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This article was submitted to Satellite Missions, a section of the journal Frontiers in Remote Sensing

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The Research Scanning Polarimeter (RSP) is an airborne along-track scanner measuring the polarized and total reflectances in 9 spectral channels. The RSP was a prototype for the Aerosol Polarimetery Sensor (APS) launched on-board the NASA Glory satellite. Currently the retrieval algorithms developed for the RSP are being adopted for the measurements of the space-borne polarimeters on the upcoming NASA’s Plankton, Aerosol, Cloud Ocean Ecosystem (PACE) satellite mission. The RSP’s uniquely high angular resolution coupled with the high frequency of measurements allows for characterization of liquid water cloud droplet sizes using the polarized rainbow structure. It also provides geometric constraints on the cumulus cloud’s 2D cross section yielding the cloud’s geometric shape estimates. In this study we further build on the latter technique to develop a new tomographic approach to retrieval of cloud internal structure from remote sensing measurements. While tomography in the strict definition is a technique based on active measurements yielding a tomogram (directional optical thickness as a function of angle and offset of the view ray), we developed a “semi-tomographic” approach in which tomogram of the cloud is estimated from passive observations instead of being measured directly. This tomogram is then converted into 2D spatial distribution of the extinction coefficient using inverse Radon transform (filtered backprojection) which is the standard tomographic procedure used e.g., in medical CT scans. This algorithm is computationally inexpensive compared to techniques relying on highly-multi-dimensional least-square fitting; it does not require iterative 3D RT simulations. The resulting extinction distribution is defined up to an unknown constant factor, so we discuss the ways to calibrate it using additional independent measurements. In the next step we use the profile of the droplet size distribution parameters from the cloud’s side (derived by fitting the polarized rainbows) to convert the 2D extinction distribution into that of the droplet number concentration. We illustrate and validate the proposed technique using 3D-RT-simulated RSP observations of a LES-generated Cu cloud. Quantitative comparisons between the retrieved and the original optical and microphysical parameters are presented.

Tomography is a retrieval technique inverting a 2D spatial density from a dataset of measured “slices” (

While topography is an analysis of essentially active measurements (when a ray sent through the object is captured by a detector on its other side), this term sometimes is used in cloud remote sensing in a wider sense: as any technique of inversion of cloud interior structure from remote optical measurements. Previously developed retrieval algorithms of this kind (e.g.,

We present a new algorithm for inversion of the internal cloud structure. It is ideologically closer than LSF approaches to “tomography” in the traditional sense, while also relying on passive optical measurements. Such measurements obviously cannot directly provide a proper tomogram (directional optical thickness as a function of angle and offset of the viewing ray), but allow for estimation of it using a nested family of “cloud shapes” corresponding to an array of thresholds in the measured total reflectance. Once such tomogram is obtained, we proceed following the standard tomographic procedure and apply inverse Radon transform to it deriving a 2D field of the extinction coefficients (which requires calibration using independent measurements). So this technique may be called “semi-tomographic”. Note that unlike the LSF techniques, our method does not require iterative 3D RT computations, but relies on a number of empirical assumptions which we validate using simulated data. In fact, we only use simulated data (LES + 3D RT) for testing of our algorithm and for selection of an appropriate “proxy” formula relating the measured reflectance to the corresponding directional COT.

The presented algorithm has been developed for the Research Scanning Polarimeter (RSP) measurements. The RSP (

This study was inspired by the RSP measurements made during NASA’s Cloud, Aerosol and Monsoon Processes Philippines Experiment (CAMP^{2}Ex) conducted in 2019. This campaign took place off Philippines coast during the monsoon season, which is characterized by abundance of moisture and strong convections producing wide variety of cloud types and shapes which present interesting cases for application of our tomographic analysis. We will report analyses of the actual RSP data from CAMP^{2}Ex in subsequent publications following this paper.

The RSP has high angular resolution of 14 mrad field of view (FOV) with measurements made at 0.8° intervals within ±60° from nadir. The translation of these parmeters into spatial resolutions of the RSP datasets depends on the speed and the altitude of the aircraft (

The RSP measures both total and polarized reflectances in nine spectral channels. Its measurement geometry and the view-line aggregation types used for different kinds of retrievals are schematically presented in

Schematic representation of the RSP’s measurement geometry

The COT retrievals from RSP-measured total reflectances at nadir view are made using a modification of the legacy bi-spectral technique (

The standard operational RSP cloud top height (CTH) product is based on a stereo block-correlation algorithm developed by A. Wasilewski. This technique has been generalized to multi-layer cloud scenes by

For illustration and validation of the proposed tomographic technique we use simulated RSP measurements generated by the 3D radiative transfer (RT) model called “Monte Carlo code for the phYSically correct Tracing of photons In Cloudy atmospheres” (MYSTIC

We selected the last (No. 3) cloud from the segment used in _{
c
} (measured in cm^{−3}) within the 2D cross section of this cloud. _{eff} (measured in _{
c
} ≥ 10 cm^{−3} for consistency with _{eff} in the RT model is constant and set to 0.1. The effective radius and variance of cloud droplet DSD are routinely retrieved from the RSP measurements of the polarized reflectance using the parametric algorithm (

_{
c
} ≥ 10 cm^{−3}.

The extinction coefficient field for the 2D cross section of this cloud is presented in _{
c
} is the droplet number concentration (in cm^{−3}), _{ext} is the extinction efficiency (^{–6} makes the units of _{ext} to be m^{−1}. However, for our purposes we can use a simplified version of _{ext} ≈ 2 for large cloud droplets and expressed the second moment ⟨^{2}⟩ of the droplet size distribution in terms of _{eff} and _{eff} [assuming that the DSD has gamma-distribution shape (

In this study we use the implementation of the Radon transform from the standard Interactive Data Language (IDL v.8.4) library (RADON function) with ramp filter added. The 2D field to be determined is the spatial distribution of the cloud extinction coefficient _{ext} (^{−1}), whose integral over a linear transect (chord) through the cloud yields the directional COT

Radon transform geometry: _{d} (_{d} (

The inverse Radon transform deriving the extinction coefficient distribution from the tomogram is called “filtered backprojection.” It consists of two parts. First, the ramp filter is applied to the tomogram _{d} (

Here

After filtering the backprojection itself is applied yielding the inverted extinction field:

Note that inverse Radon transform is not exact, so the inverted _{ext} (

The factor

Before considering the conversion of simulated remote sensing measurements into a 2D extinction distribution, we first want to check that our implementation of Radon transform works well on the LES extinction field itself. This means that the consecutive application of direct and inverse Radon transforms should yield the field (almost) identical to the initial one. This test is illustrated in _{ext} (_{d} (_{ext} (_{ext} (

Test of Radon transform on a LES-generated cloud.

The RSP’s uniquely high angular resolution coupled with the high frequency of measurements can provide geometric constraints on the cumulus cloud’s 2D cross section and yield cloud’s geometric shape estimates (

A similar “space carving” methodology has been developed in a 3D case for the Multi-angle Imaging SpectroRadiometer (MISR) measurements (

The notions of cloud shape or cloud boundary, while intuitively well-understood, are, strictly-speaking, quite ambiguous both in physical and optical sense. Physically, cloud is a spatially distributed collection of water droplets with no hard-defined surface. It can be artificially bounded by a level surface of the droplet number concentration corresponding to a threshold value chosen by the observer. Similarly, “optical surface” of the cloud can be defined using an arbitrarily chosen brightness threshold. It is affected by the solar and viewing geometries. For example, “optical surfaces” are systematically shifted relative “physical” ones towards the side of the cloud directly illuminated by the Sun. The higher is the brightness threshold, the smaller is the cloud-shape cutout derived using the technique described above.

However, while it is difficult to assign a precise physical meaning to the cloud shape corresponding to a single brightness threshold, the collection of such shapes derived for a range of thresholds may carry information about the internal structure of the cloud.

