1 ANNEX (doi: 10.4401/ag-6762) VOLCANIC ASH CLOUD OBSERVATION USING GROUND-BASED KA-BAND RADAR AND NEAR- INFRARED LIDAR CEILOMETER DURING THE EYJAFJALLAJÖKULL ERUPTION Marzano, F. S., et al. Volcanic ash plumes are formed during explosive volcanic eruptions. After advection over several thousands of kilometers, volcanic ash particles are highly fragmented, dispersed and aged with micron- sized sorting. This Annex describes the ash microphysical modeling and the simulated radar and lidar signatures. A.1. VOLCANIC ASH MICROPHYSICAL MODELING In this work all particles are assumed to be spheroids, i.e. ellipsoids with circular symmetry. This shape is, on the one hand, quite general and, on the other hand, simplifies the numerical treatment of scattering. Similarly to what described in Marzano et al. (2012), we set up a microphysical model of ash particle distributions. This means to specify: a) particle size distribution (PSD); b) particle ellipsoidal axial ratio (AR); c) particle orientation distribution (POD); d) particle density and dielectric constant. Ash PSDs are typically modeled through a normalized Gamma or Weibull size distribution [e.g., Marzano et al., 2006; 2010]. In case of a multi- mode size distribution, it is always possible to suppose more than one analytical PSD, characterized by different mean sizes and total number of particles. In this study, we have adopted the scaled Gamma (SG) PSD as a general model for all particles. If De is the diameter of a spherical volume-equivalent particle, the scaled-Gamma PSD Np [mm-1m-3] for a generic class of ash particles p can be written as:                   np e np p D D np e npep e D D NDN  )( (1) where the “intercept” parameter Nnp and the “slope” parameter np in a logarithmic plane are related to the “shape” parameter p and to the particle density p through: 1 4 6 10 4 4 6      pnp pnpp npp np D C N p    ; )( (2) with Cp the mass concentration [g/m3] and Dnp [mm] the number-weighted mean diameter. PSD in (1) is completely specified by the three parameters p, Dnp and Cp (with a particle density p between 1 and 2.5 g/cm3). If particles are sphere equivalent, their mass is mp=p(/6)De3. The number of ash classes with respect to their average size is set to 5 as follows: i) very fine ash (VA) with mean equivalent diameters uniformly distributed between 2-3-23 m (0.125-8 m); ii) fine ash (FA) between 23-26 m (8-64 m); iii) coarse ash (CA) between 26-29 m (64-512 m); iv) small lapilli (SL) between 29-212 m (0.512-4.096 mm); and finally v) large lapilli (LL) between 212-215 m (4.096-32.768 mm). With respect to previous studies [Marzano et al., 2012], we have added the class of large lapilli and very fine ash to take into account ballistic fallout as well as dispersed small particles in order to deal with a fairly general observation scenario (closer to and farther from the volcanic vent). Moreover, uniform distribution around the mean size has been assumed instead of Gaussian probability. Each diameter class may be subdivided with respect to other main parameters, e.g. the ash concentration and orientation angle [Marzano et al., 2012]. The following sub-sets have been introduced: i) 4 classes of ash concentration, i.e.: very small (VC), small (SC), moderate (MC), and intense (IC) concentrations. These sub-sets are eventually compacted into 1 class uniformly going from 10-6 g/m3 up to 10 g/m3; 2 ii) 5 classes of particle orientation with Gaussian POD characterized by a mean canting angle m, i.e.: tumbling with m=30° (TO.1), tumbling with m=45° (TO.2), tumbling with m=60° (TO.3), oblate m=0° (OO); prolate m=90° (PO). These 5 sub-sets are reducible to 3; iii) 2 classes of AR models, i.e.: ratio basaltic- andesitic (RB) and ratio rhyolitic (RR), even though we have here selected only the RB case; iv) 3 classes of VA, FA, CA spherical particles (SP) with all 5 concentrations. Particle dielectric constant models have been derived from literature; in particular, their dependence on SiO2 weight WSiO2 and relative humidity fraction (RH) is derived from the available data [e.g., Adams et al., 1996; Shettle and Fenn, 1971]. Ash classes can exhibit a mixed-phased permittivity due to liquid and ice water in the atmosphere. Particle classes with a mixed refractive index have been also introduced [Marzano et al., 2010]. In such a case, we have limited our attention to particles smaller than lapilli, so that 3 new mixed-phase classes have been labeled by VAm, FAm, CAm for all 5 concentrations. Liquid and ice water particle classes (e.g., cloud droplets, ice crystals, snowflakes) can be added to generalize the scenario. Considering all combinations, the nominal number of classes is 233 (5·4·5·2+15+15+3) which can be effectively reduced to 20 (5·1·3·1+3+2), as we will discuss later on in this work. A.2. VOLCANIC ASH PARTICLE SCATTERING AND EXTINCTION The conditions allowing for the Rayleigh scattering regime, depending on particle refraction index and the ratio between the particle diameter and the incident wavelength , are not always satisfied by the characteristics of the problem under discussion. Simulated polarimetric radar observables for spheroidal particles can be obtained by using the T-matrix numerical method [Mishchenko et al., 1996], implemented through the Hydrometeor Ash Particle Ensemble Scattering Simulator (HAPESS) [Marzano et al., 2012]. In order to simulate the optical polarimetric signatures of volcanic particles we have included within HAPESS the geometrical optical limits which cannot be evaluated by T-matrix code due to numerical convergence problems [Min et al, 2003; Gasteiger et al., 2011]. In the next sub-sections we will introduce observables related to backscattering amplitude since those related to forward scattering phase are beyond the scopes of this paper [see Marzano et al., 2012]. Polarimetric radar observables Using horizontal (h) and vertical (v) polarization states, copolar (Zhh, Zv) and crosspolar (Zvh, Zvh) radar reflectivity factors can be expressed in terms of ash PSD. If  is the wavelength associated to a given frequency f, Kp is the ash particle radar dielectric factor and  is the particle canting angle, Sxy(b) are the backscattering components at x (x=h,v for the receiving mode) and y (y=h,v for the transmitting mode) polarization of the complex scattering matrix S, we can define:     ),(sin)( )(),,()( )( )(         e b xy p ep epe b xy p xy DS K ddDp DNfDS K fZ 4 8 25 4 2 0 2 025 4 (3) In (3) angle brackets stands for average over PSD Np and POD pp, assuming a uniform azimuthally symmetric POD [Marzano et al, 2012]. Note that: i) Zhv=Zvh for cross-polarization reciprocity; ii) the canting angle  is defined in the polarization plane with respect to its vertical polarization unit-vector; iii) if Zxy is expressed in [mm6m-3], then ZXY=10log10(Zxy) is conventionally expressed in dBZ. 3 The derived differential reflectivity Zdr, which is the ratio of reflectivity at the two polarization states, and the linear depolarization ratio Ldr are defined as: )( )( )( fZ fZ fZ vv hh dr  ; )( )( )( fZ fZ fL hh hv dr  (4) where Zdr and Ldr are unitless and usually expressed in dB. The specific power attenuation hh (vv) at polarization h (v) can be obtained from:   ),,(Im)( )( fDSf efxyxy  42 (5) The apex “f” of S indicates its forward components. Note that, if xy is in [km-1], the above quantities are usually defined in dB within the radar community XY=4.343xy. Note that, if the ellipsoidal axial ratio shows an intrinsic variability, the particle non-sphericity can significantly affect the differential reflectivity Zdr. The Ldr is not dependent on the particle concentration and it is governed by the inverse of reflectivity- weigthed axial ratio within the Rayleigh scattering. The Ldr signature will then increase with increasing canting angle and axial ratio variability. Polarimetric lidar observables The lidar backscattering coefficients hh, vv and vh at horizontal (h) and vertical (v) polarization states can be expressed similarly to the corresponding radar coefficients. Indeed, from (4) their definition can be written as:     ),(sin)( )(),,()( )( )(    e b xyep epe b xyxy DSddDp DNDS 4 4 2 00 (6) where x=h,v again stands for the receiving mode and y=h,v for the transmitting mode polarization. Note that xy is usually expressed in [km-1sr-1] so that XY=10log10(xy) is here conventionally expressed in dB (in analogy to dBZ for ZXY). The same applies to the specific attenuation or extinction coefficient xy, which similarly to (5) is expressed in [km-1] and is defined as:   ),,(4Im2)( )(  efxyxy DS (7) Analogously to (7), if xy is in [km-1], XY=4.343xy is conventionally expressed in dB. Similarly to Zdr and Ldr in (4), the unitless lidar linear co-polarization ratio co and cross-polarization ratio cr can be defined by: )()( )()( )(    hhvv hhvv co    ; )( )( )(    hh vh cr  (8) Radar and lidar polarimetric signatures In order to discuss combined polarimetric signatures, we have considered existing radar and lidar instrument setups, e.g. zenith-pointing Ka- band radars and colocated lidars operating at 355 nm, 532 nm and 1024 nm wavelength [e.g., Madonna et al., 2010]. HAPESS outputs have been validated for volcanic ash with numerical simulations available in literature [e.g. Wiegner et al., 2009]. Figure 1a: Correlation between zenith Ka-band polarimetric radar observables in terms of copolar reflectivity Zhh(36GHz), linear depolarization ratio Ldr(36GHz), and specific attenuation hh(36GHz) for FA size class, using all ash concentrations classes (VC, SC, MC and IC) and for different particle orientations (OO, PO, SP and TO.2) with basaltic axial ratio model. -80 -70 -60 -50 -40 -30 -20 -10 0 10 -10 3 -10 2 -10 1 Ka: Fine Ash Z hh [dBZ] L d r h v [ d B ] -30 -20 -10 0 10 20 30 40 50 60 -10 3 -10 2 -10 1 Ka: Small Lapilli Z hh [dBZ] L d r h v [ d B ] -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 10 -8 10 -6 10 -4 10 -2 Ka: Fine Ash Zhh [dBZ]  h h [ d B /k m ] FA-OO FA-PO FA-SP FA-TO -40 -30 -20 -10 0 10 20 30 40 50 60 10 -6 10 -4 10 -2 10 0 Ka:Small Lapilli Zhh [dBZ]  h h [ d B /k m ] SL-OO SL-PO SL-SP SL-TO 4 This section illustrates examples of ash sub-classes, limited here for brevity to FA signatures in terms of Ka-band and near infrared (NIR) polarimetric observables (which are used for the combined system introduced later on). Figure 1a shows how Zhh(36GHz) correlates to linear depolarization Ldr(36GHz) and specific power attenuation ahh(36GHz). Note that the particle orientation has an impact on Ldr(36GHz), with the OO (mainly horizontal with respect to the ground) providing values of about -40 dB. Specific attenuation is pretty low for FA (lower than 0.15 dB/km), but larger of 1 dB/km for larger particles. In figure 1a we have also added the behavior of spherical particles (SP) where we expect theoretically Zhv=0 and Ldr=0. Figure 1b: Same as in the Figure 1a panels, but for the zenith lidar copolar backscatter coefficient hh(1064nm), linear cross- polarization ratio δcr(1064nm), and extinction coefficient hh(1064nm) with relative humidity ranging between 30 and 70%. In Figure 1b panels show the correlation between zenith-pointing NIR backscatter coefficient hh(λ), linear cross-polarization ratio cr(λ), and extinction coefficient ahh(λ) for fine ash. 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