Anthony B Davis
email: anthony.b.davis@jpl.nasa.gov

Nicolas Ferlay

Quentin Libois

Alexander Marshak

Yuekui Yang

Qilong Min

Cloud information content in EPIC/DSCOVR

Keywords: Multiple scattering, Oxygen A-band, Oxygen B-band, Radiative transfer, Remote sensing, Cloud top height, Geometrical cloud thickness, DSCOVR, EPIC, Retrieval uncertainty quantification

come

Introduction & Overview

Clouds are still a poorly understood element of the Earth's climate system and, consequently, they are poorly parameterized in global climate models (GCMs)-so much so that a large part of the uncertainty about predicting future climate can be traced to this gap [START_REF]Intergovernmental Panel on Climate Change, Climate Change 2013: The Physical Science Basis[END_REF]. This fact, as troublesome as well-documented, has motivated the National Academies to list clouds as top-priority for continued and enhanced observation from space in the recently released 2017 Decadal Survey [START_REF]and Medicine, Thriving on Our Changing Planet, A Decadal Strategy for Earth Observation from Space[END_REF].

The Deep Space Climate ObserVatoRy (DSCOVR) mission [START_REF]Data, and Information Service (NES-DIS), DSCOVR: Deep Space Climate Observatory[END_REF][START_REF] Burt | Deep Space Climate Observatory: The DSCOVR mission[END_REF] has pushed Earth observation from space literally to new heights by locating the platform near the Sun-Earth "Lagrange-1" (L 1 ) point, at c. 1,500,000 km from our planet in direction of the Sun-figuratively, as well. Indeed, the Earth Polychromatic Imaging Camera (EPIC) [START_REF] Marshak | Earth observations from DSCOVR/EPIC instrument[END_REF] is one of the two Earth-oboserving sensors on DSCOVR. EPIC images the Earth every 60 to 100 minutes [START_REF]DSCOVR: EPIC, Earth Polychromatic Imaging Camera[END_REF] with a 2,048×2,048 pixel camera (≈8 km pixels at the center of the disc) boosting 10 narrow spectral channels sampling the UV-VIS range. Among these, the four of interest here are at the longest wavelengths: 680, 688, 764, and 780 nm.

They are dedicated in particular to forming two in-band/out-of-band radiance ratios for di-oxygen's A-and B-bands. This is primarily for the purpose of determining the altitude at which the highest cloud layer occurs in every cloudy pixel. The present investigation, described both here and in a companion paper [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF], builds on a preliminary study by Yang et al. [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF] who proposed an elegant method for deriving both cloud top height (CTHs) and cloud geometrical thickness (CGT). For single-layer cloud cover, this is a rudimentary way of profiling the cloudy atmosphere. Such information would be very helpful to cloud scientists working on GCM cloud schemes. Following Yang et al.'s theoretical study, we start with the assumption that EPIC has access to both pieces of cloud profile information.

In the companion paper [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF], Davis et al. used a more realistic representation of the EPIC sensor. In particular, they included measurement noise, which was neglected in [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF], and applied the formal information content analysis methodology of Merlin et al. [START_REF] Merlin | Cloud information content analysis of multiangular measurements in the oxygen A-band: application to 3MI and MSPI[END_REF], itself inspired by Rodgers' [START_REF] Rodgers | Inverse Methods for Atmospheric Sounding: Theory and Practice[END_REF] theory of optimal estimation (OE). In spite of good signal-to-noise ratios (SNRs) in the four channels, and of the reduced noise level achieved by forming the in-band/continuum radiance ratios, Davis et al. conclude that there is too much alignment of the A-and B-band responses to both cloud properties to separate them in presence of the anticipated noise. Consequently, only the dominant property (CTH) can be derived with good precision. In the jargon of OE, CGT has to be moved from an unknown element of the atmospheric "state vector" argument of the forward model to the set of atmospheric properties whose values need to be determined otherwise, within known uncertainty.

In the present study, we use a closed-form analytical model for the in-band/continuum radiance ratios instead of the computational 1D radiative transfer (RT) model used in the companion paper as a proverbial "black box." Here, we derive the simpler but analytically tractable model from the first principles of 1D RT theory, gaining insights about A-and B-band cloud observations along the way. Coming from this physics-based approach, we arrive at the same conclusions as in [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF] about the retrievability of CTH, and not CGT, where a statistics-based approach was adopted in the sense of OE formalism. Moreover, the analytical model is used to quantify the bias in retrieved CTH if the impact of CGT on the signal is neglected, hence the need to make some plausible assumption about GCT to ensure not only precise but accurate retrievals of CTH.

In the next Section, we describe the 1D RT modeling framework used in this investigation, leading to our analytical approximation and shedding new light on the uncertainty budget for cloud property retrievals, with the more technical aspects described in Appendix A. The systematic (forward modeling) error incurred when neglecting incloud O 2 absorption is quantified in Section 3. In Section 4, we turn to the impact of random (measurement) error on retrievals, leading to a physics-based cloud information content analysis for EPIC's O 2 A-and B-band channels with CTH and CGT as remote sensing targets. We summarize our findings in §5 and discuss potentially farreaching connections with laboratory studies of mean pathlengths that are elucidated in Appendix B.

Radiative Transfer Framework

Atmospheric optical properties

As in the companion paper, we represent the atmosphere a priori with three layers (above, inside, and below the cloud layer); see Fig. 1 in [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF]. We denote the O 2 absorption coefficient, dependent on the ambient temperature T z and pressure P z , as k O 2 (λ; P z , T z ) at altitude z. The optical depth for absorption by di-oxygen between altitude z and the top-of-atmosphere (TOA, symbolically, z = ∞) along the vertical is

τ O 2 (λ; z) = ∞ z k O 2 (λ; P z , T z )dz. (1)
Following [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF], we make these assumptions:

(λ; z)] 0 ztop-H = τ O 2 (λ; 0) - τ O 2 (λ; z top -H) with H denoting CGT.
• In the A-and B-bands as well as in the continuum, the cloud layer is therefore the only region with a non-vanishing scattering coefficient, hence single-scattering albedo (SSA).

-Rayleigh scattering optical depths of the atmosphere, roughly 0.03 and 0.04, respectively for A-and B-band regions (reference and in-band channels), are negligible compared to all other optical depths, that is, for scattering by cloud droplets, and all the more so in the presence of absorption by O 2 . Following [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF], we neglect Rayleigh scattering.

-Aerosols will generally have optical depths that are larger than for Rayleigh scattering in these spectral regions, but still nowhere near the COT values considered here. We neglect them too in order to keep our atmospheric model as close as possible to the one used by Yang et al. [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF], as we will be comparing results quantitatively further on.

SSA in the cloud layer is therefore determined only by the value of the cloud droplet scattering coefficient σ c that, again following [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF], is assumed constant, and the gaseous absorption coefficient introduced in (1). It can therefore be estimated as needed from COT τ c , H and the values of k O 2 (λ; P z , T z ) for z top -H < z < z top .

Internal variations of the SSA matter only when H becomes commensurate with the pressure scale-height (≈8 km).

• Finally, the surface is assumed here to be black, meaning that we limit this study to clouds above water. Yang et al. [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF] assigned to the surface albedo a small but finite value for water of 0.05. In the companion paper [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF], both black and highly reflective lower boundaries were examined.

In the absence of molecular and aerosol scattering distributed across the whole atmosphere, the last simplification reduces to just two the number of atmospheric layers to consider: transparent or absorbing above-cloud layer; scattering cloud layer, with or without absorption.

Spectral considerations

Figure 1 shows the EPIC in-band filter functions f in (λ) for A-and B-bands in arbitrary units along with τ O 2 (λ; 0) from ( 1) on a log-scale, with more spectral detail than in Fig. 2 of [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF] and more focus on the EPIC in-band channels. Now let f in (λ) denote the normalized filter function, i.e., f in (λ)/ ∆λ f in (λ)dλ in nm -1 . In the companion paper, we used a standard correlated-k technique [START_REF] Domoto | Frequency integration for radiative transfer problems involving homogeneous non-gray gases: The inverse transmission function[END_REF][START_REF] Lacis | A description of the correlated-k distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres[END_REF][START_REF] Goody | Atmospheric Radiation: Theoretical Basis[END_REF][START_REF] Kratz | The correlated k-distribution technique as applied to the AVHRR channels[END_REF][START_REF] West | The correlated-k method and related methods for broadband radiation calculations[END_REF] to account for the spectral variability, as modulated by f in (λ). Here, we switch to a simpler band-average approach [e.g., [START_REF] Goody | Atmospheric Radiation: Theoretical Basis[END_REF][START_REF] Young | Band Model Theory of Radiation Transport[END_REF]. A-and B-band average quantities of interest are:

• the mean absorption optical thicknesses, ∆λ τ O 2 (λ; 0)f in (λ)dλ, yielding 1.08 for Bband and 9.56 for A-band, which seems like a good contrast in absorption strength, hence a potential for differentiating shorter and longer paths, which is precisely what is needed to sense both cloud top (with shortest paths) and cloud base (with longest paths);

• an airmass-dependent effective absorption optical thicknesses defined as

τ (∆λ) O 2 (M ) = - 1 M log ∆λ exp[-M τ O 2 (λ; 0)]f in (λ)dλ , (2)
where the so-called "airmass" factor

M = 1 cos θ 0 + 1 cos θ = 1 µ 0 + 1 µ , (3)
with θ 0 and θ denoting solar (SZA) and viewing (VZA) zenith angles, respectively.

For EPIC, we have θ ≈ θ 0 , and a typical value of M is 3, which would result from θ 0 ≈ cos -1 ( 2 /3) = 48.2 • . The nonlinear average in (2) then yields effective absorption optical thickness of τ

(B)
O 2 (3) = 0.17 for the B-band and τ

(A) O 2 (3) = 0.41
for the A-band. Figure 2 shows the relatively weak dependence of these effective O 2 absorption optical depths on M (when θ = θ 0 , over a range where sphericity effects can be safely neglected).

The latter nonlinear averaging over the orders of magnitude in spectral variability of τ O 2 (λ; 0) inside the EPIC A-and B-band filters is recommended when seeking a single absorption optical depth in the simple monochromatic representations of the EPIC signals used, as needed, further on. However, the contrast between the A-and B-bands has gone from a factor of ≈10 to a factor of ≈2, which is moreover between two relatively small optical depths characteristic of the wings of the otherwise very deep lines.

Band-average transmission models have been developed over the last decades, most famously by Elsasser, Goody, and Malkmus; see, e.g., [START_REF] Modest | Radiative Heat Transfer[END_REF]Ch. 10], and references therein. Indeed this continues to be an area of active research [e.g., [START_REF] Conley | Extension of the weak-line approximation and application to correlated-k methods[END_REF][START_REF] Davis | Generalized radiative transfer theory for scattering by particles in an absorbing gas: Addressing both spatial and spectral integration in multi-angle remote sensing of optically thin aerosol layers[END_REF][START_REF] Davis | Addendum to [JQSRT 205[END_REF]. The simple scheme that uses exp(-τ

(∆λ)
O 2 (M )) from (2) may not be the best, even for our present purposes. Nonetheless, the band-average representation of O 2 absorption in (2) and Fig. 2 will be used, for simplicity, for numerical estimations in Fig. 3 and in §4; until then, much of the reasoning remains monochromatic. The impact of this choice will be discussed where appropriate, and we differ to a future publication the quantitative comparison of (2) with more sophisticated statistical transmission models. We only need to bear in mind here that most of the radiation that matters for the sensor response is transmitted through the optically thin parts of the filter's band-pass, i.e., the wings of the spectral lines.

We now have all the required information to compute radiances observed by EPIC at all the wavelengths of interest in A-and B-band studies. However, to understand these observations, it is important to bear in mind that it is fundamentally all about the amount of absorption by O 2 cumulated along all possible paths that sunlight can follow between injection and escape at TOA. This approach will lead us to approximatebut-analytical expressions for the EPIC signals of interest that are good enough for the present uncertainty quantification exercise, but far from satisfactory in a full physicsbased retrieval algorithm.

Monochromatic Formulation

Let I TOA (λ, Ω) be radiance at wavelength λ leaving the TOA in direction Ω, in this study, toward EPIC/DSCOVR at L 1 . We will use it non-dimensional "BRF" form R TOA (λ, Ω) = πI TOA (λ, Ω)/µ 0 F 0 (λ), which is the bi-directional reflection factor (BRF) of the surface-atmosphere system. The BRF is defined as the effective planar albedo were the system Lambertian, i.e., independent of Ω 0 (Ω 0z = µ 0 ), Ω (|Ω z | = µ), for the incoming (solar) and outgoing (view) directions, respectively, and of the relative azimuth.

In the upper (above-cloud) non-scattering region, all we need to know is solar and observational geometry. Therefore, neglecting surface reflection of light transmitted through the cloud in both directions, radiance observed from space BRF form is

R TOA (λ, Ω) ≈ exp[-(1/µ 0 + 1/µ)τ O 2 (λ; z top )]R λ (Ω; Ω 0 , τ c ), (4)
where we have already referred to (1/µ 0 +1/µ) as the airmass factor M in (3). The cloud layer's BRF, R λ (Ω; Ω 0 , τ c ), depends primarily on τ c , also the scattering phase function p c (θ s ), but only weakly in most (i.e., non-rainbow) directions, past its asymmetry factor

g = π 0 cos(θ s )p c (θ s ) sin(θ s )dθ s , (5)
which is ≈0.85 for the liquid-water clouds considered in [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF][START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF] and the present study.

These cloud droplet properties vary little with λ between the A-and B-bands, let alone inside them.

Spectral dependence of R λ (• • • ) in (4)
R λ (Ω; Ω 0 , τ c ) ≡ R(k λ , Ω; Ω 0 , τ c ) = ∞ 0 exp(-k λ ct)R(ct, Ω; Ω 0 , τ c ) dct, (6)
which is simply the Laplace transform of R(ct, Ω; Ω 0 , τ c ) with k λ being the Laplace conjugate variable of path length ct. Normalized by R(0,

Ω; Ω 0 , τ c ) = ∞ 0 R(ct, Ω; Ω 0 , τ c )dct, R(ct, Ω; Ω 0 , τ c
) becomes the probability density function (PDF) of the random variable ct. This path length distribution has a mean, variance, and so on. Higher moments are out-of-scope for the present study, but it is important to know the mean in-cloud cumulative path, i.e.,

ct (Ω; Ω 0 , τ c ) = ∞ 0 ct R(ct, Ω; Ω 0 , τ c ) dct ∞ 0 R(ct, Ω; Ω 0 , τ c ) dct = -1 R(0, Ω; Ω 0 , τ c ) ∂ R ∂k λ k λ =0 , (7)
where the denominator R(0, Ω; Ω 0 , τ c ) is the steady-state radiance in the continuum, i.e., near but outside the absorption band. A priori, ct (Ω; Ω 0 , τ c ) is also a function of other basic cloud properties, such as physical thickness H, the SSA for cloud particles (assumed unity here), and the droplet scattering phase function p c (θ s ).

Starting with some version of (7), ct (Ω; Ω 0 , τ c ) has been derived in closed form for standard approximations in 1D RT in plane-parallel clouds:

• Asymptotic theory views opaque scattering plane-parallel media as a perturbation of a semi-infinite medium (τ c → ∞); it leads to [23, p. 591]

ct (Ω; Ω 0 , τ c ) ≈ (µ + µ 0 )H, (8)
for τ c 1, when the SSA is unity (non-absorbing media) and p c (θ s ) = 1/4π (isotropic scattering). 1 Originally, this expression was derived for the mean optical path length Λ = σ c ct: Λ ≈ (µ + µ 0 )τ c , which is equal to the mean number of scatterings for radiance reflected from a uniform media [23, p. 584]. Multiplying both sides by the continuum mean-free-path (MFP), which is the inverse of extinction σ -1 c = H/τ c , yields [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF].

• In the diffusion approximation, focus is exclusively on the 0 th and 1 st spherical harmonics of the radiance field and of the scattering phase function, hence it will predict reflected flux, i.e., radiance escaping at TOA weighted by |Ω z | and averaged over Ω in the upwelling hemisphere:

F TOA (Ω 0 , τ c ) = Ωz<0 I TOA (Ω; Ω 0 , τ c )|Ω z |dΩ. (9)
For conservative (SSA = 1) scattering, this model leads to [START_REF] Davis | Multiple-scattering lidar from both sides of the clouds: Addressing internal structure[END_REF][START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF] ct

F (Ω 0 , τ c ) = Ωz<0 ct (Ω; Ω 0 , τ c )I TOA (Ω; Ω 0 , τ c )|Ω z |dΩ F TOA (Ω 0 , τ c ) ≈ 2 3 + µ 0 H × [1 + C(τ c , g, µ 0 )] , (10)
where C(τ c , g, µ 0 ) is a pre-asymptotic correction term that decays as 1/τ c at large values. Now, Ref. [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF] provides only a partial derivation of (10) in its Appendix D, and [START_REF] Davis | Multiple-scattering lidar from both sides of the clouds: Addressing internal structure[END_REF] only provides C(τ c , g, 1), albeit with excellent Monte Carlo validation, in view of the multiple-scattering lidar focus. We therefore provide here, in Appendix A, the explicit expression for C(τ c , g, µ 0 ) and describe the main steps of its derivation.

Either way, we see that ct (Ω, Ω 0 , τ c ) is, to leading order, proportional to physical cloud thickness H, irrespective of scattering details, and the proportionality factor depends only on illumination and viewing geometry.

If ct (Ω, Ω 0 , τ c ) is all we know about R(ct, Ω; Ω 0 , τ c ), then our logical estimate of steady-state spectral radiance R λ (Ω; Ω 0 , τ c ) is obtained by setting

R(ct, Ω; Ω 0 , τ c ) = R(0, Ω; Ω 0 , τ c ) δ [ct -ct (Ω; Ω 0 , τ c )] , (11)
hence, from [START_REF]DSCOVR: EPIC, Earth Polychromatic Imaging Camera[END_REF],

R λ (Ω; Ω 0 , τ c ) = R(0, Ω; Ω 0 , τ c ) exp[-k λ ct (Ω; Ω 0 , τ c )], (12)
where we can use the mean absorption coefficient in the cloud:

k λ ≈ [τ O 2 (λ; z)] ztop-H ztop /H. ( 13
)
We now ask: under what conditions is the above hypothesis that "all we know about the path length distribution is its mean" is likely to be good enough? It boils down to one or both of the following requirements. First, the width (e.g., standard deviation) of the PDF for ct could be relatively small with respect to its mean ct . Second, the combined value of k λ ct may remain small enough over said width of the PDF of ct that one can invoke the 1 st -order linear approximation for exp(-k λ ct) ≈ 1 -k λ ct in [START_REF]DSCOVR: EPIC, Earth Polychromatic Imaging Camera[END_REF]. In the case of optically thick clouds, the first condition is not expected to be verified; it has indeed been shown that, at least for reflected light, the variance of ct increases with COT, while ct does not, as COT increases without bound [e.g., 24, and references therein].

Best therefore to look into the second option where, because of the spectral integration discussed in the previous and following subsections, we should consider both ct and k λ as random variables. Having just dismissed the PDF of ct as inherently broad for high-COT clouds, we are relieved that the variability of k λ is overwhelmingly dominated by small values in both O 2 absorption bands (where, at the prevailing pressures and temperatures, the wings occupy more of the spectrum than the cores of absorption lines). Although it is out of the scope of the present study, a thorough examination of the assumption of a degenerate (i.e., Dirac-δ) PDF for ct is in order with O 2 absorption in mind, and additional reasons for this are articulated further on. At present, we will be content with the fact that, in spite of all the simplifying assumptions, the analytical forward model developed herein leads to identical conclusions about EPIC's A-and B-band information content for vertical cloud profiling as in the companion paper [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF] where accurate spectral integration and 1D RT modeling was used-as a "black box"to support a sophisticated statistical estimation approach.

We can now form the observed differential optical absorption spectroscopy (DOAS) ratio: applying (4), we have

r λ (Ω; Ω 0 , τ c ) = R TOA (λ ∈ ∆λ) R TOA (λ / ∈ ∆λ) ≈ e -(1/µ 0 +1/µ)τ O 2 (λ;ztop) R λ (Ω; Ω 0 , τ c ) R(0, Ω; Ω 0 , τ c ) = e - 1 µ 0 + 1 µ τ O 2 (λ;ztop)+k λ ct (Ω;Ω 0 ,τc) , (14)
where "λ ∈ ∆λ" is shorthand for any wavelength inside the absorption band of interest, and conversely for "λ / ∈ ∆λ" (although outside, it should be as near to the absorption band as possible so that scattering properties are almost identical for both wavelengths).

