Application of the Radon transform to multi-angle measurements made by the research scanning polarimeter: a new approach to cloud tomography, Part III—accuracy and applicability

1 Introduction

The field of cloud tomography is rapidly developing mostly with the focus on algorithms using extensive 3D radiative transfer (RT) computations within a minimization problem in a very multi-dimensional space of cloud configurations (Martin and Hasekamp, 2018; Levis et al., 2020; Doicu et al., 2022a; Doicu et al., 2022b). In our previous study (Alexandrov et al., 2021), hereinafter referred to as Part I, we introduced a different approach, which is based on the original Radon transform (Radon, 1917; Radon and Parks, 1986) that constitutes the foundation of the traditional computer (X-ray) tomography widely used in medical testing (CT scan). This approach, while being semi-empirical, is much less computationally intensive than 3D RT inversions and, unlike them, is algorithmically transparent to a human researcher. This allows for the identification of the uncertainty sources and the quantification of the retrieval accuracy. This method has been successfully validated in Part I using 3D RT simulations based on realistic LES-generated clouds. This gives us reason to believe that its accuracy and sources of errors are similar to those of 3D RT inversion algorithms. Our algorithm was successfully applied to real measurements made by the research scanning polarimeter (RSP) during NASA’s Cloud, Aerosol and Monsoon Processes Philippines Experiment (CAMP 2Ex) (Reid et al., 2023) conducted in the vicinity of the Philippines in August–September 2019. The results were reported by Alexandrov et al. (2025) (this article is hereinafter referred to as Part II).

In this study, we evaluate requirements for measurement accuracy and geometry necessary for tomographic retrievals. We will demonstrate the presented accuracy analysis techniques with examples of synthetic and real clouds from our previously published articles.

2 The tomographic algorithm: outline and challenges

Our tomographic algorithm consists of several steps. First, a family of “cloud shapes” is derived using the “cut-out” technique (Alexandrov et al., 2016a), similar to “space carving” (cf. Levis et al., 2020). In this approach, a 1D cloud mask (as a function of view angle) is constructed for each scan using a specified brightness threshold applied to the measured reflectance. Knowing the sensor’s position relative to the cloud, the view angles of the edges of this cloud mask are used to construct view rays tangent to the (threshold-dependent) “surface” of the cloud. The collection of all available view rays defines a polygon containing this surface. The cloud shape is then inscribed into this polygon as a smooth curve formed as the common boundary of a collection of disks fitted inside the polygon (see Alexandrov et al. (2016a) and Part I for details). This procedure (illustrated in Figure 1) is used to produce a family of such cloud shapes corresponding to a range of reflectance thresholds (Figure 2, left). This family is expected to be nested.

By construction, this method is limited to single-layer isolated clouds, such as cumulus (Cu) or cumulus congestus (CuCg, also called towering cumulus, Tcu). Such clouds are usually optically thick, so we do not introduce formal limits on cloud optical thickness (COT) for the applicability of tomographic analysis. Optically thinner clouds are not as bright as thicker ones, so for the same number of brightness thresholds, the difference between the threshold values will be smaller, and the corresponding cloud shapes will be closer to each other. This can be illustrated by the examples from Part II, where we compared a relatively small and optically thin Cu cloud (Cloud 1) with a COT of approximately 20 and a larger and thicker CuCg Cloud 2 with a COT of almost 60. However, particularly thin parts of clouds can be missed in brightness-based retrievals. For example, a thin “tail” of Cloud 2 in Part II was not detected by our algorithm, while being seen in lidar and radar data.

The above-mentioned inscription method is our own original invention, first published by Alexandrov et al. (2016a). It was successfully validated in Alexandrov et al. (2016a) and Part I using simulated data. In addition, cloud top height profiles derived using this technique were validated in Part II by comparison with correlative lidar and radar measurements. Due to the complexity of this method, its accuracy is difficult to quantify, so its reliable applicability limits are based on a series of our empirical tests showing that the polygon aspect ratio should be between 0.8 and 2.

We are open to exploring other curve-fitting methods, in particular, Bezier curve interpolation (see, e.g., Farin et al., 2002). This technique (of which we were not aware until now) was initially developed for car body engineering and is currently widely used in various areas of graphic design. It is a proven, robust method for inscribing a smooth curve into a polygon, so we expect it to work well for our cloud applications.

On the second step, the value of the corresponding reflectance threshold is assigned to all points of each cloud shape in the family. These values are then interpolated between the shapes forming 2D “reflectance density” (RD, an abstract object shown in Figure 2, right). This RD is then converted into a directional cloud optical thickness (dCOT) tomogram using a proxy formula. On the last step, the inverse Radon transform (Radon, 1917; Radon and Parks, 1986) is applied to this tomogram, yielding a 2D extinction coefficient density within the cloud.

The most challenging part of the described tomographic algorithm is the construction of cloud shapes using view rays tangent to the cloud surface, so in this study, we focus on the quantification of the accuracy and limitations of this particular part. The ability to construct such shapes imposes certain requirements on the measurements in terms of their frequency, angular range, and resolution, as well as the platform’s speed and altitude. Some basic accuracy estimates will be presented in Section 3.

One of the problems affecting cloud shape derivation is the lack of near-horizontal viewing rays defining a cloud’s top and bottom. This particularly affects retrievals from high-altitude platforms because the measurements from them with high view angles are made at long distances from the cloud. Such measurements are very sensitive to the sensor’s attitude stability and the cloud’s movements (because the cloud can easily move out of the sensor’s field of view). In this geometry, the signal can also be significantly attenuated by the atmosphere, while the sensor’s field of view can be obstructed by other clouds. Reduction of the admissible angular range in such cases could lead to the creation of cut-out polygons with much higher top vertices than the actual cloud tops. Such polygons by themselves would not be representative of the cloud dimensions, while the inscription of round cloud shapes into them might be difficult and uncertain. Alternatively, cloud top height can be obtained from independent lidar or radar measurements (preferably made from the same platform). We will address this issue in Section 4.

Another obstacle affecting measurements made from high-altitude platforms is associated with long cloud overflight times between the points corresponding to the edges of the sensor’s angular viewing range. During such times, the cloud of interest can move (along-track or away from the field of view), change, or even entirely disappear. Estimates of cloud overpass times will be presented in Section 5, while the effects on the retrievals of cloud advection during the overpass will be discussed in Section 6. In Section 7, we will present numerical estimates of the retrieval accuracy for clouds described in several of our previously published studies.

3 Measurement resolution and retrieval accuracy

The accuracy of our algorithm is defined by its ability to produce a nested family of N s h p 2D cloud shapes having a certain spatial resolution. The minimal N s h p needed for a meaningful analysis is three. