Modeling the depolarization of space-borne lidar signals

Kaori Sato; Hajime Okamoto; Hiroshi Ishimoto

doi:10.1364/OE.27.00A117

1. Introduction and physical model

The linear depolarization ratio δ from cirrus clouds mainly reflects the single scattering properties of non-spherical ice particles [1,2], due in part to the small cloud optical thickness τ at the lidar wavelength. If the particle shape and orientation can be specified, δ can be calculated by the recently developed physical optics method [3–6]. For water clouds, the cloud τ is large, and δ is affected by multiple scattering events. The δ observed by cloud-aerosol lidar with orthogonal polarization (CALIOP) [7] at 532 nm can exceed 40% due to its large footprint [8], which is larger than δ typically measured by ground-based lidars with small fields of view (FOVs). Besides the FOV, vertical cloud microphysical profiles have strong influences on the observed δ profiles [1]. A quantitative analysis of δ, in addition to the backscattering coefficient, is expected to lead to better characterization of water cloud and super-cooled microphysics, which currently possesses one of the largest uncertainties in future climate predictions [9].

To investigate the information content of δ, a double-scattering depolarization model [1,10], a triple backscattering model obtained via a time-dependent transfer model [11], and a depolarization parameter model [12] have been developed for application to ground-based lidar and multiple FOV lidar measurements. However, fewer space-borne lidar studies have analyzed δ from water clouds with regard to higher-order scattering. Based on Monte Carlo simulations for vertically homogeneous cloud profiles, ref. [13] investigated the relationship between the layer-integrated backscattering coefficient and the layer-integrated linear depolarization ratio δ_layer.

The recently developed physical model (PM) [14] uses high-order scattering phase functions p_n [15,16] and path integral formulation [17,18] to capture the transition of the time-dependent backscattered irradiance from the single-scattering regime to the multiple-scattering and diffusion regimes, without tracking complete scattering processes. Error analyses indicated a mean relative error of about 15% for the estimated backscattering coefficient by the PM overall, in reference to the Monte Carlo approach of [19] (referred to as MC model). Here, we introduce a vectorized physical model (VPM). Specifically, the PM [14] was extended with a polarization function to create a VPM to analyze the vertical profile of the observed depolarization ratio due to multiple scattering from water clouds by space-borne lidar.

The paper is organized as follows. In Section 2.1, the basis of the PM [14] is provided. In Section 2.2, the n-th order scattering matrix p_n and the mechanisms by which the polarization state is changed according to the number of scattering events, microphysical properties, and FOV are introduced to vectorize the PM. The backscattered irradiance observed parallel and perpendicular to the incident electric field of the transmitted polarized light are estimated for single scattering events and on-beam and off-beam multiple scattering components to derive δ. In Section 3, δ is estimated using the proposed VPM and compared with results from the literature and those obtained using MC simulations. Section 4 provides a summary of our work and concluding remarks.

2. Vectorized physical model

2.1 Overview of the scalar PM

Here, we briefly describe the PM approach. In an xyz Cartesian coordinate system, the cloud top, cloud bottom, and the satellite are located at z = z_1/2 ( = 0), z = z_imax_+1/2, and z = −z_sat, respectively. The lidar onboard the satellite with beam divergence (θ_bd) and FOV (θ_fov) emits an x-polarized laser beam in the positive z direction. The cloud is divided by the scattering mean free path 1/σ_sca,i, where σ_sca,i is the scattering coefficient at cloud layer i (i = 1,2,…,imax). z_i corresponds to the midpoint of the i-th cloud layer (z_i-_1/2 < z ≤ z_i+_1/2). The geometric thickness of layer Δz_i equals l_i = 1/σ_sca,i, corresponding to a unit optical thickness by definition, except at the boundaries of inhomogeneity, where the optical thickness of such a layer τ_i ≤1 because Δz_i ≤ l_i. The total backscattered irradiance I_tot is estimated at discrete time steps of 2t_j (j = 1,2,…, jmax), and t_j = z_{i = j}/c, where c is the speed of light in vacuum. Subscript i = j denotes the cloud layer property corresponding to the layer indexed by an integer equal to j.

The PM decomposes I_tot(t_j) into three components: the single scattering component I_ss(t _j), the on-beam multiple scattering component I_ms,tot,on(t_j), and the off-beam multiple scattering component I_ms,off(z_i, t_j), corresponding to the pulse stretching effect with z and t dependence. Thus, I_tot(t_j) is given by

I_{tot} (t_{j}) = I_{ss} (t_{j}) + I_{ms,tot,on} (t_{j}) + \sum_{i = 1}^{j} I_{ms,off} (z_{i}, t_{j}) .

I_ss is obtained by the single scattering approximation for a layer of finite thickness as [14]

I_{ss} (t_{j}) = C_{s} I_{o} σ_{sca, i = j} p_{ss, i = j} (π) / (4 π) \exp [- 2 τ (z_{i = j - 1 / 2})] [\frac{\exp {- 2 σ_{ext} (z_{i = j}) Δ z_{i = j}} - 1}{- 2 σ_{ext} (z_{i = j}) Δ z_{i = j}}],

where σ_ext(z_i) and p_ss,_i(θ) are the single-scattering extinction coefficient and single-scattering phase function at scattering angle θ, respectively. C_s is a factor that depends on the lidar specifications and range from the source to z_{i = j} [14]. Multiple scattering components I_ms,tot,on and I_ms,off are treated similarly to Eq. (2), but with the introduction of effective extinctions σ_ext,on and σ_ext,off, respectively [14]:

I_{ms,tot,on} (t_{j}) = J_{on} (t_{j}) σ_{sca, i = j} p_{ss, i = j} (π) / (4 π) \exp [- 2 \sum_{i = 1}^{j - 1} (σ_{ext,on} (z_{i}) Δ z_{i})] [\frac{\exp {- 2 σ_{ext,on} (z_{i = j}) Δ z_{i = j}} - 1}{- 2 σ_{ext,on} (z_{i = j}) Δ z_{i = j}}],

\begin{array}{l} I_{ms,off} (z_{i}, t_{j}) = \sum_{k = 1}^{i} I_{ms,off, k} (z_{i}, t_{j}) \\ = \sum_{k = 1}^{i} [C_{s} J_{off} (z_{k}, z_{i}, t_{j}) σ_{sca, i} p_{n = i} (π) / (4 π) \exp (- 2 \sum_{q = k}^{i} σ_{ext,off} (z_{q}, z_{k}, z_{i}, t_{j}) Δ {z^{'}}_{q})] . \end{array}

J_on is the source term for I_ms,tot,on, which results from the ss component scattered at layer 1, as described in Fig. 1. J_off is the source term for I_ms,off, resulting from the ms,on component scattered at layer k (1 ≤ k ≤ i) and incident on z_i at t_j. The exact expressions for J_on and J_off, as well as σ_ext,on and σ_ext,off, are provided in [14], and all are functions of the n-th order scattering phase function p_n(θ) (n = 1, 2,…,N) [15,16]. The scattering angle θ of p_n(θ) is defined with respect to the initial incident direction. Therefore, the incident direction of light after the (n−1)-th scattering event is taken to be in the z direction. Δz_q´ in Eq. (4) is the photon path length in layer q, where Δz_q´>Δz_q due to multiple scattering.

Fig. 1 (a) Relationships among ① the single scattering component I_ss (red), ② the on-beam multiple scattering component I_ms,tot,on (blue), and ③ the off-beam multiple scattering component I_ms,off (green). The source of the ms,on component (J_on) is the ss component scattered out of the initial incident direction of the beam at layer z₁. The sources of the ms,off components (J_off) are basically the ms,on components scattered at previous layers at earlier times. (b) Schematic of layer i, where z_i denotes the z axis value corresponding to the center of the layer. z_i-1/2 and z_{i + 1/2} are the bottom and top boundaries of the layer, respectively.

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2.2 The VPM

The basic concept of the VPM is as follows. A general expression for lidar multiple scattering of N-th order is described in [11] as an N-th order volume integration (dv_N) of the product of the N set of 4 × 4 scattering phase matrices and the transformation matrix for the Stokes parameter for an angle between the (n − 1)-th and n-th scattering planes (L_n′) [11]:

I_{t o t} (t_{j = N}) = I_{o} f \underset{v_{1}}{∭} \dots \underset{v_{N}}{∭} {L^{'}}_{N} p (θ_{N}) \dots {L^{'}}_{3} p (θ_{2}) {L^{'}}_{2} p (θ_{1}) {L^{'}}_{1} d v_{1} \dots d v_{N} .

f is a function of the scattering geometry and scattering coefficients at the scattering volumes [11]. Unlike the PM, the resulting matrix in Eq. (5) is not directly related to the N-th order 4 × 4 scattering phase matrix p_N. In the VPM, multiple scattering processes are modeled by considering one set of transformation matrices between the incident and the N-th scattering plane, in a manner similar to that used for a single scattering framework. In addition to introducing p_N, the VPM accommodates the following two mechanisms that produce δ; (1) depolarization of the incoming wave for the N-th scattering event due to rotation of the scattering plane during multiple scattering processes and (2) geometrical effect which collects photons scattered in near backward directions at the N-th scattering event. To treat such mechanisms, I_tot(t_j _{= N}) is given by:

I_{tot} (t_{j = N}) = I_{o} F ({L^{'}}_{N} p_{N} {L^{'}}_{1}) .

The function F in Eq. (6) includes integration over the backscattering angles and represents the estimation of the geometric effects due to the ms,on and ms,off processes. It also considers the horizontal spread of the photons for the on-beam and off-beam components to determine the depolarization of the incoming wave at the layer of interest as a function of scattering order (subsection 2.2.4). Equation (6) corresponds to an extension of the formulation of the PM in Eqs. (1)-(4).

2.2.1 Introduction of a high-order Mueller matrix

It was noted in [15] that the average value of a polynomial after n scattering events equals the n-th order of the average value of the polynomials after a single scattering event when scattering is cylindrically symmetric. Based on this property, the backscattered irradiance was formulated by using the n-th order scattering phase function p_n rather than tracking the complete scattering processes in [14]. p_n is expressed in the Legendre polynomial expansion form with the l-th order expansion coefficient (2l + 1)(A_l)ⁿ, where A_l is the l-th order expansion coefficient of the single scattering phase function [14]. Hereafter we denote A_l as A_l,₁₁. In this study, we consider polarimetric property of the lidar backscattering by replacing p_n with the n-th order 4 × 4 scattering phase matrix p_n. p_n is given as follows:

p_{n} = \sum_{l = 0}^{\infty} (2 l + 1) B_{l,}_{n} P_{l} (\cos θ),

P_l is the Legendre polynomial of the l-th order. The ij element of B_l_,n is (A_l,ij)ⁿ and A_l,ij corresponds to the expansion coefficient of the single scattering phase matrix. In analogy to single scattering, the relationship between the scattering phase matrix p_n elements and the Mueller scattering matrix S_n elements is given by S_n = σ_scak²p_n/(4π) where k = 2π/λ and λ is the wavelength. The p_n elements that are essential to I_tot are shown in Fig. 2. A transition from an initial anisotropic shape to an isotropic shape occurs as the scattering order increases.

