## 1. Introduction

[2] The April–May 2010 eruption of the Eyjafjallajökull volcano in Iceland happened during a period characterized by high atmospheric pressures over Northern Europe and the North Atlantic [*Petersen*, 2010]. This created the conditions for long-range transport of the volcanic ash plumes to Northern, Central and Southern Europe and to the Atlantic Ocean, with atmospheric residence times of the order of many days. A major air traffic disruption happened as a consequence of this [*Gertisser*, 2010]. Observations carried out during that period are of great interest for the validation and improvement of numerical dispersion models used for the prediction of volcanic ash concentrations. They have moreover proven to be a useful tool in the nearly real-time decision-making process concerning the opening or closure of airspace. Lidar has been one of the most successful remote-sensing instruments for locating the plumes: several ground-based stations have carried out observations [*Ansmann et al.*, 2010; *Flentje et al.*, 2010; *Pietruczuk et al.*, 2010; *Mona et al.*, 2011; R. J. Hogan et al., Lidar and Sun photometer retrievals of ash particle size and mass concentration from the Eyjafjallajökull volcano, manuscript in preparation, 2011] and moreover observations of volcanic ash have also been obtained by airborne lidar [*Schumann et al.*, 2011; *Marenco et al.*, 2011]. In this paper we present an inversion method useful for separately retrieving the contribution of two aerosols, using a dual-polarization backscatter lidar operating at an ultraviolet wavelength.

[3] Several papers have appeared in the last decades discussing the solution to the lidar equation for an elastic-backscattering system [see, e.g., *Fernald et al.*, 1972; *Fernald*, 1984; *Klett*, 1985; *Kovalev*, 1993; *Takamura et al.*, 1994; *Marenco et al.*, 1997; Hogan et al., manuscript in preparation, 2011]. This amount of literature is justified by the fact that the equation is underdetermined, i.e. for any measured lidar profile infinite mathematical solutions are possible, whereas only one will have to be retained as the best estimate of the state of the atmosphere. In the assumption of single elastic scattering, monochromatic radiative transfer applied to lidar returns leaves us with backscatter and extinction tied together in an integral equation: with two unknowns and only one equation, insufficient information is available for finding a deterministic solution.

[4] In the absence of a definite physical principle to be used for the purpose of constraining the solution, since *Fernald et al.* [1972] the scientific community has often resorted to the assumption that the lidar ratio (extinction-to-backscatter ratio) is constant. This assumption could in principle be relaxed as in the work by *Klett* [1985] provided that the vertical profile of the lidar ratio is known, but if one is left with an elastic-backscatter lidar alone there is of course no simple way of establishing that profile. Note that on one side the assumption of a constant lidar ratio is made because there are often no other practical possibilities for inverting the lidar equation, and on the other side this seems a reasonable assumption for an aerosol layer from a given source and for which microphysical properties such as composition and particle size/shape may be assumed homogeneous.

[5] Even in the assumption of a constant lidar ratio, the lidar retrieval remains underdetermined since two parameters need be set. One of them is the value of the lidar ratio itself, and the other is the lidar calibration constant; the latter is usually determined by comparison of the signal with what is expected from a known target, most often Rayleigh scattering by an aerosol-free layer. This framework is often referred to as the Fernald-Klett approach [*Fernald*, 1984; *Klett*, 1985]. The lidar constant is usually not determined *a priori* in the laboratory, for two reasons: the first being that the laser output and receiver efficiency may vary (i.e. the lidar constant is so-called because of its independence upon range, but it still may vary with time), and the second, more fundamental, is that as shown by *Fernald* [1984] the mathematical solution is unstable for a near-range calibration, so that the reference range used for calibration must be taken at the far end of the lidar profile. In the latter case, as we proceed inward from the reference range the solution becomes more and more independent from the calibration assumptions.

[6] Note that the Fernald-Klett approach is not the only one that is based on the assumption of a constant lidar ratio. For an elastic-backscattering system two independent constrains to the lidar equation must still be given (this being required mathematically), and several approaches have appeared in the literature. *Kovalev* [1993], for instance, proposed a solution based on the knowledge of the lidar ratio and of the aerosol optical depth; *Di Girolamo et al.* [1994] used a solution based on *two* aerosol-free calibration ranges, one below and one above the aerosol layer; and *Takamura et al.* [1994]; *Marenco et al.* [1997] proposed a solution based on an independent optical depth measurement and an aerosol-free calibration range.