In this study we assume that the reflectance measured by the RSP at a certain viewing angle is quantitatively one-to-one related to the directional COT (dCOT) along the corresponding view ray. This assumption allows us to use the measured reflectance to build dCOT tomogram, which will be then converted (using inverse Radon transform) into a 2D distribution of the cloud extinction coefficient. This assumption is, of course, a simplification in our essentially 3D setting, where multiple scattering within the whole cloud contributes to the reflectance at any particular direction. However, we will show below that this simplified assumption is sufficient for our purposes. When reflectance becomes a proxy for dCOT, it loses its optical properties and gains physical ones. Having this in mind, we introduce “reflectance-proxy” (RP) which coincides with the measured reflectance where it is present, while can be computed for other (virtual) view rays using inter- or extrapolation (in the same way as dCOT can be extended). Some of such view rays (e.g., horizontal ones) are inconsistent with the observation geometry and do not exist in real observation datasets. Also, the extended RP values are not expected to comply with the RT laws, so the RP is a rather abstract dataset that gets its physical meaning only after conversion to dCOT. This conversion will be discussed in

Following the above assumptions we create an intermediate reflectance-proxy tomogram by interpolating the reflectances observed at the actual view rays to all possible view rays from a high-resolution grid of angles and offsets (relative to a chosen cloud center). This construction utilizes the collection of cloud shapes corresponding to a range of brightness thresholds. In a realistic analysis, the brightness threshold array consists of discrete values. The corresponding cloud shapes also form a discrete family, which is expected to be nested. _{1} and _{2} respectively. These distances are then used to compute the weights _{1} = _{2}/(_{1} + _{2}) and _{2} = _{1}/(_{1} + _{2}) with which the respective level-curves’ RP values contribute to the linearly interpolated value assigned to

Given the way the cloud shape corresponding to a certain brightness threshold is constructed, all the actual view rays tangent to it correspond to the same reflectance value equal to this of the threshold. We extend this property to all possible (not only actual) view rays tangent to this cloud shape. This, of course, is an abstraction not necessarily consistent with the nature of realistic light scattering within the cloud. This extension allows us to assign a RP value to any view ray (which from now on we will also call “chord”) crossing the RPD domain. The RP value assigned to a chord is this of the RPD’s level curve to which this chord is tangent. For the functional shape of the RPD in our case (which monotonically increases towards the cloud center) it is easy to see that this value is simply the maximum of the RPD along the chord.

In more complicated cases, such as e.g., two partially merged clouds, a chord may be tangent to two level curves with different values (each surrounding a different cloud center). In this case the along-chord RPD would have two local maxima, and the value of the larger of them should be assigned to the chord. This follows from the general structure of RPD with higher-value level curves being inside the interiors of the lower-value ones (since clouds are brighter at their center(s) than at their edges). So a chord coming from an observation with a lower threshold

While the RPD can be defined on a 2D grid with very fine resolution (we use 1 m, which certainly is an excess), the chords’ RP values can be also combined into a function of angles and offsets on a very fine grid (we use 1° resolution in angle and 1 m—in offset). This function is the RP tomogram _{tom} (_{tom} (_{tom} (

_{tom} (_{tom} (

On the next step of our analysis we need to find out how to relate the dCOT tomogram to the RP- and the chord-length-tomograms from _{tom} alone as the dCOT tomogram would result in a constant extinction value within the implied cloud domain. Indeed, having a constant extinction coefficient makes the dCOTs to be proportional to the corresponding chord lengths. This suggests that inclusion of _{tom} into analysis together with _{tom} would provide us with a better representation of the spatial configuration of the cloud.

Finding a good empirical proxy formula relating _{tom} and _{tom} to the dCOT tomogram _{tom} is a trial and error process, in which the LES data plays an active role (since we know what the result should be). After several trials we have stopped at the formula relating the reflectance

Here the denominator 2 max _{tom} is actually irrelevant (since the inverse Radon transform of _{tom} is defined up to a constant factor) and is included only to make _{tom} dimensionless as dCOT should be. The dCOT tomogram for our cloud (with an arbitrary value scale) is presented in

_{tom} (_{tom} and _{tom} (_{tom} using inverse Radon transform and calibrated by the initial LES COT.

