrieving P and A in extended tropical anvils, only marginal benefit is obtained for this σV. Of course, that is not to say that Vd would not be beneficial in other cloud types and for other purposes, such as retrieving air motions, and for inferring precipitation phase and even the bulk density of precipitating particles.

For these case studies, the algorithm that uses the most information, ZVaw, tends to give the best overall results, and dual-frequency algorithms perform better than single-frequency algorithms. While the two dual-frequency algorithms, Zaw and Zua, tend to perform similarly, Zaw performs better in retrieving A when accounting for forward model error.

b. Overall statistics

We use approximately 17 000 5-s averages collected in the extended anvils of TC4 to conduct the summary analysis. Beginning with P, we examine the retrieval uncertainty represented as a fractional quantity, , where the double overbar distinguishes the uncertainty from the prior variance of P. Figure 6 shows results for the low bulk density assumptions and Fig. 7 shows results for the high bulk density assumption. We first focus on Fig. 6. Neglecting the FME initially, there is a distinctly lower uncertainty when a very precise (and technologically unrealistic) Vd is assumed. The error in that case is nearly half the value of that when more realistic Vd uncertainties are assumed. Interestingly, Fig. 6a also suggests that very accurate Vd in a single-frequency algorithm nearly negates the advantage of dual-frequency retrievals with Vd. However, when a more realistic Vd uncertainty is assumed, the advantage to using Vd vanishes and is nearly identical for all algorithms.

Fig. 6.
Fig. 6.

Precipitation-rate median uncertainty statistics for low-density ice created by applying the algorithms listed in the legend and described in the appendix and main text to 17 000 5-s precipitating-anvil measurements. (a) The when neglecting forward model error, (b) when accounting for forward model error, and (c) Navg to bring to 5% when accounting for forward model error. The vertical bars show the 25th and 75th percentiles of the ZVw algorithm to illustrate the width of the distributions in each P bin.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for high-density ice particles.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

When accounting for FME in Fig. 6b, however, is always larger than a factor of 2 regardless of algorithm. Any advantage found from Vd, regardless of the accuracy of Vd, is lost and instead of decreasing with increasing P, increases to ~200% for P > 10 mm day−1 where it levels off and decreases slightly for higher P. We also find that the Zua and ZVw algorithms have too few cases with I > 0 to compute a reasonable average. Indeed, only fewer than 10% of the actual occurrences in each P bin, regardless of algorithm, have I > 0 when accounting for FME. Therefore, when taking into account reasonable FME, individual retrievals of P are likely to contain little actual information for low bulk density ice. Averaging several hundred like volumes, however, does recover information (Fig. 6c). We also find that single-frequency Doppler algorithms (ZVw in this case) require about a factor-of-2 more instances to attain a 5% uncertainty than are needed for dual-frequency algorithms.

For comparison, results for a high bulk density ice crystal assumption are shown in Fig. 7. We find substantial sensitivity in to the bulk density assumption. The value of decreases substantially when the ice crystals become denser. While the advantages and disadvantages identified in Fig. 6 still hold, we see substantial spread in the skills of various algorithms for FME = 0. The Zua algorithm that mimics the frequency combination on GPM shows that stays consistently in the 50% range as a function of P for FME = 1. However, from the Zaw algorithm shows consistently more skill at the higher precipitation rates. We also find (Fig. 7c) that very little averaging is required to recover P of 5% when the ice crystals are known to have higher bulk density.

For the uncertainty in the aggregation rate, (Figs. 8 and 9), the results are similar to the case study. There is generally not enough information in the measurements for individual retrievals of A to have I > 0 and averaging is required. The amount of averaging decreases substantially for high bulk density ice. When we neglect FME, the results are much like what we found for except that the single-frequency Doppler algorithm ZVw demonstrates less skill at retrieving A. Overall, approximately 103 like volumes must be averaged to bring to 5% for FME = 0 although for high-density ice, we find a significant spread in skill as the A increases. When we allow for FME to contribute to uncertainty, that number tends to increase by about a factor of ~20. We find a significant divergence among the algorithms when allowing for FME. There is a clear advantage to dual frequency and there seems to be some advantage to W band over Ku band as compared with the Zau algorithm. This result is especially true for higher-density ice and larger A. We reason that this is so because the non-Rayleigh effects of W band occur for smaller sizes and the PSDs are better resolved, and this is accentuated for higher-density ice crystals. While P depends on the larger end of the N(D), A requires reasonable representation of not only the particles doing the collecting but those that are being collected, thus the advantage of the combination of W and Ka bands for these precipitating ice clouds.

Fig. 8.
Fig. 8.

As in Fig. 6c, but for the aggregation rate A when forward model error (a) is and (b) is not accounted for.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

Fig. 9.
Fig. 9.

As in Fig. 8, but for high-density ice.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

4. Further discussion and conclusions

We adopt the view that any instantaneous characterization of a cloud—be it from a model, a measurement in situ, or from remote sensing—is really just a snapshot of time-dependent processes. Clouds, regardless of type, are in a constant state of change. In this study we have investigated the degree to which radar remote sensing at vertical incidence can provide insights into one of the processes that cause precipitation to form and evolve in extended tropical anvil clouds. The findings of this study can be understood by considering the terms in the Jacobian matrices, and , that contain the first derivatives of the measurables to either the state parameter terms for (P and A in our case) or the forward model assumptions (the M-D power law) as shown in Fig. 10 for low-density ice. The following are the principal findings of this study, with context provided by Fig. 10:

  1. Retrieval of A in extended tropical anvils such as those measured during TC4 is challenging when using vertically pointing multifrequency radar. Averaging would allow for information to be derived from the measurements if enough data were available to create such averages and if assumptions such as the backscatter cross sections could be considered unbiased.

    This result can be understood by comparing the magnitudes of for P and A as depicted in Fig. 10. The challenge in retrieving A is obvious since Z is on the order of 10 db less sensitive to A than to P. Significant averaging of the data is required to bring this weak signal out of the noise because of measurement error and other uncertainties.

  2. Accounting for uncertainties in forward models significantly impacts retrievals of both P and A for all combinations of frequencies with and without Vd. For instance, the uncertainty in P degrades from approximately 30% to 40% when neglecting FME to approximately 100% to 200% when allowing for the backscatter cross-section uncertainty due to M-D for low-density ice crystals.

    This finding is a direct result of the large sensitivity of Z to the M-D assumption as shown in Fig. 10. We find that the sensitivity of Z to am is on the order of 6.5 dB. In other words, Z is almost as sensitive to the uncertainty in M-D as it is to the precipitation rate and Z is much more sensitive to uncertainty in am than Z is sensitive to A. This basic result extends to Vd. While Vd is sensitive to A and P, this sensitivity is completely dominated by the sensitivity of Vd to am.

  3. Doppler velocity with realistic measurement uncertainty adds no quantifiable advantage when retrieving P and A. This is true whether or not we account for FME in the retrieval.

Fig. 10.
Fig. 10.

Frequency distributions of sensitivity terms that are found in the Jacobian matrices and . These histograms were created by calculating the change of the measurable quantities (Z and V for W band in this example) with respect to the change of the quantities of interest (P, A, or am in these examples). These were then multiplied by the scaled standard deviation of P, A, or am of the TC4 dataset as appropriate. For the sensitivity of Z with respect to P, e.g., we calculate and plot the histogram of that quantity. For the V sensitivity, we plot the histograms of , where P is replaced with A or am as appropriate. (a) Sensitivity of Z to precipitation rate, (b) sensitivity of Z to aggregation rate, (c) sensitivity of Z to M-D prefactor, (d) sensitivity of V to aggregation rate, (e) sensitivity of V to precipitation rate, and (f) sensitivity of V to M-D prefactor.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0083.1

This finding can be understood by considering the magnitude of the Vd sensitivity to P and A in Fig. 10. At of 2 cm s−1 the uncertainty in Vd was smaller than the sensitivity and Vd was able to provide information to the microphysical retrieval. When = 20 cm s−1, the uncertainty is much larger than the Jacobian terms, and V provides no advantage to the microphysical retrieval except to provide information on air motions or perhaps hydrometeor type diagnostics—topics we have not considered in this study.

An important additional issue regarding point 2 is that the result depended upon our assumed uncertainty in the M-D power-law parameters, and these values of uncertainty are a function of the M-D power law. The comparisons of and for low and high bulk density ice suggest that skill increases substantially for both quantities as bulk density increases. As discussed, FME = , and the covariance in the power-law parameters is not well known. On the basis of comparisons of studies in which the power laws were derived for different cases, we assumed that and . However, were we to have some way of knowing how to reduce these variances we could significantly reduce the FME. Not knowing how to parameterize the microphysical ice crystal habits dominates the retrieval uncertainty of both P and A with the increasing from a few tens of a percent to factors ranging between 2 and 4 for low bulk density ice. These results demonstrate that if knowledge of microphysical properties and processes in ice-phase clouds is deemed important, then not only is a dual-frequency radar measurement system necessary, but some means of identifying the important distinguishing characteristics of ice crystal populations like M-D relationships is also required.

Acknowledgments

TC4 data were acquired from https://espo.nasa.gov. We acknowledge the efforts of the engineers and scientists that participated in the TC4 project. In particular, we thank Drs. P. Lawson, G. M. Heymsfield, and A. J. Heymsfield for use of their data in this study. Support was provided under NASA Grants NNX10AM42G, NNX15AK17G, and NNX13A169G.

APPENDIX

Equations for ZVw, Zaw, and Zua Measurement Combinations

Expressions that vary because of measurement combinations are listed here. For ZVw,
eq2
eq3
eq4
eq5
  • For Zaw,

eq6
eq7
eq8
eq9
  • For Zua,

eq10
eq11
eq12
eq13

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  • Adams-Selin, R. D., S. C. van den Heever, and R. H. Johnson, 2013: Impact of graupel parameterization schemes on idealized bow echo simulations. Mon. Wea. Rev., 141, 12411262, doi:10.1175/MWR-D-12-00064.1.