[7] All of the solutions to the lidar equation mentioned above are based on a two-component atmosphere, where the components are the molecular free-atmosphere (Rayleigh scattering) and the aerosols, respectively; the first component being assumed to be fully known from a model or a radiosonde profile, and the second component being assumed to have a constant lidar ratio. If distinct aerosol layers are observed at the same time, which are believed to have different properties, a two-component scheme can still be used by dividing up the lidar profiles into separate sections. This approach will however be insufficient for cases when two different aerosol types co-exist in a same atmospheric layer: if knowing in advance how the proportion of the two aerosols varies with range, one could use one of the schemes that assumes a variable lidar ratio, but this cannot be expected to work if that proportion has to be determined from the lidar observations.

[8] Other approaches to the problem exist, namely using systems such as Raman lidar [*Ferrare et al.*, 1998; *Tesche et al.*, 2009] or high spectral resolution lidar [*Shipley et al.*, 1983; *Rogers et al.*, 2009], which permit one to independently evaluate backscattering and extinction. The study of the simple elastic-backscattering lidar remains relevant, however, because many systems not featuring these additional detection channels can still provide useful information. Raman lidars are moreover strongly limited by the amount of daylight, so that some widely used systems can only be fully exploited at nighttime, whereas high resolution lidar requires hyperspectral measurements which may complicate the experimental apparatus. On the other hand, elastic-backscatter lidar (with or without depolarization) has the advantages of simplicity and lower cost, two features that encourage the development of monitoring networks.

[9] The addition of a depolarization channel has proven in many cases to be the most reliable way to distinguish volcanic ash from other, more common, aerosols, and a strong depolarization is considered a good tracer for ash [*Hoffmann et al.*, 2010; *Ansmann et al.*, 2011] and dust [*Freudenthaler et al.*, 2009; *Tesche et al.*, 2009]. In particular, the ash layers have sometimes been observed at a sufficiently high altitude, well above the boundary layer (BL) aerosol, and thus could be treated separately. At other times, however, the ash has reached the BL top, and as a consequence has mixed with the BL aerosol. In such observations, the lidar signal can be inverted into quantitative measurements (backscatter and extinction coefficients), and the depolarization signal can be separately used as a qualitative indicator to distinguish the ash from the BL aerosol background; but working within a two-component atmosphere framework it may be difficult in some scenes to quantify the contribution of both aerosol types separately.

[10] An important advance has been brought by *Sugimoto et al.* [2003] and *Tesche et al.* [2009], where an algorithm is described that permits separately quantifying the contribution of two externally mixed aerosols, once total backscatter and extinction profiles have been computed. This algorithm is optimal, for instance, for use with a Raman lidar, where the lidar ratio assumption is not needed, whereas for an elastic-backscatter lidar it relies on assuming that the two aerosol types have an identical lidar ratio.

[11] The present paper originates from the remark that if depolarization is good at identifying the contribution of ash rather than locally produced aerosols, then the depolarization information should be incorporated into the aerosol inversion scheme. As a matter of fact, having a depolarization lidar channel permits having an additional and independent equation, and thus permits including additional unknowns to the lidar problem. We take this opportunity for testing a three-component atmosphere reduction scheme, where the three components, assumed externally mixed, will be respectively: Rayleigh scattering, volcanic ash (depolarizing), and BL aerosols (assumed non-depolarizing).

[12] Section 2 describes the origin of the data; section 3 recalls the preliminary standard depolarization lidar processing that is applied throughout this article; section 4 treats selected lidar profiles, where ash is physically separated from other aerosols, in a two-component atmosphere scheme, for the purpose of characterizing volcanic ash in terms of lidar ratio and depolarization ratio; section 5 illustrates the mathematical details of the three-component atmosphere approach for externally mixed aerosols; and section 6 presents its application and results. Finally, section 7 presents a discussion of the observations and some conclusions.