As we mentioned above, the extinction distribution derived from the dCOT tomogram using inverse Radon transform is defined up to an arbitrary constant factor and has to be calibrated. The calibration factor can be determined by iteratively computing the 3D-RT-simulated reflectances for scaled extinction fields until they match the RSP measurements. This way is computationally expensive and also involves uncertainties associated with the 2D nature of our retrievals that do not provide the full extent of the 3D cloud structure. A better alternative to this is to rely on additional measurements such as the COT derived from the RSP’s nadir reflectances using a LUT, or cloud-top extinction coefficient retrieved from the measurements made by the High Spectral Resolution Lidar (HSRL) which is often deployed on the same airborne platform as the RSP during field campaigns.

The algorithm for COT retrievals from the RSP’s nadir total reflectances uses a LUT computed assuming a plane-parallel geometry. Thus, while working fairly well for stratiform clouds, the retrievals for small popcorn Cu can be significantly affected by 3D effects such as leaking of scattered light through the cloud’s sides and shadowing (see e.g.,

An alternative calibration method can be based on the extinction coefficient values at cloud top presumably known from HSRL retrievals.

We continue to explore the potential of the RSP’s own measurements of the polarized reflectance to provide the calibration constant. They are dominated by single scattering and, therefore, are less affected by 3D effects. However, for now, we want to separate calibration issues from assessing the accuracy of the characterization of the 2D cloud structure. For this purpose we adopt the calibration made by matching the tomographic COT maximum to the LES COT maximum. The tomographic COT then corresponds to the red COT-curve in

_{ext} = 0.02 m^{−1}, black—to _{ext} = 0.07 m^{−1} (these values are represented by blue and red respectively in the color plot). We see quite good spatial agreement between the two fields in

Comparison between the tomographic retrievals of the extinction coefficient from _{ext} = 0.02 m^{−1}, black—to _{ext} = 0.07 m^{−1} (these values are represented by blue and red respectively in the color plot).

_{ext} field (with 1 m × 1 m spatial resolution) is interpolated to the LES data points (located on 100 m × 40 m grid). Then only the data points with positive values in both datasets are taken into comparison (to avoid the influence of zero-value points outside the cloud). The direct (as in _{ext} is presented in _{
k
} of the difference, which is 0.015 m^{−1} (20.5% of the LES dataset’s maximum) for the un-shifted field and improves with the 50 m shift to 0.011 m^{−1} (15.1% of the LES dataset’s maximum). The correlation between the two datasets also improves with the shift: from 73 to 84%. The LES extinction values used in this comparison have the mean of 0.023 m^{−1}, the median of 0.017 m^{−1}, and the maximum of 0.073 m^{−1}. The width of the scatter plot in _{ext} magnitude, so the best overall accuracy assessment is that 65% of the retrieval datapoints have extinction values within _{
k
} = ±0.01 m^{−1} from their LES counterparts, and 96%—within 2_{
k
} = ±0.022 m^{−1} (the corridor shown by dotted lines in _{
k
}-corridor with the half-width of 0.030 m^{−1}.

Quantitative comparison of the retrieved extinction coefficient with the initial data from the LES model. The retrieved _{ext} field is interpolated to the LES data points. Only the data with above-zero values from both datasets are taken into comparison. _{ext} from both datasets. _{
k
}-corridor around 1–1 solid line.

The RSP-measured polarized reflectances are dominated by single scattering and are representative of the cloud layer of unit COT or about 50 m into the cloud (

_{eff} and _{eff} = 0.1). Hence, the non-parametric RFT retrieval technique (

Derivation of droplet effective radius and variance profiles from RSP polarized reflectances aggregated to points along the cloud’s side. The cloud shape here is taken from

Cloud-side profiles of DSD parameters appear to provide a good estimate of their inside-cloud counterparts both in simulated and real datasets. The 2D cross section of the _{eff} distribution in our LES output presented in ^{2}Ex demonstrated very good agreement (on average) between the DSD parameters derived from the RSP measurements at cloud tops and those measured _{eff} profile and the same _{eff} value of 0.1 throughout the cloud. While any LSF algorithm generally works as a “black box” not revealing dominant sources of specific retrievals, the tests performed by

In our case _{eff} in _{eff} in _{eff} (blue curve) has a modest increase with height from 14 to 16 _{eff} = 15.7 _{eff} = 0.14 throughout the cloud.