    • Crossref
    • Search Google Scholar
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  • Austin, R. T., and G. L. Stephens, 2001: Retrieval of stratus cloud microphysical parameters using millimeter-wave radar and visible optical depth in preparation for CloudSat: 1. Algorithm formulation. J. Geophys. Res., 106, 28 23328 242, doi:10.1029/2000JD000293.

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  • Berry, E., and G. G. Mace, 2014: Cloud properties and radiative effects derived from A-Train satellite data in Southeast Asia. J. Geophys. Res. Atmos., 119, 94929508, doi:10.1002/2014JD021458.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Christensen, M. W., G. L. Stephens, and M. D. Lebsock, 2013: Exposing biases in retrieved low cloud properties from CloudSat: A guide for evaluating observations and climate data. J. Geophys. Res. Atmos., 118, 12 12012 131, doi:10.1002/2013JD020224.

    • Crossref
    • Search Google Scholar
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  • Cooper, S. J., T. S. L’Ecuyer, P. Gabriel, A. J. Baran, and G. L. Stephens, 2007: Performance assessment of a five-channel estimation-based ice cloud retrieval scheme for use over the global oceans. J. Geophys. Res., 112, D04207, doi:10.1029/2006JD007122.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Delanoe, J., and R. J. Hogan, 2008: A variational scheme for retrieving ice cloud properties from combined radar, lidar, and infrared radiometer. J. Geophys. Res., 113, D07204, doi:10.1029/2007JD009000.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Deng, M., G. G. Mace, Z. Wang, and H. Okamoto, 2010: Tropical Composition, Cloud and Climate Coupling Experiment validation for cirrus cloud profiling retrieval using CloudSat radar and CALIPSO lidar. J. Geophys. Res., 115, D00J15, doi:10.1029/2009JD013104.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Field, P. R., A. J. Heymsfield, and A. Bansemer, 2006: A test of ice self-collection kernels using aircraft data. J. Atmos. Sci., 63, 651666, doi:10.1175/JAS3653.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hammonds, K. D., G. G. Mace, and S. Y. Matrosov, 2014: Characterizing the radar backscatter-cross-section sensitivities of ice-phase hydrometeor size distributions via a simple scaling of the Clausius–Mossotti factor. J. Appl. Meteor. Climatol., 53, 27612774, doi:10.1175/JAMC-D-13-0280.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hashino, T., and G. J. Tripoli, 2011: The Spectral Ice Habit Prediction System (SHIPS). Part III: Description of the ice particle model and the habit-dependent aggregation model. J. Atmos. Sci., 68, 11251141, doi:10.1175/2011JAS3666.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Henderson, D. S., T. L’Ecuyer, G. Stephens, P. Partain, and M. Sekiguchi, 2013: A multi-sensor perspective on the radiative impacts of clouds and aerosols. J. Appl. Meteor. Climatol., 52, 853871, doi:10.1175/JAMC-D-12-025.1.

    • Crossref
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  • Fig. 1.

    Frequency distributions of various observed and derived quantities from the TC4 dataset that we analyzed. Included here are ~24 000 5-s averages. Shown are (a) temperature, (b) pressure, (c) ice water content, (d) number concentration, (e) effective radius, (f) derived Ku-band dBZe, (g) derived Ku-band V, (h) precipitation rate, (i) aggregation rate derived using Eq. (1), and (j) the correlation between the precipitation rate and the aggregation rate in the ice cloud data collected during TC4.

  • Fig. 2.

    Visible GOES image that overlays a short segment of the flight track of the DC-8 on 17 Jul 2007. The arrow marks the location of the case study discussed in the text. The image was obtained through the courtesy of Dr. P. Minnis and L. Nguyen, NASA Langley Research Center.

  • Fig. 3.

    The X-band radar reflectivity cross section collected by the EDOP instrument on the ER-2 between 1526 and 1532 UTC 17 Jul 2007 along the flight track depicted in Fig. 2. The period we analyze is between 1528 and 1529 UTC. The linear feature at 9 km is the reflection of the NASA DC-8 in the EDOP radar beam.

  • Fig. 4.

    Particle size spectra measured by the (a) PIP and (b) 2DS probes on the NASA DC-8 during the period represented in the radar cross section in Fig. 3.

  • Fig. 5.

    Particle size distributions measured by the 2DS and PIP on the NASA DC-8 at the point shown by the arrow in Fig. 2 at (a) 1528:00 and (b) 1528:35 UTC.

  • Fig. 6.

    Precipitation-rate median uncertainty statistics for low-density ice created by applying the algorithms listed in the legend and described in the appendix and main text to 17 000 5-s precipitating-anvil measurements. (a) The when neglecting forward model error, (b) when accounting for forward model error, and (c) Navg to bring to 5% when accounting for forward model error. The vertical bars show the 25th and 75th percentiles of the ZVw algorithm to illustrate the width of the distributions in each P bin.

  • Fig. 7.

    As in Fig. 6, but for high-density ice particles.