For ct (Ω; Ω 0 , τ c ) in ( 14), we take the best of both predictive models in ( 8) and [START_REF] Rodgers | Inverse Methods for Atmospheric Sounding: Theory and Practice[END_REF]:

ct (Ω; Ω 0 , τ c ) ≈ (µ + µ 0 )H × [1 + C(τ c , g, µ 0 )] . (15)
Indeed, asymptotic theory provides the desirable angular reciprocity upfront in the prefactor of (8) while diffusion theory in [START_REF] Rodgers | Inverse Methods for Atmospheric Sounding: Theory and Practice[END_REF] restores the anticipated weak dependence on COT and g at the cost of a minor violation of reciprocity confined to the preasymptotic correction term.

Invoking ( 12)-( 13), the negative natural logarithm of the DOAS ratio in ( 14)-( 15)

is

-log r λ (Ω; Ω 0 , τ c ) ≈ (1/µ 0 + 1/µ)τ O 2 (λ; z top ) + (µ + µ 0 ) [τ O 2 (λ; z)] ztop ztop-H × (1 + C(τ c , g, µ 0 )) , (16)
which is a simplified monochromatic forward model for DOAS ratios. To get the required O 2 optical depths at or between arbitrary altitudes, we assume a simple exponential profile in atmospheric pressure, hence of O 2 . Therefore, a first-order approximation for

τ O 2 (λ; z) is τ O 2 (λ; 0
) times e -z/H mol , where we can set the pressure scale height H mol to ≈8 km.

Absorption-Band Integration

We recall from Fig. 1 that the EPIC sensor is performing weighted spectral integrations over the A-and B-bands. For that, we introduce a new notation for [START_REF] Lacis | A description of the correlated-k distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres[END_REF]:

R ∆λ (Ω; Ω 0 , τ c ) = R(0, Ω; Ω 0 , τ c ) ∆λ exp[-k λ ct (Ω; Ω 0 , τ c )]f in (λ)dλ, (17)
where the broad spectral variability stems from [START_REF] Goody | Atmospheric Radiation: Theoretical Basis[END_REF]. In view of [START_REF]DSCOVR: EPIC, Earth Polychromatic Imaging Camera[END_REF], that means that there is actually a double integration going on, both in the spectral domain and over path lengths. The above "all-we-know-is-the-mean-path-length" assumption will be more accurate if the PDF of the random variable ct is skewed toward short paths where Beer's exponential transmission law is nearly linear in path length, at least for the smaller values of k λ representative of the wings of the O 2 spectral lines. For light reflected by optically thick clouds, numerical simulations show that this is indeed the case, e.g., [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF].

Similarly, the DOAS ratio observed by EPIC is a ratio of spectral integrals over [START_REF] Kratz | The correlated k-distribution technique as applied to the AVHRR channels[END_REF]. However, because scattering properties vary little in the continuum (λ / ∈ ∆λ), one can formulate r ∆λ (Ω; Ω 0 , τ c ) simply as the in-band integration of r λ (Ω; Ω 0 , τ c ) from [START_REF] Kratz | The correlated k-distribution technique as applied to the AVHRR channels[END_REF]. From there, taking logs, log r ∆λ (Ω; Ω 0 , τ c ) will look like ( 16) but using the effective optical depth from (2) for both A-and B-bands.

To get the effective O 2 optical depths at or between arbitrary altitudes based on its sea-level value in (2), we can assume the same exponential profile as for τ O 2 (λ; z). Hence

τ (∆λ) O 2 (z) = τ (∆λ)
O 2 (M ) exp(-z/H mol ).

Figure 3 shows the B-and A-band DOAS ratios r ∆λ (Ω 0 ; Ω 0 , τ c ) from ( 16 To summarize our stance on spectral integration, by considering only the effective absorption optical thickness in (2), we have implicitly ignored the impacts of spectral correlations. Because transmission laws that account for spectral correlations tend to raise the value of transmission for a given path length, making it less exponential and more linear, we expect this future improvement will reinforce not only the robustness of the spectral integral but also the validity of our other key assumption in [START_REF] Domoto | Frequency integration for radiative transfer problems involving homogeneous non-gray gases: The inverse transmission function[END_REF], namely, that all we know is the first moment of the path length within clouds.

Forward Model Bias Estimation

Following [START_REF] Ferlay | Toward new inferences about cloud structures from multidirectional measurements in the oxygen A-band: Middle-of-cloud pressure and cloud geometrical thickness from POLDER-3/PARASOL[END_REF], we can divide ( 16) by the known airmass factor (1/µ 0 + 1/µ) in (3), and thus define implicitly the "apparent" or "centroid" cloud-top altitude z

(app) top in τ O 2 (λ; z (app) top ) = -log r (obs) λ /(1/µ 0 + 1/µ) (observed) τ O 2 (λ; z top ) + µµ 0 [τ O 2 (λ; z)] ztop ztop-H × (1 + C(τ c , g, µ 0 )) (modeled) (18)
≈ z top -H mol log 1 + µ 0 µ e H/H mol -1 × (1 + C(τ c , g, µ 0 )) , (19)
irrespective of λ or ∆λ, i.e., whether using the A-or B-band (in this approximation). If, moreover, the cloud is geometrically thin in the sense that H H mol , then a 1 st -order

Taylor expansion leads to z (app) top ≈ z top -µ 0 µH × (1 + C(τ c , g, µ 0 )), now irrespective of H mol as well.

For EPIC, we have Ω ≈ -Ω 0 , hence µ ≈ µ 0 . We then see in [START_REF] Davis | Generalized radiative transfer theory for scattering by particles in an absorbing gas: Addressing both spatial and spectral integration in multi-angle remote sensing of optically thin aerosol layers[END_REF] that the expected negative bias is clearly at its greatest, for given H, at sub-solar observation geometry

(µ = µ 0 = 1): z (app) top ≈ z top -H × (1 + C(τ c , g, µ 0
)). See Fig. 2 in [START_REF] Ferlay | Toward new inferences about cloud structures from multidirectional measurements in the oxygen A-band: Middle-of-cloud pressure and cloud geometrical thickness from POLDER-3/PARASOL[END_REF] and Fig. 6 in [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF] for qualitative confirmation on observational and computational grounds, respectively.

In reality and in exact 1D RT, the maximum bias of predicted in ( 19) is a slight overestimation. Our Fig. 4 shows z top -z (app) top normalized to H from [START_REF] Davis | Generalized radiative transfer theory for scattering by particles in an absorbing gas: Addressing both spatial and spectral integration in multi-angle remote sensing of optically thin aerosol layers[END_REF] with µ = µ 0 and H mol = 8 km. We see that for oblique solar/view geometry, the bias is between 0 (at grazing sun/views) and H.

It is somewhat disconcerting that, in its final form [START_REF] Davis | Generalized radiative transfer theory for scattering by particles in an absorbing gas: Addressing both spatial and spectral integration in multi-angle remote sensing of optically thin aerosol layers[END_REF], our simplified model has no explicit spectral dependence and, therefore, cannot be used to distinguish A-and B-band cloud information contents, irrespective of how the spectral band integration is modeled. We would normally expect that access to two wavelengths/bands would lead to a way of extracting both z top and H, the parameters used here to define the cloud profile. Instead, they combine invariably into z (app) top in [START_REF] Davis | Generalized radiative transfer theory for scattering by particles in an absorbing gas: Addressing both spatial and spectral integration in multi-angle remote sensing of optically thin aerosol layers[END_REF]. Several approximations were made along the way, and the one where spectral dependence was all but lost is not immediately apparent. Careful scrutiny however leads back to the early step in [START_REF] Domoto | Frequency integration for radiative transfer problems involving homogeneous non-gray gases: The inverse transmission function[END_REF] where only the mean path length was assumed known for the sunlight that entered the cloud and was eventually reflected back to space. This lead to a simple exponential dependence on both above-cloud and in-cloud absorption by O 2 . If only two path length moments were known, e.g., mean and variance, then the in-cloud term R λ (Ω; Ω 0 , τ c ) obtained from [START_REF]DSCOVR: EPIC, Earth Polychromatic Imaging Camera[END_REF] would not be an exponential [START_REF] Bakan | Path length distributions of photons scattered in turbid atmospheres[END_REF].

Consequently, the dependence on τ O 2 (λ; 0) would not cancel out after the later-and quite reasonable-approximation that τ O 2 (λ; z) has the same exponential profile in z for all λ.

The insight gained here is that a diversity of path lengths in the scattering medium is crucial to the access to more than one cloud parameter. One straightforward way of diversifying in-cloud path length, even with a single O 2 channel, is by varying view angle, i.e., µ in expression [START_REF] Davis | Generalized radiative transfer theory for scattering by particles in an absorbing gas: Addressing both spatial and spectral integration in multi-angle remote sensing of optically thin aerosol layers[END_REF] for z (app) top . This is the option that was exercised by the POLDER/PARASOL mission, as previously mentioned in connection with [START_REF] Ferlay | Toward new inferences about cloud structures from multidirectional measurements in the oxygen A-band: Middle-of-cloud pressure and cloud geometrical thickness from POLDER-3/PARASOL[END_REF][START_REF] Desmons | Improved information about the vertical location and extent of monolayer clouds from POLDER3 measurements in the oxygen A-band[END_REF], but it is not available for EPIC. The alternative, this one available for EPIC, is to have more than one spectral channel. However, the forward model needs more sophistication than used here, where the goals are only (i) to estimate CTH retrieval bias when in-cloud scattering is neglected, and (ii) to demonstrate that instrument error needs to be small enough to distinguish the spectral signals. Just how small in EPIC's A-and B-band case is the topic we now broach.

14

The simplified DOAS ratio model in [START_REF] Young | Band Model Theory of Radiation Transport[END_REF] can also be used to translate measurement error on r ∆λ (• • • ) into uncertainty in the inferred z (app) top , which is obtained by inverting said model without the second (in-cloud path length) term, cf. ( 18)- [START_REF] Davis | Generalized radiative transfer theory for scattering by particles in an absorbing gas: Addressing both spatial and spectral integration in multi-angle remote sensing of optically thin aerosol layers[END_REF]. Taking differentials on both sides of ( 16), and using (1), we obtain

δr λ r λ ≈ 1 µ 0 + 1 µ k λ (z top ) + (µ + µ 0 ) k λ (z top -H) -k λ (z top ) δz (app) top ≈ (µ 0 + µ) 1 µ 0 µ -1 k λ (z top ) + k λ (z top -H) δz (app) top , (20)
where, for simplicity, we have neglected the pre-asymptotic correction term that multiplies the term in (µ 0 + µ) in the upper expression. Invoking the approximately exponential profile of

k λ (z) = - d dz τ O 2 (λ, z) ≈ τ O 2 (λ, 0) H mol exp(-z/H mol ), (21)
the band-effective O 2 absorption optical depth at sea level in (2), hence symbolically λ becomes ∆λ, and EPIC's viewing geometry (µ = µ 0 ), we have

δr ∆λ r ∆λ ≈ τ (∆λ) O 2 (2/µ 0 ) H mol 2µ 0 1 µ 2 0 + e H/H mol -1 exp (-z top /H mol ) δz (app) top . (22)
As discussed in [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF], a common assumption for the relative error in DOAS ratio, ∆r λ /r λ , is 1 to 1.5%. This contrasts with the ≈3% error assumed for absolute radiometry, but some of that error (e.g., calibration drift) cancels in the ratio. Thus, setting the right-hand side of ( 22) to 0.015 (1.5%), and solving for δz

(app)
top , we obtain the following estimate of retrieval error:

∆z (app) top ≈ 0.015 H mol τ (∆λ) O 2 × e ztop/H mol 2µ 0 µ -2 0 + e H/H mol -1 , (23)
where H mol = 8 km. For specificity, we take an airmass factor of 5: µ 0 = 0.4, θ 0 ≈ 66 • , which is approaching the upper limit of 6 in Fig. 2 for the A-and B-bands. These LUTs are displayed in Fig. 9 of [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF] (reproduced as Fig. 6 in the companion paper [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF]) for θ 0 = 40 • and three different choices of τ c .

The ranges of h sum go from ≈2 km to 20 km or more. In contrast, the full ranges of h diff are only 0.18, 0.20, and 0.55 km, respectively for τ c = 30, 10, and 5. In the joint retrieval, we would want to be able to locate the observation coordinate (h sum , h diff ) as precisely as possible to infer (z top , H) accurately. To be quantitative, we would want the measurement uncertainties (error bars) on (h sum , h diff ) to be, say, 5-to-10 times smaller than the overall range of the relevant LUT. For h diff that translates to an uncertainty (quantified by, e.g., "2 sigmas") no larger than ≈0.05 km.

We can now bring to bear the estimate of uncertainty on retrieved centroid cloud top height z

(app) top in [START_REF] Van De Hulst | Multiple Light Scattering -Tables, Formulas and Applications[END_REF]. The resulting uncertainty on h sum and h diff is the root-mean-square (RMS) sum of ∆z

(app) top

for the A-and B-bands. Figure 5 shows the outcome (i.e., "1 sigma") expressed in km. We clearly see that these sensor-induced uncertainties may be very reasonable for the horizontal axes (h sum ) in Fig. 9 of [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF], better still, if divided by √ 2 (i.e., the estimate of the uncertainty on the mean of z

(app) top

for the A-and B-bands.

However, the numbers in Fig. 5, especially if doubled to estimate the "2 sigma" range, will overwhelm the full extent of the vertical axes (h diff ) of [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF]'s Fig. 9. This is in spite of the undeniable sensitivity of h diff to H in the LUTs.

Conversely, to reduce uncertainty on z 5 Closing Remarks

Summary

By invoking the signal physics underlaying EPIC's differential optical absorption spectroscopic (DOAS) ratios for the A-and B-bands, we have shown that these two measurements are too redundant to allow for retrieving both cloud top height (CTH) and cloud geometrical thickness (CGT). In the absence of any independent knowledge of CTH, this important cloud property can be inferred from either or, better, both Aand B-band channels (adopting the average, weighted as needed). However, if CGT is neglected altogether, these estimates will be biased low due to the O 2 absorption path length cumulated inside the cloud. A bias-correction scheme is in order, and a reasonable approach used by others assumes an adiabatic lapse rate to relate CTH and cloud base height (i.e., CTH minus CGT). This finding is consistent with the outcome of the investigation by Xu et al. [START_REF] Xu | Passive remote sensing of altitude and optical depth of dust plumes using the oxygen A and B bands: First results from EPIC/DSCOVR at Lagrange-1 point[END_REF] of dust plume height retrieval using EPIC's A-and B-band channels.

The above conclusion is also completely aligned with that of the companion paper [START_REF] Davis | Cloud information content in EPIC/DSCOVR's oxygen A-and B-band channels: An optimal estimation approach[END_REF] by the same authors as here along with others in the same journal where the statistical formalism of optimal estimation was used. The two-pronged research described in the present pair of articles builds on a preliminary investigation by Yang et al. [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF] of EPIC's A-and B-band cloud sensing capability prior to DSCOVR's launch to the L 1 point, where the possibility of a joint retrieval of CTH and CGT was envisioned with an idealized instrument. Taking into account realistic levels of sensor noise makes it clear that only a robust retrieval of CTH should be targeted, and that is already a major contribution to cloud science for years to come.

A remarkable invariance property of mean path length

Before closing, we take note of a very interesting development in the optics of purely scattering media that relates directly to our present work with clouds and di-oxygen absorption, to diffuse optical tomography [START_REF] Arridge | Optical tomography in medical imaging[END_REF] in biomedical imaging [START_REF] Boas | Imaging the body with diffuse optical tomography[END_REF], and to many other applications. Using thermodynamical and weak light absorption arguments Blanco and Fournier [START_REF] Blanco | An invariance property of diffusive random walks[END_REF] showed theoretically that, in our notations,

ct (all) F = 4V /S, (24)
where:

• the superscript "(all)" means isotropic (Lambertian) illumination of the whole surface of the arbitrarily-shaped medium M, and we recall that the subscript "F " means integration over all escape angles to obtain hemispherical flux, in this case, perpendicular to the local outgoing normal, and averaging covers the whole boundary ∂M of the medium;

• V is the volume of M;

• S is the surface of M.

nor on the opacity of the medium: it can be void of scatterers (i.e., the uniform extinction coefficient σ vanishes, hence only ballistic trajectories occur); or it can be very opaque (σV /S 1); or anything in between. This mean path invariance property was extended by Pierrat et al. to multiple scattering theory using both physical optics and time-dependent 3D radiative transfer [START_REF] Pierrat | Invariance property of wave scattering through disordered media[END_REF]. Recently, Savo et al. [START_REF] Savo | Observation of mean path length invariance in light-scattering media[END_REF] demonstrated its experimental validity in the laboratory.

In our present study, we use plane-parallel media that can be viewed, e.g., as rectangular parallelepipeds of height H and square horizontal section of side L H.

We then have V = HL 2 and S = 2L 2 + 4HL, hence 4V /S = 2H/(1 + 2H/L) and lim L→∞ 4V /S = 2H is the corresponding prediction for ct

(all) F = R ct R + T ct T , (25)
where R 1 is the optically thick cloud's reflectivity, and T = 1 -R 1 is its transmittivity, ct R is the mean path for reflected light (any direction), denoted ct F in the main text, and ct T is its counterpart for transmitted light. With R approaching unity, hence vanishingly small T , our estimates of ct R in [START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF] and in [START_REF] Rodgers | Inverse Methods for Atmospheric Sounding: Theory and Practice[END_REF], after averaging over µ 0 (leading term only in the latter case) and over µ (in the former case), fall significantly short of 2H. Although, they at least have the linear dependance H, and none on any other cloud optical property to first order. Consequently, ct T in (25) must be large enough compared to H to compensate for the diminutive T in the weighted average. Appendix B draws on previously published results akin to the ones derived and used herein to show that ( 24) is indeed verified for light diffusing through plane-parallel slabs. To the best of our knowledge, therein is the first proof of the mean path length invariance property in [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF] that uses formal diffusion theory.

Outlook

Future work will focus, on the one hand, on supporting the operational implementation of an algorithm to obtain CTH from EPIC's A-and B-bands using an informed guess for CGT to minimize retrieval bias; uncertainty on the retrieved CTH will be reduced by using both oxygen bands, and stated as part of the retrieval. On the other hand, we will be investigating applications of [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF] to remote sensing and radiant energy budget estimation.

and

p 1 (τ, g, µ 0 ) = 24µ 0 1 -3µ 2 0 (2 -3(1 -g)µ 0 ) + 2 24 + µ 0 8 -18g -9(10 -3(2 -g)g)µ 0 -18(1 -3g)µ 2 0 + 81(1 -g) 2 µ 3 0 τ + 6(2 -3µ 0 )(3 + (4 -3µ 0 )µ 0 )(1 -g)τ 2 + 9µ 0 (2 -3µ 0 )(1 -g) 2 τ 3 .
(A.16) ) as function of (log 10 τ, µ 0 ) for g = 0.85, with µ 0 = 2 /3 highlighted since it is often used in the main text as a typical solar/viewing geometry for EPIC. Right: Same as left panel but using δ-Eddington rescaling [START_REF] Joseph | The delta-Eddington approximation for radiative flux transfer[END_REF], that is, to use g = (g -0.5)/(1 -0.5) = 0.7 and log 10 τ = log 10 τ + log 10 0.5.

We note, finally, that the denominator in (A.14) is a 3 rd -order polynomial in τ while p 0 (τ, g, µ 0 ) is only 2 nd -order, hence a slow decay of C(τ, g, µ 0 ) in 1/(1 -g)τ with a prefactor given by lim The prefactor (A.17) of the slow decay of C(τ, g, µ 0 ) in 1/(1 -g)τ is plotted versus (g, µ 0 ) over their full ranges, noting that g < 0 is associated with backward -peaked phased functions. The value at (0, 1) for isotropic scattering and normal incidence is highlighted, and it results in a slightly negative prefactor in [START_REF] Rodgers | Inverse Methods for Atmospheric Sounding: Theory and Practice[END_REF], that is, a pre-asymptotic approach to (15) from below rather than from above. This is consistent with the trend displayed in [START_REF] Van De Hulst | Multiple Light Scattering -Tables, Formulas and Applications[END_REF]p. 590,Fig. 17.8] where the case of µ = µ 0 = 1 is studied for isotropic (g = 0) scattering. Right: Zoom of the l-h panel into the region where 0.5 < g < 1; specific ranges for g = 0.85 and g = 0.7 are highlighted, corresponding respectively to the standard value of g for warm clouds and its (δ-Eddington) rescaled counterpart for f = 0. We note that the latter case has a narrow range of prefactor values, as it is close to the saddle point of the function of two variables. In both panels, the typical value of µ 0 = 2/3 (used in the main text) is highlighted.

τ →∞ (1 -g)τ × C(τ, g, µ 0 ) = (1 -g) 2 - 1 2 + 3µ 0 -(2 -3g)µ 0 , (A.

A.4 Verification & Validation

For a Monte Carlo solution of the time-dependent 1D RT equation for the light pulse starting at cloud top under normal incidence (µ 0 is unity), we refer to the Davis' [21, Fig. 2]. Agreement with the diffusion approximation on the predicted values of ct F /H is excellent over a large range of τ and two relevant values of g. Moreover, that is in spite of neglecting the time-derivative in (A.4), i.e., using bone fide time-dependent diffusion theory, not the improvement afforded by using the telegrapher's equations solved here. This is tantamount to a validation of the diffusion-type approximations used here because validation is about "solving the right equations" [START_REF] Roache | Verification and Validation in Computational Science and Engineering[END_REF]. Therefore, a model such as time-dependent 1D RT (implemented numerically with a Monte Carlo scheme) that is inherently more realistic than diffusion can provide validation data for a diffusion model. That is of course pending its own validation based on confrontation with realworld observations, even if they are laboratory-controlled instantiations of plane-parallel optical media [START_REF] Mishchenko | Radiative transfer theory verified by controlled laboratory experiments[END_REF].

Another check that is more qualitative, but brings more physical insight, is to show that ct F ∝ H follows from random walk theory, including the fact that dependencies on other cloud properties (τ, g) will at best be weak. Since we are now invoking a model that has less fidelity than our ODE-based model, we can only describe this exercise as a preliminary form of verification, which has been defined as "solving the equations right" [START_REF] Roache | Verification and Validation in Computational Science and Engineering[END_REF].

We start by recalling that the space-time Green function G(t, x) of the abovementioned diffusion problem in the absence of boundaries is the solution of ∂ t -D∇2 = δ(t)δ(x). It's expression for t > 0 is exp(-x 2 /4Dt)/(π4Dt) 3/2 . From there, two useful relations follow.

First comes, by integration, the well-known law of diffusion:

R 3 x 2 G(t, x)dx = x 2 = 6Dt. (A.18)
We recall here that D = c t /3, with t = 1/(1-g)σ being the so-called "transport meanfree-path," that is, the cumulative distance covered on average before the memory of the original direction has been all but erased in spite of the forward scattering tendency.

It is the mean length of each step in an equivalent isotropic random walk, as is implicit in the diffusion model.

Second is the lesser known "law of first return," which answers the question: What is the probability p 1st (t)dt of a random walk returning to its plane of origin between time t and t + dt? It can be shown [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF][START_REF] Redner | A Guide to First-Passage Processes[END_REF] that

p 1st (t)dt = c √ π t t ct 3/2 exp(-t /2ct), (A.19)
equivalently, p 1st (ct) = p 1st (t)/c. This PDF has a sharp increase from 0 at short times/paths, thanks to the exponential term, a peak at ct = t / √ 3, and a very long tail, decaying in ct -3/2 . In particular, both the mean and variance of ct are divergent.

Physically, a pulse of light entering a semi-infinite scattering medium will keep coming back for a very long time due to the multiple scattering that sends it very deep in the absence of any absorption.

What happens it the medium has a finite geometric thickness (H < ∞), but also a very large optical thickness, even scaled for the forward scattering tendency? In other words,

τ t = (1 -g)τ = H/ t 1.
Basically, there is a characteristic time/path ct H that the light takes to reach the cloud base in a substantial amount vis-à-vis the sum total of transmitted light. From (A. where ct H / t = ( 1 /2)(H/ t ) 2 , which is indeed ≫ 1 since H/ t is already 1 in the diffusion limit. Therefore, ct F (H) ≈ 2/π H, irrespective of the other length-scale in the problem, namely, t that, in turn, is determined by τ and g (along with H). In spite of all the approximations, even the prefactor, 2/π ≈ 0.8, is not far from the proper hemispherical average of (µ + µ 0 ), which is unity (hemispherical averages of µ 0 and µ contribute each 1/2).

≈) H for optically thick clouds, irrespective of τ and g.

B Verification of Mean Path Invariance for Diffuse

Propagation Through Plane-Parallel Slabs

In §5 we noted that a remarkable invariance property expressed in [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF] was recently uncovered for the mean path length ct (all) F that corresponds to isotropic irradiance uniformly distributed over every boundary point of the uniform but arbitrarily-shaped optical medium M, as well as directionally-integrating sensors covering the boundary ∂M . What is truly remarkable about the result is that it holds independently of the opacity. It can even be void of scattering particles, and all trajectories are ballistic. If not, then their density is immaterial, as is their phase function, as long as their is no absorption (i.e., conservative scattering, SSA is unity).

What makes the fact that ct (all) F is invariably four times the volume V of M divided by its surface S relevant to our study is that, by definition, "reflected" light comes from the illuminated surfaces of the medium [START_REF] Davis | A primer in 3D radiative transfer[END_REF]. So, in this case, all the light is reflected, even if it originated on the opposite side of the medium. A key element of the present study is that the mean path length for light reflected from a plane-parallel slab is proportional to its geometrical thickness H. What is V /S, if not a generalization of H (actually H/2) to arbitrary cloud geometry?

We noted in ( 25) that for a plane-parallel slab, ct (all) F is a weighted sum of the mean paths for reflected and transmitted light. However, we also noted that our present diffusion-theoretical estimates of ct R for the reflected light, which for the optically thick media amenable to the diffusion approximation are the most heavily weighted, fall short of the expected 2H from the invariance rule, even after the required angular integration (hemispherical averages of µ 0 and, as needed, µ are 1/2).

Here, we revisit some previous work [START_REF] Davis | Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Greenfunction theory in the diffusion domain[END_REF][START_REF] Davis | Space-time characteristics of light transmitted through dense clouds: A Green's function analysis[END_REF] on path length statistics for both reflected and transmitted light in the diffusion limit. However, in contrast with App. A, isotropic illumination is used rather than a collimated incoming beam, which is consistent with the set-up for the invariant mean path length result in [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF].

B.1 Reflection

Davis et al. [START_REF] Davis | Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Greenfunction theory in the diffusion domain[END_REF] formulated the concept of "off-beam," or "multiple-scattering," or "wide-field-of-view" lidar as a novel tool for cloud remote sensing, specifically targeting where τ t = H/ t = (1-g)τ c is the scaled optical thickness and χ is the so-called "extrapolation scale" expressed in units of the transport mean-free-path t = 1/(1 -g)σ, which factor χ, in the limit of vanishingly small τ t it increases from 2/3 to 4/3 (but not in this

Figure 1 :

 1 Figure 1: In-band EPIC filter functions (in arbitrary units), with full-column O 2 absorption optical depth for the standard mid-latitude summer atmosphere in the "line-by-line" limit (wavenumber increment δν = 0.01 cm -1 ): Left: B-band (δλ ≈ 0.5 pm). Right: A-band (δλ ≈ 0.6 pm).

Figure 2 :

 2 Figure 2: Plots of τ (∆λ) O 2 (M ) in (2)-(3) for A-and B-bands, with a reminder of the far more different filter-averaged τ O 2 (λ), and correspondence between M and θ 0 for EPIC.

  comes entirely from the absorbing gas inside the cloud, that is, [τ O 2 (λ; z)] ztop-H ztop . However, estimation of cumulative path inside the cloud is a nontrivial question best framed in the time domain. See[21, and references therein] for investigations of cumulative path length in the framework of multiple scattering lidar, with active cloud remote sensing in mind at VNIR wavelengths. Just as for EPIC, a monostatic lidar system is in very nearly backscattering observation geometry, but the incidence is invariably quasi-normal to the cloud's upper boundary. Imagine a short "pulse" of sunlight impinging on cloud top at time/path ct = 0. What is the distribution in time/path ct of the light emerging from the cloud? Let R(ct, Ω; Ω 0 , τ c ) be that impulse response expressed in km -1 . Then, assuming uniform gaseous absorption coefficient k λ in the cloud, one can compute the steady-state absorption spectrum R λ (Ω; Ω 0 , τ c ) from the so-called equivalence theorem[22, among others]:

  Figure3shows the B-and A-band DOAS ratios r ∆λ (Ω 0 ; Ω 0 , τ c ) from (16) for EPIC as functions of z top and H in [km] when µ = µ 0 = 2/3, hence an intermediate airmass of 3, and (τ c , g) = (30,0.85), scaled to (15,0.7), as explained in Appendix A, for improved accuracy of the diffusion approximation. We note the sensitivity in both bands to both cloud properties. Apparently, however, only the range of DOAS values seems to change, not the pattern of dependence on cloud parameters. Further on, we discuss the ramifications of this similarity-but not identity, due to the nonlinear dependence on τ (∆λ) O 2 from (2).

Figure 3 :

 3 Figure 3: DOAS ratios for EPIC's B-(left) and A-band (right) as functions of the two cloud properties of interest: CTH z top and CGT H, both in [km]. Other parameters are provided in main text.

Figure 4 :

 4 Figure 4: Normalized CTH bias z top -z (app) top /H from (19) for EPIC's solar/viewing geometry (µ 0 = µ), and H mol = 8 km.

  enough to fit 5-to-10 observed "points" into the vertical axes of[START_REF] Yang | A method of retrieving cloud top height and cloud geometrical thickness with oxygen A and B bands for the Deep Space Climate Observatory (DSCOVR) mission: Radiative transfer simulations[END_REF]'s Fig.9, we would need to reduce the ranges obtained in our Fig.5to values no larger than ≈0.025 km, i.e., 10-to-20 times smaller. That would require relative uncertainties on the DOAS ratios to be 10-to-20 times smaller than 1.5%, which is not feasible with current or foreseeably futuristic space-based sensors.

Figure 5 :

 5 Figure 5: Estimated uncertainties on h sum or h diff in Fig. 9 of [8]-equivalently, Fig. 6 of [7]-induced by instrumental error, modeled as the RMS sum of uncertainties on cloud centroid heights from the A-and B-bands. Left: Dependence on µ 0 in safe zone ( 1/3) for sphericity effects, and on z top in km when H = 2 km (µ 0 = cos 40 • highlighted), using τ (∆λ) O 2 (5) values from Fig. 2. Right: Dependence on H in km and z top ≥ H + 0.5 when µ 0 = cos 40 • (as in [8]).

Figure A. 1

 1 Figure A.1 shows log 10 C(τ, g, µ 0 ) as a function of log 10 τ and µ 0 when g = 0.85, the canonical value for liquid water clouds, in the l-h panel. In the r-h panel, the same function is plotted, but for scaled cloud properties using the so-called δ-Eddington scheme[START_REF] Joseph | The delta-Eddington approximation for radiative flux transfer[END_REF]: (τ, g) becomes (τ , g ) = ((1 -f )τ, (g -f )/(1 -f )), with f = 0.5. The rationale for this rescaling is that the accuracy of the present diffusion-type approximation is improved dramatically, especially at low τ , by removing the ≈50% of the single-scattered light that goes into the forward peak of the phase function and putting it back into the non-scattered beam, and then recomputing the asymmetry factor as g to ensure that (1 -g )τ = (1 -g)τ .

Figure A. 1 :

 1 Figure A.1: Left: log 10 C(τ, g, µ 0) as function of (log 10 τ, µ 0 ) for g = 0.85, with µ 0 = 2 /3 highlighted since it is often used in the main text as a typical solar/viewing geometry for EPIC. Right: Same as left panel but using δ-Eddington rescaling[START_REF] Joseph | The delta-Eddington approximation for radiative flux transfer[END_REF], that is, to use g = (g -0.5)/(1 -0.5) = 0.7 and log 10 τ = log 10 τ + log 10 0.5.

17

 17

  ) which is plotted and discussed in Fig. A.2. We can clearly see the decay in τ -1 as τ increases without bound in Fig. A.1.

Figure A. 2 :

 2 Figure A.2: Left:The prefactor (A.17) of the slow decay of C(τ, g, µ 0 ) in 1/(1 -g)τ is plotted versus (g, µ 0 ) over their full ranges, noting that g < 0 is associated with backward -peaked phased functions. The value at (0, 1) for isotropic scattering and normal incidence is highlighted, and it results in a slightly negative prefactor in[START_REF] Rodgers | Inverse Methods for Atmospheric Sounding: Theory and Practice[END_REF], that is, a pre-asymptotic approach to (15) from below rather than from above. This is consistent with the trend displayed in [23, p. 590, Fig.17.8] where the case of µ = µ 0 = 1 is studied for isotropic (g = 0) scattering. Right: Zoom of the l-h panel into the region where 0.5 < g < 1; specific ranges for g = 0.85 and g = 0.7 are highlighted, corresponding respectively to the standard value of g for warm clouds and its (δ-Eddington) rescaled counterpart for f = 0. We note that the latter case has a narrow range of prefactor values, as it is close to the saddle point of the function of two variables. In both panels, the typical value of µ 0 = 2/3 (used in the main text) is highlighted.

H

  and τ c . One goal is to obtain τ c without requiring absolute radiometric calibration nor loss of sensitivity at large optical thickness. The loss of sensitivity to τ c can be clearly seen, e.g., in the well-known two-steam/diffusion expression for the albedo of a plane-parallel cloud: R(τ t ) = τ t 2χ + τ t , (B.1)

Figure ) .Figure B. 1 :

 )1 Figure). The right panel shows the transmitted and reflected contributions to ct (all) F

  , past which sphericity effects start to matter. With that assumption, hence τ

			(∆λ) O 2	≈ 0.33 and 0.14 for A-band and B-band,
	respectively. This yields ∆z	(app) top	≈ 0.14 and 0.32 km respectively for A-and B-band
	estimates for a low cloud layer, (z top , H) = (2.5,2) [km]. For the same cloud at a height
	of ≈8 km, we will have roughly e ≈ 2.7 times larger uncertainties. At smaller (larger)
	airmasses, we will have somewhat larger (smaller) uncertainties, due to the decreased
	(increased) amounts of O 2 absorption. In the 1-4 km range, H has a minor role in (23)
	since e H/8 -1 has to compete with 1/µ 2 0 .
	The joint (z top , H) retrieval method proposed by Yang et al. [8] uses two-entry look-

up tables (LUTs) for those cloud parameters as a function of two observable quantities denoted h sum and h diff , which are in essence the sum and difference of the estimates of z (app) top

  [START_REF] Conley | Extension of the weak-line approximation and application to correlated-k methods[END_REF], we can estimate it intuitively as2 ct H ≈ cH 2 /6D = H 2 /2 t = Hτ t /2,which is H and thus ≫ t . That characteristic time/path acts in essence as a truncation for the heavy tail of p 1st (ct) in (A.19). We thus seek the mean ct F (H) of this truncated PDF: p H (ct) ≈ p 1st (ct) if ct < ct H , and 0 otherwise. That mean path can be estimated as ct F(H) = ∞ 0 ct p H (ct)dct ≈ π -1/2 ct H 0 ct (ct/ t ) -3/2 dct/ t , ignoringthe exponential "build-up" term. This leads to ct F (H) ≈ ( t /

	√ π) [ √ x ]	ct H / t x=0 /( 1 /2),

Although van de Hulst[START_REF] Van De Hulst | Multiple Light Scattering -Tables, Formulas and Applications[END_REF] provides neither proof nor speculation, we confidently surmise that (8) holds for any phase function in the limit τ c 1/(1 -g). This assertion is based on the fact that, in the diffusion approximation, g only appears in the pre-asymptotic correction term in[START_REF] Rodgers | Inverse Methods for Atmospheric Sounding: Theory and Practice[END_REF].

In solid-state physics, 2t H = H 2 /c t is known as the Thouless dwell time[START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF].

Acknowledgments

This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA). ABD, QM and YY acknowledge partial financial support from NASA's ROSES program element for DSCOVR Earth Science Algorithms managed by Dr. Richard Eckman. AM is supported by the NASA/GSFC DSCOVR mission funded by NASA Earth Science Division. ABD acknowledges support from the MEASURES project at JPL managed by Dr. Joshua K. Willis. NF acknowledges financial support from CNES, Université de Lille and Région Haut-de-France. Moreover, part of the discussion and work between authors has been made possible though support from the Programme National de Télédétection Spatiale (PNTS, http://www.insu.cnrs.fr/pnts), under grant #PNTS-2017-04. We thank Jay Herman, Marine Desmons, François Thieuleux, Jérôme Riédi, Céline Cornet, Philippe Dubuisson, Claudine Vanbauce, Frédéric Parol, Klaus Pfeilsticker, Alex Kokhanovsky, Lee Harrison, Ralf Bennartz, Dennis O'Brien, Graeme Stephens, Suniti Sanghavi, Mark Richardson, Dave Crisp, Feng Xu, Sid Redner, and others for many fruitful discussions on probing clouds with the O 2 A-band. We thank Stéphane Blanco,

Richard Fournier in particular for very interesting and stimulating discussions on the invariance property of mean path length for diffusion random walk. Finally, penetrating commentary by the anonymous reviewers of this manuscript was highly appreciated.

We need an explicit expression for mean in-cloud pathlength ct as a function of cloud properties, namely, CGT H, COT τ , phase function asymmetry factor g, when the solar beam is either normal or oblique, which we represent with µ 0 = cosSZA.

In the diffusion approximation, we limit the spherical harmonic expansion of the time/path-dependent diffuse radiance field I(ct, z, Ω) to the 0 th -and 1 st -order terms. In plane-parallel geometry (hence 1D RT), letting µ = Ω z , we have:

irrespective of the azimuthal angle, where J(ct, z) is the scalar (a.k.a. actinic) flux and F (ct, z) the net vertical vector flux. Note that J(ct, z) is closely related to the mean (directionally-averaged) radiance J(ct, z)/(4π) and radiant energy density

The phase function expansion in spherical harmonics is similarly truncated:

where µ s = Ω • Ω is the cosine of the scattering angle, and g is its mean value, viewing p(µ s ) as a probability distribution; see [START_REF] Marshak | Earth observations from DSCOVR/EPIC instrument[END_REF] in main text. Note that µ (s) in (A.1) and (A.2) is simply the 1 st -order Legendre polynomial while F (ct, z) and g are its respective coefficients in the two expansions.

Substitution of these truncated expansions in spherical harmonics into the timedependent 1D RT equation leads to two coupled partial differential equations (PDEs); see, e.g., [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF]. First, we have the exact "continuity" equation that expresses the conservation of radiant energy:

where σ s/a is the scattering/absorption coefficient and F 0 is the time-integrated incoming irradiance. Second, we have the approximate "constituent" equation that expresses how the radiant energy density flow (net flux) is driven by the gradient in its density:

(A.4) Remark: Neglecting ∂F/∂ct in the above, as is usual in the time -dependent diffusion approximation per se, and noting that q F (ct, z) vanishes far from sources, we recognize

as the diffusion coefficient in Fick's law:

Substituting Fick's law into the continuity equation (A.3) with no absorption (σ a = 0) and, again, far from sources (q J ≈ 0), yields the classic parabolic PDE for diffusion processes such as heat transfer:

Now, not neglecting ∂F/∂ct leads to the so-called "telegrapher's PDEs," which is a more accurate representation of the original linear transport problem than the diffusion approximation, especially at early times (short paths) [START_REF] Weiss | Some applications of persistent random walks and the telegrapher's equation[END_REF], and that is the course we keep in the following.

The source terms q J (ct, z) and q F (ct, z) specified in the above PDEs model a pulsed version of solar irradiance impinging on the plane-parallel optical medium at ct = z = 0.

As in the main text, we will not consider absorption by the scattering particles (water does not absorb much at the O 2 A-and B-band wavelengths), hence σ a = 0 and the (total) extinction coefficient is σ = σ s .

The required boundary conditions (BCs) express that no diffuse radiation is entering the plane-parallel medium at its top (z = 0), nor at its base (z = H), at any time past the instant the pulse arrives at the upper boundary. Rephrasing this as a statement for incoming hemispherical fluxes, and going back to (A.1), these BCs can be expressed simply as:

for all ct > 0.

For A-and B-band studies it is legitimate to neglect cloud droplet absorption: (σ s , σ a ) becomes (σ, 0). In that context, we can take Laplace transforms of (A.3)-(A.6), recalling from (6) that the Laplace conjugate variable of path ct is gaseous absorption k, and this leads to an equivalent steady-state problem expressed with coupled ordinary differential questions (ODEs):

To complement these coupled ODEs, Laplace-transformed BCs in (A.5)-(A.6) are:

This completes the definition of the boundary-value problem to solve.

A.2 Solution

Standard methods can be used to solve the above problem of coupled 1st-order ODEs. In the present study, however, we are not interested in the solution [J, F ](k, z) everywhere.

Specifically, we require reflected flux at z = 0, which is Ĵ(k, 0)-2 F (k, 0)]/4 = Ĵ(k, 0)/2, using the upper BC in (A.9). Normalized by the incoming downwelling flux, we obtain the cloud's reflectivity:

This partial answer to our question about ct is given explicitly in [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF] as a complicated function of four variables (kH, τ, g, µ 0 ) where σ and H have been combined into the (nondimensional) COT, τ = σH (denoted τ c in the main text), and we note that k and H always occur as a non-dimensional product.