Fig. 2 n-th order scattering phase matrix elements at 532 nm for a liquid particle with an effective radius (r_eff) of 10 µm.

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2.2.2 Formulation of the VPM

Here I_tot(t_j) is given as a function of the high-order intensity functions $i_{w, h, j}^{s} (θ)$ , which depend on S_n as well as the polarization state of the incoming wave at the layer of interest and provide the ratio of the scattered to incident light at scattering angle θ (where subscript w stands for one of ss, ms,on, or ms,off, and subscript h = x or y).

The parallel and perpendicular components of I_tot(t_j) in Eq. (6) in respect to the incident electric field of the emitted light are denoted as I_tot,x_, and I_tot_,y, respectively. From Eq. (1), these components can be further expressed by I_ss,h(t_j), I_ms,tot,on,h(t_j), and I_ms,off,h(z_i, t_j) as:

\begin{array}{l} I_{t o t} (t_{j}) = I_{tot, x} (t_{j}) e_{x} + I_{tot, y} (t_{j}) e_{y} \\ = [I_{ss, x} (t_{j}) + I_{ms,tot,on, x} (t_{j}) + \sum_{i = 1}^{j} I_{ms,off, x} (z_{i}, t_{j})] e_{x} + [I_{ss, y} (t_{j}) + I_{ms,tot,on, y} (t_{j}) + \sum_{i = 1}^{j} I_{ms,off, y} (z_{i}, t_{j})] e_{y} \\ = [I_{ss, x} (t_{j}) + I_{ms,tot,on, x} (t_{j}) + \sum_{i = 1}^{j} \sum_{k = 1}^{i} I_{ms,off, k, x} (z_{i}, t_{j})] e_{x} + [I_{ss, y} (t_{j}) + I_{ms,tot,on,y} (t_{j}) + \sum_{i = 1}^{j} \sum_{k = 1}^{i} I_{ms,off, k, y} (z_{i}, t_{j})] e_{y} . \end{array}

In the VPM, I_ss,h(t_j), I_ms,tot,on,h(t_j), and I_ms,off,h(z_i, t_j) in Eq. (8) are expressed in terms of

i_{w, h, j}^{s}

as follows:

I_{ss, h} (t_{j}) = C_{s} I_{o} \cdot \frac{1}{k^{2}} i_{ss, h}^{s} (π) \exp [- 2 τ (z_{i = j - 1 / 2})] [\frac{\exp {- 2 σ_{ext} (z_{i = j}) Δ z_{i = j}} - 1}{- 2 σ_{ext} (z_{i = j}) Δ z_{i = j}}] (h = x or y),

I_{ms,tot,on,h} (t_{j}) = C_{s} J_{on} (t_{j}) \cdot C_{on} \frac{1}{k^{2}} \frac{\int_{θ_{\min, on, j}}^{θ_{\max, on, j}} i_{on, h, j}^{s} (θ_{bk}) p_{s s, i = j} (θ_{in}) \sin θ_{in} d θ_{in}}{\int_{θ_{\min, on, j}}^{θ_{\max, on, j}} p_{s s, i = j} (θ_{in}) \sin θ_{in} d θ_{in}} \exp [- 2 \sum_{i = 1}^{j - 1} (σ_{ext,on} (z_{i}) Δ z_{i})] [\frac{\exp {- 2 σ_{ext,on} (z_{i = j}) Δ z_{i = j}} - 1}{- 2 σ_{ext,on} (z_{i = j}) Δ z_{i = j}}],

\begin{array}{l} I_{ms,off, h} (z_{i}, t_{j}) = \sum_{k = 1}^{i} I_{ms,off, k, h} (z_{i}, t_{j}) \\ = \sum_{k = 1}^{i} [C_{s} J_{off} (z_{k}, z_{i}, t_{j}) \cdot C_{off, k, i, j} \frac{1}{k^{2}} \frac{\int_{θ_{\min,,off, j}}^{θ_{\max,off, j}} i_{off, h, j}^{s} (θ_{bk}) p_{n = j} (θ_{in}) \sin θ_{in} d θ_{in}}{\int_{θ_{\min,,off, j}}^{θ_{\max,off, j}} p_{n = j} (θ_{in}) \sin θ_{in} d θ_{in}} \exp (- 2 \sum_{q = k}^{i} σ_{ext,off} (z_{q}, z_{k}, z_{i}, t_{j}) Δ {z^{'}}_{q})] . \end{array}

_{$i_{w, h, j}^{s}$} are given as a function of S_n and the depolarization of the incoming wave in subsection 2.2.3. The decay rate is assumed to be independent of the polarization state. C_on and C_off_,k,i,j are proportionality factors to ensure the same I_tot for the PM and VPM. The angular integrations over θ_in are introduced to accommodate the geometric effect. The backscattering angle to the detector θ_bk in Eqs. (10) and (11) is given by the incident angle θ_in and the geometric position of the cloud layer and satellite [10], as shown in Fig. 3 as

θ_{bk} = π - (θ_{in} (t_{j}) - θ_{sat}),

θ_{sat} = \tan^{- 1} [(z_{i} - z_{i - 1}) \tan (θ_{in} (t_{j})) / (z_{sat} + z_{i})] .