The effective radius and variance profiles derived from the polarized reflectances at the cloud side are assumed to be good representations of the corresponding profiles inside the cloud (as we discussed in the previous section). Then the vertical profiles _{eff}(_{eff}(_{
c
}(_{ext} (

Here _{
c
} is measured in cm^{−3}, _{ext}—in m^{−1}.

In our case droplet size profiles are considered to be altitude-independent (_{eff} = 15.7 _{eff} = 0.1), so _{
c
}(_{ext} (_{
c
} with the initial LES field (shown in color). Over-plotted white curve is the level curve of the retrieved _{
c
} (^{−3}, black curve corresponds to the 60 cm^{−3} level (these level values are represented by blue and red respectively in the color plot). As in _{
c
} field in both bottom plots is also shifted by 50 m to the right.

_{eff} = 15.5 _{eff} = 0.1 (from _{
c
} with the initial LES field (shown in color); over-plotted white curve corresponds to retrieved _{
c
} = 20 cm^{−3}, black—to _{
c
} = 60 cm^{−3} (these values are represented by blue and red respectively in the color plot); both retrieved curves are shifted to the right by 50 m. _{
c
} fields in both bottom plots are shifted by 50 m to the right.

_{
c
} values with those in the LES dataset. As in _{
c
} > 1 cm^{−3} and positive _{ext} (i.e., positive _{eff}) in both datasets were selected for the comparison. The LES _{
c
} values have the mean of 28.75 cm^{−3}, median of 28.36 cm^{−3}, and maximum of 71.74 cm^{−3}. _{
c
} field. There are very small negative biases in both plots (less than 0.5 cm^{−3}), while the standard deviation _{
n
} provides more definitive accuracy estimate. As expected, the accuracy improves with the shift from 17.6 to 12.8 cm^{−3}, while the correlation increases from 65 to 81%. In _{
n
}-corridor (depicted by two dotted lines). The shift does not change this number much (97.7%) while the corridor is visibly narrower in _{
c
} retrievals as ±13 cm^{−3}, regardless of the _{
c
} value (within the range of our model), which in this case constitutes 17% of the maximal _{
c
} and 43% of its mean or median.

Quantitative comparison between the retrieved droplet number concentration and the initial one from the LES model. The retrieved _{
c
} field is interpolated to the LES data points. Only the data points with _{
c
} > 1 cm^{−3} in both datasets are taken into comparison. _{
c
} from both datasets.

The ability to derive 2D droplet number concentration field opens possibility to calibrate tomographic retrievals using

We presented a new remote sensing tomographic technique allowing for retrieval of cloud internal structure from external measurements made by the Research Scanning Polarimeter. While tomography (in narrow sense) is an active technique incompatible with the geometry of atmospheric remote sensing, we developed a “semi-tomographic” approach in which the tomogram of the cloud is estimated from passive measurements instead of being measured directly. This tomogram is then converted into spatial distribution of extinction coefficient using inverse Radon transform (filtered backprojection), which is the standard procedure in the actual tomography (used e.g., in medical CT scans). This procedure is computationally inexpensive compared to approaches relying on multi-dimensional least-square fitting since it does not require iterative 3D RT simulations.

We illustrated and validated the proposed technique using simulated RSP observations of a LES-generated Cu cloud. The radiation field was computed using MYSTIC 3D RT code. On the first step of our algorithm we use the RSP’s view rays grazing the cloud “surface” to create a nested family of cloud shapes corresponding to a range of radiometric thresholds. This family is used for estimation of the “reflectance tomogram” which is then converted into COT tomogram using the proxy relation