Although it can be done in closed form [START_REF] Polonsky | Off-beam cloud lidar: A new diffusion model and an analysis of LITE returns[END_REF], we will not need to compute here the inverse Laplace transform of R(k,

The last step to obtain ct is to expand A.11 into a 1 st -order Taylor series in k:

Finally, we invoke (7) from the main text, which is a consequence of the fundamental equivalence relation in [START_REF]DSCOVR: EPIC, Earth Polychromatic Imaging Camera[END_REF]. In short, we have:

The outcome, using R(kH, τ, g, µ 0 ) from [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF], is thus the product of H and a complicated non-dimensional function of (τ, g, µ 0 ). Basically, we are looking at a rational function in multiple variables with additional polynomials multiplying exp(-τ /µ 0 ) in both numerator and denominator.

A.3 Result

The mean path length in (A.13) for cloud-reflected light in non-dimensionalized form is ct F /H, which remains a function of (τ, g, µ 0 ). Then, taking a cue from [START_REF] Rodgers | Inverse Methods for Atmospheric Sounding: Theory and Practice[END_REF] in the main text, we can write it as (2/3 + µ 0 ) × [1 + C(τ, g, µ 0 )]. We have thus identified explicitly the leading term for ct F when τ → ∞, namely, (2/3 + µ 0 )H. The pre-asymptotic correction term is then isolated, and it reduces to the following expression:

was used extensively in §A.4 on the approach to ct R via random walk statistics rather than PDEs and numerical simulation (generally with Monte Carlo techniques). The numerical value of χ is determined by comparison with a more accurate representation of the transport problem at hand.

Primarily with the inference of H in mind, the authors used the simplest possible model for the ratio ct R /H based on isotropic illumination in the diffusion limit that comes hand-in-hand with the simple expression for R in (B.1). They show that

where C R (x) = (x + 3/2)/2x(x + 1).

B.2 Transmission

For other purposes, such as predicting the waveform from cloud-to-ground lightning flashes as observed from space through an optically thick cloud, Davis and Marshak [START_REF] Davis | Space-time characteristics of light transmitted through dense clouds: A Green's function analysis[END_REF] did the same as in [START_REF] Davis | Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Greenfunction theory in the diffusion domain[END_REF], but for transmitted light, thus obtaining a simple expression for the ratio ct T /H based on isotropic illumination:

where C T (x) = (4x + 3)/2x(x + 1).

B.3 Reflection & Transmission

Detailed derivations of all of the above expressions follow the same steps as laid out in Appendix A, and can be found in the review paper by Davis, Polonsky, and Marshak [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF] on diffusion of solar and laser radiation in plane-parallel media, as an idealized model for optically thick stratiform clouds. Apart from neglecting the ∂F/∂ct term in (A.4) to be in the framework of bone fide diffusion theory, the only differences are:

• no source terms in the continuity (A.3) or constituent (A.4) equations;

• instead the boundary/initial conditions in (A.5) and (A.6) become respectively J(ct, 0) + 3χF (ct, 0) = 4δ(ct) and J(ct, H) -3χF (ct, H) = 0. Collecting all the above expressions pulled from [START_REF] Davis | Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Greenfunction theory in the diffusion domain[END_REF][START_REF] Davis | Space-time characteristics of light transmitted through dense clouds: A Green's function analysis[END_REF], or [START_REF] Davis | Space-time Green functions for diffusive radiation transport, in application to active and passive cloud probing[END_REF], and recalling that T (τ t ) = 1 -R(τ t ) in the absence of absorption, we can compute ct


Louis Hennet

James W E Drewitt

Daniel R Neuville

Viviana Cristiglio

Jad Kozaily

Séverine Brassamin

Didier Zanghi

Henry E Fischer

Neutron diffraction of calcium aluminosilicate glasses and melts

The combination of neutron diffraction with aerodynamic levitation and laser heating, pioneered by Neville Greaves and co-workers about 15 years ago, is an important tool for studying the structure of liquid melts. Since the first work on liquid Al2O3 published in 2001, the technique has been largely improved and experiments are now routinely performed at neutron sources, providing interesting structural information on various materials.

In this paper, the structure of glass-forming compounds in the system CaO-Al2O3-SiO2 was measured by applying neutron diffraction with aerodynamic levitation. Results obtained in the liquid state above the melting point and from the glass at room temperatures are presented. Various compositions were studied by increasing the silica content and by changing the ratio CaO/Al2O3. As observed using other methods, the main structural changes relate to modification of the Al-O short range order.

Introduction

Calcium aluminosilicate, CaO-Al2O3-SiO2 (CAS), glasses and melts are of both technological and scientific interest. In particular their good optical and mechanical properties [1,2] and their refractory nature make them attractive for a wide range of industrial applications, including the production of concrete [3,4]. CAS melts are also the main components of slags produced in furnaces in coal and steel industries. In geosciences, they are of fundamental importance as they form a significant fraction of the composition of geological magmas [5,6,7].

The thermodynamic and dynamic properties of CAS show a strong dependence with the microscopic structure [START_REF] Mysen | Structure and Properties of Silicate Melts[END_REF]. It is, therefore, important to study in detail the high temperature properties of the melts. In order to study the liquid state, we used the aerodynamic levitation technique combined with CO2 laser heating. This technique was pioneered about 15 years ago by Neville Greaves with teams from the University of Aberystwyth (UK) and the CNRS-CEMHTI in Orléans (France) [START_REF] Landron | [END_REF]10].

In this paper, we present results obtained from various CAS glass compositions at both room temperature and in the liquid state. Some of these glass compositions have already been studied by x-ray diffraction at room temperature [11,12] but only a few have been analyzed by neutron diffraction, in particular along the join R=1.57 [11,13]. Here and in the following, the R value is defined as the ratio CaO/Al2O3. In the liquid state only the compositions without silica have been studied extensively [14,15]. Our first results obtained for CAS liquids with low silica content have been reported recently in combination with Molecular Dynamics (MD) simulations. [16]. A recent work is also devoted to the study of CAS slags with other compositions [17]. The objective of the present investigation is to complete these previous works by extending the range of CAS compositions studied by neutron diffraction in order to better understand the structure of the glasses and melts, in relation with the observed properties.

Technical details

Sample preparation

The samples were prepared from high purity crystalline CaCO3, Al2O3 and SiO2 powders. The appropriate quantities of powders were melted at 1600°C for 4 hours in a platinum crucible and then quenched to room temperature. The resulting materials were grinded and re-melted several times in order to ensure a good homogeneity of the glasses. A full description of the sample preparation is described in Neuville et al. [18]. The required quantity of material for each sample was then levitated in air using an argon-oxygen (~3%) gas flow and melted by laser heating. Rapid quenching, achieved by switching off the laser power, resulted in the formation, on cooling, of solid spherical samples with diameters of 4 mm (for the glass measurements) and 2.7 mm (to investigate the liquid state under levitation). The samples studied are listed in Table I. As in previous works, we employ the notation Cax.y, where x and y refer to the molar concentration of SiO2 and Al2O3, respectively, and 1-x-y is the CaO molar content. The precise molar compositions of the glasses are reported in Neuville et al [7,19].

Neutron diffraction

The neutron diffraction (ND) measurements were made using the aerodynamic levitation and laser heating device installed on the D4C diffractometer [20] at the Institut Laue-Langevin (Grenoble, France), as described in ref [21]. Diffraction measurements were taken for the liquid samples at temperatures above the melting point for all studied compositions, for the empty levitation device inside the diffraction chamber, and for vanadium samples with the aim of making an absolute normalization of the scattering intensity to barns.st -1 .atom -1 . The neutron wavelength and zero-angle shift were determined by fitting the positions of the Bragg peaks obtained from a reference nickel sample. The neutron wavelength used during the experiment was 0.4969 Å, giving a scattering vector Q-range of 0.3-23.6 Å -1 . The data processing was performed using the program CORRECT [START_REF] Howe | NFL Studsvik Internal Report CORRECT: a correction programme for neutron diffraction data[END_REF] which corrects the data for attenuation, background, multiple scattering and inelasticity (Placzek) effects. Since, for various technical reasons, there can be appreciable uncertainties with respect to the relative beam-illuminated volumes of the vanadium samples compared to the CAS samples using the levitation technique, we found it more accurate and reliable to perform an auto-normalization of the scattered intensity from the CAS samples. Specifically, the high-Q asymptote of the sample's intensity (after accounting for the Placzek inelasticity effect) was scaled so as to be equal to the theoretical self-scattering that depends only on the known composition of the sample. We then also confirmed that the atomic number density required for accurate normalization of the r-space function G(r) (see equation in section 3.2) was equal to the theoretical atomic number density, thereby acting as an r-space consistency check of our auto-normalization performed in Q-space.

Experimental results

Structure factors

The total structure factors S(Q) of all studied samples at room temperature and in the liquid state are presented in figure 1. For all compositions, room and high temperature measurements are very similar with only a few differences due principally to thermal effects which increase the structural disorder. This means that the short range order is very close in the glasses and melts. As observed for the compositions along the composition lines R=1 and 3, by increasing the SiO2 content, the first peak (FSDP) in ) (Q S shifts towards lower Q values with a higher intensity. This shows that the degree of intermediate-range order, higher in the glassy state, is improved by incorporating silica into the structure. Although the FSDPs are less visible in the liquid state, they seem to be at the same positions found for the glasses and the same Q shift is observed with increasing SiO2 content.

Pair distribution functions

The total pair distribution function is calculated from the structure factor by the Fourier transform:

     0 0 2 d ) sin( ] 1 ) ( [ 2 1 1 ) ( Q Qr Q Q S r r G   ,
where 0 denotes the atomic number density. All values are listed in Table 1. For glasses they were calculated using densities measured with the Archimedean method applying toluene as immersion liquid [7,[START_REF] Neuville | [END_REF]. For melts, they were derived from densities determined by Courtial and Dingwell [START_REF] Courtial | [END_REF].. Figure 2 shows the total pair distribution functions G(r) for all the compositions studied along the joins 1 and 3.

The calcium aluminate compositions Ca0.50 and Ca0.25 have previously been studied in detail using neutron and x-ray scattering techniques in combination with MD and RMC simulations [14,25]. For these two compositions, the first peak found respectively at 1.76 and 1.74Å in the glass arises purely from the contribution of the nearest neighbour Al-O correlations. On melting, the observed shift of its position towards higher values is characteristic of an increase of the proportion of AlO5 units in the structure (see data reported in Table I). In all G(r)s, the peak at around 2. largely overlaps the peak and make the analysis more difficult. For composition along the join R=1.57, a detailed structural study have been made using x-ray and neutron diffraction experiment [13] and with Reverse Monte Carlo and MD simulations [27].

NMR experiments performed on all these glass compositions show that Al is 4-fold coordinated along the join R=3 up to 20% silica content and with small amounts of AlO5 units above and along the join R=1 (values are reported in table I for all compositions) [19].

For all liquids, only a small enlargement is observed on the high r side of the first peak. This is in accordance with an increase of the number of the AlO5 units observed to elongate the Al-O distance in the Ca0.50 melt [26]. With CAS compounds, cation-cation correlations contribute much less to the total G(r) than cation-O bonds and give rise only to small features found around 3.5 Å.

The determination of the coordination numbers and bond distances is performed using a Gaussian fit to the correlation function T(r)=4r0g(r) as described in Hannon and Parker [28]. Results of the fits are presented in figure 3 for glasses (a) and melts (b). In each case, only the two extreme compositions along the lines R= 1 and 3 are presented. 2. This shows that the shape of the wide O-O correlations which give distances between 2.5 and 3.5 Å can be reasonably described by the contribution of these 3 pairs. However, higher coordinated Ca found in previous works [13,29] would give rise to O-O correlations at positions between those for AlO4 and CaO6 polyhedra and cannot be excluded. Indeed, the Ca-O correlations for 7-and 8fold coordinated Ca are expected around 2.4 and 2.5 Å and if present they are thus difficult to resolve.

As seen previously, accurate information about the Ca-O local structure is difficult to obtain due to large overlapping contributions from the O-O correlations and temperature effects which broaden the peaks. This precludes the determination of Ca-O coordination number from our total structure measurements. By comparison to XANES experiments at the Ca-Kedge, Ca was estimated to be in regular 6-fold coordination in C3A glass (Ca0.25), 7fold coordination in CA (Ca50.00) with localized 6-to 7-fold coordination sites in CAS glasses [18,30]. A recent NDIS study of the Ca50.00 composition found a Ca coordination number of 6.2 in the liquid state and around 6.4 in the glass [26]. This shows that the use of more selective techniques than conventional ND are important in order to more precisely determine the atomic arrangement around Ca atoms. Recently, NDIS experiments have been made to investigate the structure of liquid Ca0.25 [31].

Discussion

As shown in various papers [26,29,30], the Ca0.25 glass (R=3) has a structure composed by tetrahedraly coordinated Al, Q 2 species with 2 bridging (BO) and 2 nonbridging (NBO) oxygen atoms, and Ca in a regular six fold coordination. With increasing the Al2O3 content, Al changes from Q 2 to a mixture of Q 3 and Q 4 species for Ca0.39 (R=1.57) [27] and Q 4 in the Ca0.50 glass (R=1) associated to some proportion of AlO5 units to ensure the cationic mobility as shown in [19,30]. This change in the Q species distribution implies a decrease of the number of NBO leading to the observed increase of the viscosity with the Al2O3 content [30].

With adding silica along the join R = 3, the distribution of Al into Q species increases also from Q 2 to Q 4 for the Ca23.21 (not studied in this paper) as shown in Al NMR, Raman spectroscopy and XANES at the Al K-edge [19]. In this case, the increase of the SiO2 content, produces an increase of the polymerization of Al from Q 2 to Q 4 because Si atoms are in Q 0 species and concentrate all NBOs around them. For the Ca33.18 composition, the Q species of Si increase from Q 0 to Q 1 whereas Al Q species stay in Q 4 and some AlO5 units appears.

Along the join R=1, which is the charge compensation line, the glasses are then fully polymerised and by using XANES experiments at the Ca absorption edge, Neuville et al explained the evolution of the FSDP by an increase of the ring size from 4 to 5 members for low and high silica content [18]. When R is increased, Ca atoms play the role of network modifiers with the creation of NBOs leading to less ordered structure.

In the silicate glasses, the first peak in G(r) is a weighted sum of the Si-O and Al-O correlations. By fixing the tetrahedral configuration of the SiO4 units during the fit to T(r), it is possible to derive reasonable Al-O bond lengths of 1.74-1.76 Å and 1.76-1.78 Å in the glasses and melts, respectively. The Al-O coordination numbers are slightly higher in the melts, reflecting an increased proportion of AlO5 units for most of the compositions. It can be noted that with an error bar of ± 0.5 neutron scattering technique is not sensitive enough to reproduce the evolution of the coordination numbers determined with NMR at room temperature (see table I). For Ca0.25, if the distance Al-O at room temperature found at 1.74 Å is characteristic of purely 4-fold coordinated Al in accordance with the amount of AlO5 (Table1), the found coordination number of 4.2 is a bit high, although it is within the error bar of ±0.5. This could be explained with a possible small influence of the Ca-O correlation (highest concentration) but more probably by an overestimation of the density, measured on a sample using a different cooling rate. New density measurements will be performed directly on the measured sample and ongoing x-ray scattering experiment on theses glass compositions will also enable to conclude. At high temperatures, there is no precise determination of the AlO5 content and the main results come from MD simulation (some of them are reported in Table I). For C3A, one can see that different values can be obtained, depending on the calculation method.

The study of Ca-O correlations is more difficult due to large overlapping O-O contributions. From a simple Gaussian fit, it seems that the bond distances (summarized in Table I) and thus the coordination number, doesn't change by increasing the silica content along the joins R=1 and 3. Depending on the temperature and the composition, the Ca-O coordination number is then expected to be between 6 and 7 by comparison with a previous study [26,30]. However, a more sophisticated fit including all contributions is needed to get more accurate information on the Ca-O correlation and take into account existing Ca with higher coordination.

The small effect of the increase of the silica content was also observed by Cormier et al along the join R=1.57 They found that even at low silica contents, silicon becomes part of the aluminate network and thus attributed the changes observed on the dynamics to structural rearrangements at medium range order.

The large number of atomic pairs in the CAS system complicates the data analysis. In order to have a full description of the short range order for these compositions, additional high temperature EXAFS measurements at the Ca-K absorption edge would be useful. Indeed, existing works are limited to very few melt compositions [30]. Using diffraction, it is necessary to use more selective techniques. The Ca X-ray absorption edge at 4.038 keV precludes the use of anomalous scattering experiments as performed with liquid Y2O3 [32].

NDIS is then a good solution, despite the long counting times required due to the relatively small size of the levitated spheres.

In addition of these experimental techniques, interesting information can be also derived from MD and/or RMC simulations. Drewitt et al studied various calcium aluminate melts including Ca0.50 and Ca0.25 [14]. From these simulations, they derived information about the distribution of coordination numbers for Ca and Al and the various bond angle distributions provide insight into the connectivity of polyhedra. L. Cormier et al studied in details some glasses along the join R=1.57 [27] and in addition of the previous informationthey got also information on the ring statistics, the distribution of the Q species, the number of NBOs, or the existence of triclusters. This interplay between triclusters, NbOs and AlO5 units was also studied by MD simulation for the low silica content composition along the join R=1 [16] enabling to derive also valuable information on the dynamics [33].

Conclusion

This study shows that the use of neutron diffraction combined with levitation techniques is an important tool for deriving structural information from liquid materials and to obtain useful information about coordination numbers and bond lengths.

In this work we showed that limited structural changes are observed by increasing the ratio CaO/Al2O3 or the SiO2 content in CAS glasses. At high temperatures a small increase in the Al-O coordination number is observed. This is due to the presence of a larger fraction of AlO5 units. The short range order around Ca remains almost unchanged over the whole composition range studied. These observations are in good agreement with previous results

obtained with other techniques. The combination of these results with simulations will make it possible to derive reliable information in the liquid state and in particular about the dynamics.

Figure 1 .

 1 Figure 1. Total structure factors S(Q) of the CAS glasses and liquids for R=0.8 and 1 (a) and R=1.57 and 3 (b). The liquid temperatures are reported in the figure.

Figure 2 .

 2 Figure 2. Total pair distribution functions G(r) for the CAS glasses and liquids along the composition lines R=1 (a) and R=3 (b). The positions of the various O-O correlations, as calculated assuming SiO4 and AlO4 tetrahedra and CaO6 octahedra, are shown as a guide. The oscillations above r=4 Å arise from correlations between second neighbours.

  3-2.35 Å arises from Ca-O bonds and is less resolved in the melt. In particular, the Ca-O peak broadens in the Ca0.25 composition and becomes barely resolvable from the next O-O correlations in the Ca0.50 melt, thus complicating the structural analysis. Further measurements of the Ca0.50 glass and melt have been made by applying the method of neutron diffraction with isotopic substitution (NDIS) [26], which combined with the use of MD simulations, provides much more detailed structural information on the Ca-O coordination environment. With the introduction of silica, the first peak corresponds to an overlapped contribution from both the Al-O and the Si-O nearest neighbour pairs. Its position moves gradually toward the Si-O bond distance as the SiO2 content is increased. It becomes more difficult to determine the respective contributions of the Si-O and Al-O bonds, and the Ca-O correlations are also affected by the associated increase of the O-O contribution from SiO4 tetrahedra that

Figure 3 .

 3 Figure 3. Total correlation function T(r) for the compositions bearing 0 or 33% of SiO2 along the joins R=1 and 3 for glasses (a) and melts (b). The partial fits and the final result are compared with the experimental data.

  Å in the liquid state. For all samples, the average coordination numbers of Al-O bonds are found to be around 4 in accordance with NMR results[19]. Assuming perfect tetrahedra (angle of 109.47), it is possible to estimate the O-O bond length for SiO4 and AlO4 as 2.62

	and 2.83 Å, respectively, in the glass. The Ca-O distance found around 2.32 Å is
	characteristic of 6-fold coordinated Ca as found in previous works [13,14,16,30]. Considering
	perfect CaO6 octahedra, an O-O distance of 3.28 Å is calculated (2.32 x √2). The positions of
	these three calculated O-O bond distances are indicated in figure

The complete results are summarized in table I.

For the silicate samples, the Gaussian fit was performed by fixing the coordination number of the Si-O bond around 4 leading to a Si-O bond length of 1.62 Å in the glasses and 1.65

Corresponding author: Louis.hennet@cnrs-orleans.fr

Acknowledgements

We thank Alain Bertoni for help with the diffraction experiments and the Institut Laue-Langevin for the beamtime allocation. We also thank the French National Research Agency for its financial support under the grant No. ANR-13-BS08-0012-01.

R

T (°C) 0 (Å -3 ) r Si-O ± 0.02 Å r Al-O ± 0.02 Å r Ca-O ± 0.05 Å CN Si-O ± 0.