θ_in will be discussed in subsections 2.2.4.1 and 2.2.4.2.

Fig. 3 Geometry for the definition of θ_bk in Eq. (12).

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2.2.3 Procedure to estimate the parallel and perpendicular components of irradiance

In the following, we consider the electric fields used to estimate I_ss,h(t_j), I_ms,tot,on,h(t_j), and I_ms,off,h(z_i, t_j).

_{$i_{w, h, j}^{s}$}in Eqs. (9)–(11) are the x and y components of the azimuthally integrated normalized intensity functions between θ and θ + dθ (dθ → 0) of a concentric cone [20], as shown in Fig. 4, defined as:

i_{w, h, j}^{s} (θ) = {〈 {| E_{w, h, j}^{s} |}^{2} 〉}_{ϕ} / ({| E_{w, x, j}^{in} |}^{2} + {| E_{w, y, j}^{in} |}^{2}) .

< >_ϕ indicates average over 0<ϕ≤2π.

E_{w, j}^{s}

represents the scattered electric fields at the n = j-th scattering event and is related to the incident fields after the n = (j − 1)-th scattering event

E_{w, j}^{in}

by the following [10,20]:

E_{w, j}^{s} = L_{- ϕ} D s_{n = j} L_{ϕ} E_{w, j}^{in} .

Here,

L_{ϕ} = (\begin{matrix} \cos ϕ & \sin ϕ \\ - \sin ϕ & \cos ϕ \end{matrix})

is used to work in the scattering plane,

D = (\begin{matrix} \cos θ & 0 \\ 0 & 1 \end{matrix})

expresses the fact that the perpendicular polarization is maintained in the process, while the paralell component is obtained by projection of the scattered light vector in the xy plane, and s_n is the n-th order amplitude scattering matrix. The multiplicative factor [21] is omitted in Eq. (15).

Fig. 4 General geometries of the incident and scattering planes. θ is the zenith angle from the incident direction z and ϕ is the azimuthal angle. The parallel and perpendicular components of the scattered electric fields are taken in the xy plane.

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In this study, we considered randomly oriented particles in space. Therefore, the Mueller matrix S_n can be written as [21]:

S_{n} = [\begin{matrix} S_{11, n} (θ) & S_{12, n} (θ) & 0 & 0 \\ S_{12, n} (θ) & S_{22, n} (θ) & 0 & 0 \\ 0 & 0 & S_{33, n} (θ) & S_{34, n} (θ) \\ 0 & 0 & - S_{34, n} (θ) & S_{44, n} (θ) \end{matrix}]

From Eqs. (14)–(16), the intensity functions are derived as follows [20]:

i_{w, x, j}^{s} (θ) = \frac{(m_{11, n} + m_{22, n}) {| E_{w, x, j}^{in} |}^{2} + (m_{11, n} - m_{22, n}) {| E_{w, y, j}^{in} |}^{2}}{{| E_{w, j}^{in} |}^{2}} = \frac{(m_{11, n} + m_{22, n}) + (m_{11, n} - m_{22, n}) Q_{w, j}^{in}}{1 + Q_{w, j}^{in}} (n = j),

i_{w, y, j}^{s} (θ) = \frac{(m_{11, n} - m_{22, n}) {| E_{w, x, j}^{in} |}^{2} + (m_{11, n} + m_{22, n}) {| E_{w, y, j}^{in} |}^{2}}{{| E_{w, j}^{in} |}^{2}} = \frac{(m_{11, n} - m_{22, n}) + (m_{11, n} + m_{22, n}) Q_{w, j}^{in}}{1 + Q_{w, j}^{in}} (n = j),

where

Q_{w, j}^{in} = {| E_{w, y, j}^{in} |}^{2} / {| E_{w, x, j}^{in} |}^{2}, (w = s s, o n, o f f)

are the depolarization of the incoming wave at the layer of interest, and

Q_{ss, j}^{in} = 0

. The parallel and perpendicular components in Eqs. (17) and (18) are given by:

m_{11, n} + m_{22, n} = A (θ) [2 S_{11, n} (θ) + S_{22, n} (θ)] + 3 B (θ) S_{12, n} (θ) - 2 C (θ) S_{33, n} (θ),

m_{11, n} - m_{22, n} = A (θ) [2 S_{11, n} (θ) - S_{22, n} (θ)] + B (θ) S_{12, n} (θ) + 2 C (θ) S_{33, n} (θ) .

The coefficients A, B, and C are as follows:

A (θ) = (1 + \cos^{2} θ) / 8,

B (θ) = - \sin^{2} θ / 8,

C (θ) = - \cos θ / 8 .

For x-polarized light, when single scattering by spherical particles (s₃ = s₄ = 0) is considered, Eqs. (17) and (18) reduce to the familiar expressions given in Eqs. (A15) and (A14) of [10], respectively. Equations (17) and (18) describe the relationships among I_ss,h(t_j), I_ms,tot,on,h(t_j), and I_ms,off,h(z_i, t_j) in Eqs. (9)–(11) and S_n. Note that, for ss and ms,on, S for single scattering is used for m₁₁ and m₂₂ at a backscattering angle of θ_bk in Eqs. (17) and (18), to estimate

i_{w, h, j}^{s} (θ_{bk})

in Eqs. (9) and (10).

Once we obtain $Q_{w, j}^{in}$ in Eqs. (17) and (18) as well as θ_in in Eq. (12), I_ss,h(t_j), I_ms,tot,on,h(t_j), and I_ms,off,h(z_i, t_j) can be derived.

2.2.4 Depolarization of the incoming wave

In the VPM, mechanisms to produce non zero $Q_{w, j}^{in}$ due to rotation of the scattering plane without tracking every scattering event as in Monte Carlo method are introduced separately for ms,on and ms,off. Here we describe how $Q_{on, j}^{in}$ and $Q_{off, j}^{in}$ are produced and evlove with scattering order.

2.2.4.1 On-beam scattering

_{$Q_{on, j}^{in}$}is estimated by considering a mean forward incident angle θ_ref determined by the mean horizontal spread of the photons after n = j-th scattering event and FOV. To obtain $Q_{on, j^{'}}^{in}$ , depolarization of the incoming wave at each cloud layer $Δ_{j, j^{'}}$ (1≤ j ≤ j′-1) contributing to $Q_{on, j^{'}}^{in}$ is estimated by the following recurrenece relation derived by using Eqs. (17) and (18):

Δ_{j + 1, j^{'}} = \frac{[m_{11, n} (θ) - m_{22, n} (θ)] + [m_{11, n} (θ) + m_{22, n} (θ)] Δ_{j, j^{'}}}{[m_{11, n} (θ) + m_{22, n} (θ)] + [m_{11, n} (θ) - m_{22, n} (θ)] Δ_{j, j^{'}}} (j^{'} - 1 \geq j \geq 1, n = j),

From Eq. (25),

Δ_{j, j^{'}}

is obtained by the recurrence relation beginning with

Δ_{1, j^{'}} = 0

and with a constant θ = θ_ref throughout the recurrence computation.

θ_{ref} = \tan^{- 1} [r_{ref, i} / l_{i}] (i = j^{'}),

where index i in Eq. (26) is fixed to the maximum j-index value of the sequence j′. l_i is the scattering mean free path and r_ref,_i is taken to be the minimum of half r_fov or the mean horizontal spread of the photons, as follows:

r_{ref, i} = \min [0.5 r_{fov} (z_{i}), \sum_{k = 0}^{i - 1} (l_{k + 1} \sqrt{1 - g_{n = k}^{2}})] (i = j^{'}) .

r_fov(z_i) is the FOV footprint sizes at z_i. g_n is the n-th order asymmetry factor [14]. Finally,

Q_{on, j^{'}}^{in} = Δ_{j, j^{'}} δ_{j j^{'}},

and

δ_{j j^{'}}

is the Kronecker delta.

Q_{on, j}^{in}

for different θ_ref and scattering order is shown in Fig. 5. For a given θ_ref,

Q_{on, j}^{in}

increases with scattering order, i.e., optical thickness. For constant scattering order,

Q_{on, j}^{in}

increases with θ_ref, since larger θ_ref indicates larger FOV or σ_ext. At small incident scattering angles,

Q_{on, j}^{in}

does not develop much during near-forward multiple scattering events.

Fig. 5 Depolarization of the incoming wave for ms,on $Q_{on, j}^{in}$ as a function of θ _ref and the scattering order n ( = j) for r_eff = 10 µm at λ = 532 nm.

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To determine the geometrical effect for ms,on, the minimum and maximum ranges of θ_in in Eq. (10) are estimated from the FOV and by the diffraction widths θ_df as follows:

θ_{\max,on, j} = \tan^{- 1} [r_{fov} (z_{i}) / l_{i}] (i = j),

θ_{\min, on, j} = {\begin{matrix} θ_{df} (θ_{\max, o n, j} > θ_{df}) \\ 0 (θ_{\max, o n, j} \leq θ_{df}) \end{matrix} .

θ_max,on,j is depicted in Fig. 6. In Eq. (30), the second minima of the diffraction wave is considered for θ_df, where θ_df = sin⁻¹(2.22λ/2r_eff) [22,23], and r_eff is the effective radius [6].

Fig. 6 Geometry of defining θ_max,on,j and θ_ref, where i = j and when r_ef,i in Eq. (27) equaled 0.5r_fov.

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2.2.4.2 Off-beam scattering

_{$Q_{off, j}^{in}$}is estimated similarly to $Q_{on, j}^{in}$ , but by considering the horizonal spread of the photons scattered at large scattering angles due to mutiple scattering for the off-beam component. $Q_{off, j^{'}}^{in}$ is defined by the following integration over 0≤θ ≤ π as:

Q_{off, j + 1}^{in} = \frac{\int_{0}^{π} {(m_{11, n} - m_{22, n}) + (m_{11, n} + m_{22, n}) Q_{off, j}^{in}} p_{n} (θ) \sin θ d θ}{\int_{0}^{π} {(m_{11, n} + m_{22, n}) + (m_{11, n} - m_{22, n}) Q_{off, j}^{in}} p_{n} (θ) \sin θ d θ} (j^{'} - 1 \geq j \geq 1, n = j),

where

Q_{off, 1}^{in} = 0

and j′ is the maximum j-index value of interest.

Q_{off, j^{'}}^{in}

in Eq. (31), shown in Fig. 7, gradually increases with the scattering order.

Fig. 7 Depolarization of the incoming wave for ms,off $Q_{off, j}^{in}$ as a function of scattering order n ( = j) for r_eff = 10 µm at λ = 532 nm.

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To determine the geometrical effect for ms,off, the range of θ_in in Eq. (11) is specified in the following. The source term J_off(z_k,z_i, t_j) for I_ms,off,h,_k(z_i, t_j) in Eq. (11) is determined from the fraction of the ms,on component scattered at z_k within the angle θ_off,_k_→_i(t_j-1) < θ < θ_off,_k_→_i(t_j) [14]. θ_off,_k_→_i(t_j) is defined in [14] as:

θ_{off, k \to i} (t_{j}) = \cos^{- 1} [(z_{i} - z_{k}) / {(t_{j + 1 / 2} c - z_{k})}] .

The range of θ_in is given by:

θ_{\min, off, j} = θ_{off, k \to i} (t_{j - 1}),

θ_{\max, off, j} = θ_{off, k \to i} (t_{j}) .

It is noted that

Q_{on, j}^{in}

estimated at θ_ref = θ_in is taken as

Q_{off, j}^{in}

when it is larger than

Q_{off, j}^{in}

given by Eq. (31).

In summary, to estimate I_ss,h(t_j), I_ms,tot,on,h(t_j), and I_ms,off,h(z_i, t_j), we first estimate the ratios of the parallel and perpendicular components of the incoming wave at the n = j-th scattering event $Q_{w, j}^{in}$ (w = ss, on, off) using Eq. (28) and Eq. (31) to calculate the corresponding normalized intensity functions $i_{w, h, j}^{s} (θ_{bk})$ from Eqs. (17) and (18). $i_{w, h, j}^{s} (θ_{bk})$ are then substituted into Eqs. (9)–(11) to obtain I_ss,h(t_j), I_ms,tot,on,h(t_j), and I_ms,off,h(z_i,t_j). Finally, the depolarization ratio δ(t_j) is obtained by substituting these components into Eq. (35):

δ (t_{j}) = \frac{I_{tot, y} (t_{j})}{I_{tot, x} (t_{j})} = \frac{I_{ss, y} (t_{j}) + I_{ms,tot,on, y} (t_{j}) + \sum_{i = 1}^{j} I_{ms,off, y} (z_{i}, t_{j})}{I_{ss, x} (t_{j}) + I_{ms, tot,on, x} (t_{j}) + \sum_{i = 1}^{j} I_{ms,off, x} (z_{i}, t_{j})} .

3. Simulation of δ with the VPM for spaceborne lidar

The applicability of the VPM to lidar is studied in three subsections: (1) the Lidar In-space Technology Experiment (LITE) [24] case, (2) ground-based lidar case, and (3) CALIOP case and characterization of δ in terms of microphysics and FOVs obtained by the VPM (δ_VPM) with respect to MC (δ_MC) [19].

3.1 LITE case

Figure 8(a) shows the normalized backscattered returns from the C1 cloud [25] for the LITE case (λ = 1064 nm), in which θ_FOV ranged from 0.2 to 3.5mrad, as described and simulated by The National Aeronautical and Space Administration’s Langley Monte Carlo model in [24]. Simulations using the MC model and VPM are also provided for the same cases. Based on Monte Carlo simulations, ref [13] derived a relationship between the layer-integrated multiple scattering factor η and the layer-integrated linear depolarization ratio δ_layer. η is the ratio of the layer-integrated single-to-total attenuated backscattering coefficients; η becomes smaller as multiple scattering increases relative to the single-scattering component. The η factor is a useful concept for deriving cloud microphysics [26]. The MC model showed good agreement with the results of [24] and [13] in terms of the backscattered returns and η–δ_layer relation for the C1 cloud at smaller δ_laye_r; MC results in Fig. 8(c) showed a slightly larger value for η given the same δ_laye_r and VPM overlaid the MC results (not shown).

Fig. 8 (a) Normalized backscattered returns estimated using Monte Carlo (MC) [19] simulations (black lines) and the vectorized physical model (VPM; blue symbols), and produced by [24] (gray open squares and vertical bars). Note that the values of the gray open squares and the range of the literature values shown by vertical bars were extracted from Fig. 3 of [24]. (b) Linear depolarization ratio δ estimated by the VPM and MC simulations corresponding to (a) for the C1 cloud [25] at 1064 nm for the Lidar In-space Technology Experiment (LITE) case with z_sat = 293 km. (c) Layer-integrated η–δ_layer relations by [13] (black line) and by MC [19] (colored lines).

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In Figs. 8(a) and 8(b), MC results show larger backscattered returns and δ_MC values for the larger FOV cases. A comparison of δ_VPM and δ_MC showed that these trends were captured well by the VPM. The mean relative error of the VPM results was less than about 3.6% overall, with respect to that of MC simulations.

3.2 Ground-based lidar case

A simulation was also performed for a ground-based lidar case to test the applicability of the VPM. In Fig. 9, δ_VPM is compared with δ_MC and literature values of δ obtained by other models reported in [27], in which a ground-based lidar (λ = 700 nm; θ_FOV ~1.75mrad) located 2 km from the cloud base was considered. δ from other models [27] (i.e., the 3DMcPOLID model, the ECSIM model, and a semi-analytical approach with small-angle approximation, described in [27] and in the references therein) had about 10% relative variability in the predicted range. The δ_MC generally showed good agreement with other models in [27] and was located at the upper boundary of the variability range, corresponding to the results obtained with the 3DMcPOLID model [27]. The VPM and MC showed good consistency, with the exception at small τ, where δ_VPM was within the variability range but slightly underestimated δ_MC.

Fig. 9 δ from the C1 cloud [25] obtained by VPM and MC for a ground-based lidar at 700 nm. Ranges of the mean δ values predicted by different methods described in [27] and as shown by vertical bars in the figure were extracted from Fig. A4 of [27].

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3.3 CALIOP case and VPM characterization

Sensitivity studies of δ_VPM were performed, following [14]. The accuracy of the VPM was investigated by conducting 17 experiments with varying parameter settings (Table 1). The relative errors (|ERR|) of the VPM with respect to MC were calculated as |ERR| = (|δ_VPM−δ_MC|)/δ_MC. The overall relative errors 〈ERR〉 (%) were calculated for different τ ranges by averaging |ERR| estimated at all data points.

Table 1. Experimental setting.

View Table

For Experiments 1–13, the reference simulations were performed using CALIOP specifications (λ = 532 nm; θ_fov,1 = 0.13mrad; 702km to the cloud top) [14]. Experiments 1–12 corresponded to homogeneous cloud profiles with different σ_ext, r_eff, and FOVs [σ_ext = 3, 15.7, and 40km⁻¹; r_eff = 5, 10, and 20µm; and θ_fov = θ_fov,1 and 10θ_fov,1]. A comparison of δ_VPM and δ_MC for the abovementioned experiments is shown in Fig. 10(b). δ_VPM showed good agreement with δ_MC, with 〈ERR〉 values of less than about 3.2% on average.

Fig. 10 (a) δ obtained by VPM (δ_VPM) and MC (δ_MC) for Experiment 6. In the figure, δ_VPM estimated without considering the depolarization of the incoming wave (δ_P_n) is also shown for Experiment 6 (black star). (b) δ_VPM and δ_MC for Experiments 2, 4, 6, 8, 10, and 12.

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To analyze the importance of considering the depolarization of the incoming wave on δ, δ_VPM with $Q_{w, j}^{in}$ = 0 (labeled as δ_P_n) is presented in Fig. 10(a) for Experiment 6. δ_P_n is the δ produced by p_n and the geometrical effect. δ_P_n experiment underestimated δ_MC from τ = 3 (z~191m). Without considering $Q_{w, j}^{in}$ , the difference δ_MC−δ_P_n was about 0.2 at τ = 8. In [14], I_tot was categorized into ss-dominant, ms,on-dominant, and ms,off-dominant regimes, bounded at about τ = 2 (z~127m) and τ = 8 (z~510m) for Experiment 6. The slope of δ_P_n also changed according to the three regimes; however, much better agreement between δ_VPM and δ_MC was achieved when $Q_{w, j}^{in}$ was considered.

The dependence of δ on r_eff for a given σ_ext is further examined in Experiments 1–3, 5–7, and 9–11 for σ_ext = 3km⁻¹, σ_ext~15.7km⁻¹, and σ_ext = 40km⁻¹, respectively. The δ for r_eff = 5, 10, and 20µm (denoted as δ₅, δ₁₀, and δ₂₀, respectively) are shown in Figs. 11(a-c), where it is seen that the sensitivity of δ to r_eff decreased as τ increased for a constant σ_ext. The difference |δ₅-δ₂₀| was about 0.12, 0.12, and 0.1 at τ = 2, τ = 5, and τ = 10 for σ_ext = 3km⁻¹, respectively. These differences corresponded to a relative difference Δδ = |δ₅-δ₂₀|/δ₂₀ of about 63%, 37%, and 1.2% for τ = 2, τ = 5, and τ = 10, respectively. Both MC and VPM also indicated that the sensitivity of δ to r_eff decreased as σ_ext increased for a given τ. For σ_ext~15.7km⁻¹, |δ₅-δ₂₀| was about 0.11, 0.11, and 0.06 corresponding to Δδ of about 39%, 23%, and 8% at τ = 2, τ = 5, and τ = 10, respectively. In the case of σ_ext = 40km⁻¹, |δ₅-δ₂₀| and Δδ were about 0.08, 0.09, 0.05 and 23%, 17%, 6% at τ = 2, τ = 5, τ = 10, respectively. From Figs. 11(a-c), the retrieval uncertainty in τ by δ due to the variation in r_eff and σ_ext can also be inferred. It was found that when δ = 0.3, the corresponding τ had about 2 times, 1.7 times, and 1.4 times difference among r_eff considered at σ_ext = 3km⁻¹, σ_ext~15.7km⁻¹, and σ_ext = 40km⁻¹, respectively. At δ = 0.7, there was about 1.1~1.2 times difference in τ despite the σ_ext considered.

Fig. 11 δ_VPM and δ_MC for the (a) σ_ext = 3km⁻¹ case corresponding to Experiments 1,2 and 3, (b) σ_ext~15.7km⁻¹ case corresponding to Experiments 5,6 and 7, (c) σ_ext = 40km⁻¹ case corresponding to Experiments 9,10 and 11.

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Experiment 13 corresponds to the case with vertically inhomogeneous cloud particle size profiles, specifically, r_eff = 20, 5, and 10 µm from the cloud top bounded at 240 and 480 m with a liquid water content of 0.1g/m³. 240m corresponded to the vertical resolution of the global cloud phase product of [8] (KU product). Inhomogeneous structures were evident due to the change in microphysics, as observed in δ_MC in Fig. 12. Such behavior was well captured by δ_VPM; the 〈ERR〉 value was 3.76% ± 3.22%.

Fig. 12 δ_VPM and δ_MC for the inhomogeneous case (Experiment 13). τ is around 1.8 and 9.5 at z = 240m and z = 480m, respectively.

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Experiments 14–16 and 17 corresponded to the cases studied in Figs. 8 and 9, respectively. The 〈ERR〉 values for all 17 cases are summarized in Fig. 13. The overall 〈ERR〉 was 2.32% on average, and was less than 5% for all optical thickness ranges.

Fig. 13 Relative error of δ_VPM with respect to δ_MC for the 17 experiments summarized in Table 1. The results are categorized according to the optical thickness: (a) 0 ≤ τ, (b) τ ≤ 2, (c) 2 < τ < 8, and (d) 8 ≤ τ. The maximum τ values considered are the same as those used in [14] for Experiments 1–13; about 48 were used for the σ_ext = 40 km⁻¹ cases and about 18 were used for the others. The overall mean relative errors 〈ERR〉 and standard deviations (%) listed in the upper right corner of each figure panel are estimated from all 17 profiles.

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4. Summary

In our proposed method, the PM developed in [14] for the total backscattered irradiances I_tot from spaceborne lidars is vectorized to create a VPM to simulate polarization properties under multiple scattering conditions effectively for the first time. The VPM models vectorized lidar multiple scattering with one set of n-th order Mueller matrices S_n and a transformation matrix for Stokes parameters L connecting the initial incident plane and n-th scattering plane. Additional treatment was introduced to implement the accumulated effects on the depolarization. This consists of two mechanisms: (1) depolarization of the incoming wave at the cloud layer of interest during multiple scattering in the forward direction due to rotation of the scattering plane and (2) depolarization produced by angular distributions of the backward scattering processes within the FOV (geometrical effect). The depolarization of the incoming wave was determined by considering the horizontal spread of the photons for the on-beam and off-beam components. These mechanisms were modeled separately for I_ss(t_j), I_ms,tot,on(t_j), and I_ms,off(z_i, t_j) to estimate their parallel and perpendicular components in respect to the incident electric field of the emitted light to obtain δ(t_j). These formulations accommodate the dependence of δ on the scattering order, microphysical properties, and FOV.

Evaluation of δ from VPM against MC simulations and, in part, by literature values was performed for a total of 17 cases. The effect of the depolarization of the incoming wave became important as τ increased. The comparisons showed overall agreement between the VPM and MC simulations, with a mean relative error of about 2% ± 3%. The VPM was more efficient than the MC approach. The sensitivity of δ to r_eff was also investigated, where the relative difference in δ among r_eff considered was generally larger than the relative error of the VPM against MC.

In future studies, the dependence of the vertical profiles of δ and I_tot on cloud microphysics will be investigated comprehensively using the VPM, based on measurements from space-borne lidars [7,28], as well as ground-based multiple scattering lidars [29], for satellite algorithms. The extension of the VPM to circular polarization used in multiple FOV measurements [10] should be a straightforward process.

Funding

Japan Society for the Promotion of Science (JSPS KAKENHI Grant Numbers JP18K03745, JP17H06139); The Japan Aerospace Exploration Agency (EarthCARE Research Announcement); Ministry of Education, Culture, Sports, Science and Technology (The Arctic Challenge for Sustainability (ArCS)); Collaborated Research Program of Research Institute for Applied Mechanics, Kyushu University (Fukuoka, Japan).

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Experiment	1,2,3	4	5,6,7	8	9,10,11	12	13	14,15,16	17
σ_ext[km⁻¹]	3	3	15.7	15.7	40	40	7.7/32/15.7	17.25	25
r_eff [µm]	5,10,20	10	5,10,20	10	5,10,20	10	20/5/10	C1	C1
θ_fov	θ_f_ov,1	10θ_f_ov,1	θ_f_ov,1	10θ_f_ov,1	θ_f_ov,1	10θ_f_ov,1	θ_f_ov,1	0.2, 1.1, 3.5 mrad	1.75 mrad
wavelength [nm]	532	532	532	532	532	532	532	1064	700
homogeneous	yes	yes	yes	yes	yes	Yes	no	yes	yes

Modeling the depolarization of space-borne lidar signals

Abstract

1. Introduction and physical model

2. Vectorized physical model

2.1 Overview of the scalar PM

2.2 The VPM

2.2.1 Introduction of a high-order Mueller matrix

2.2.2 Formulation of the VPM

2.2.3 Procedure to estimate the parallel and perpendicular components of irradiance

2.2.4 Depolarization of the incoming wave

2.2.4.1 On-beam scattering

2.2.4.2 Off-beam scattering

3. Simulation of δ with the VPM for spaceborne lidar

3.1 LITE case

3.2 Ground-based lidar case

3.3 CALIOP case and VPM characterization

4. Summary

Funding

References

Cited By

Figures (13)

Tables (1)

Equations (35)

Optics Express