In the second step Radon transform is applied to the COT tomogram resulting in 2D distribution of the extinction coefficient defined up to an unknown constant factor. To determine this factor a calibration procedure based on additional data should be used. While the RSP’s own measurements would be preferable for the calibration, this appeared to be problematic. Our first choice of the calibration dataset was the RSP-derived nadir COT (given that the LES-derived COT was successfully used for self-test of Radon transform on the LES data). However, unfortunately, the COT derived from the RSP’s nadir measurements assuming plane-parallel geometry appeared to have a significant low bias due to 3D effects (e.g., leaking of light through the cloud’s sides) and could not be used for calibration. We consider the RSP’s polarized reflectance as a potential calibration source since it is dominated by single scattering and is less sensitive to 3D effects. Alternatively, the values of extinction coefficient at cloud top derived from correlative lidar measurements can be also used (this is a relatively new data product developed by the HSRL team). In our simulations we found that this type of calibration leads to a rather modest 16% overestimation of the overall 2D extinction values and the COT. While more work is needed to resolve the calibration issues, in this study we wanted to estimate the accuracy of the method apart from the calibration uncertainty. To do this we used the “ideal” calibration based on the LES-derived COT (assuming that it is known from a hypothetical independent measurement).

The reflectance-based cloud shapes at the first step of our algorithm are naturally shifted towards the bright side of the actual cloud and this bias propagates to the retrieved 2D extinction field. Thus, a point-by-point comparison of this field with the initial LES-derived distribution showed notable improvements after 50 m shifting away from the bright side of the cloud. We understand that at different cloud and illumination conditions the optimal shift may be also different. However, in real remote sensing the retrievals of cloud’s physical structure are more important than that of its exact location, so an error in the latter can be tolerated (see _{
k
} of the difference with the LES values. We found that for the shifted dataset _{
k
} = 0.01 m^{−1} (13.7% of the LES dataset’s maximum) and 98% of the retrieved values of _{ext} lie within 2_{
k
} = 0.02 m^{−1} from their LES counterparts.

In the next step we converted the extinction retrievals into these of cloud droplet number concentration _{
c
}. In order to do this we first estimated the droplet size distribution parameters (effective radius and variance) within the cloud. We assume (based by the LES data) that the DSD-parameters profiles do not significantly vary in the horizontal direction (which may not be the case for other real or simulated clouds). This allows us to use throughout the cloud the profiles of _{eff} and _{eff} derived from the RSP’s polarized reflectance measurements made along the cloud side. These retrievals were made using the rainbow (cloud bow) fitting technique (_{ext} field into the spatial distribution of _{
c
} particularly straightforward (just a rescaling). As in the extinction case the comparison of the _{
c
} retrievals with the corresponding LES values improves with the 50 m shift achieving the accuracy (the standard deviation of the difference) _{
n
} = 12.3 cm^{−3} (17% of the maximal _{
c
}) while 95.3% of the points in the scatter plot lie within the 2_{
n
}-corridor. The correlation between the two datasets is 80%.

In the upcoming continuation of this series we will present examples of tomographic analyzes of real clouds observed during NASA’s Cloud, Aerosol and Monsoon Processes Philippines Experiment (CAMP^{2}Ex) conducted in 2019. During this campaign the RSP was deployed on-board of NASA’s P3-B aircraft together with a number of other remote-sensing and

A detailed analysis of the requirements to the sensor’s angular range and resolution will be also presented in our future publications (with different RSP platforms in mind, as well as other airborne and satellite instruments). Some preliminary estimates of the tomographic retrieval accuracy are presented in

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

MA conceived, designed and applied the tomographic algorithm to sample synthetic and real data. CE performed 3D RT computations using LES dataset created by A. Ackerman. BC and BD provided expertise on Radiative Transfer, the RSP’s measurements and operations as well as on other available field-campaign datasets. They also advised on the requirements to the RSP-based algorithms for their implementation into the upcoming NASA PACE satellite mission.

This research was funded by the NASA Radiation Sciences Program managed by Hal Maring and NASA grants NNH16ZDA001N-CAMP2EX and NNH19ZDA001N-PACESAT.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

We would like to thank A. S. Ackerman for generating the LES datasets that we keep using in our studies for many years, including this publication. We also want to thank L. Di Girolamo for encouragement of this work and for useful discussions.

The Supplementary Material for this article can be found online at: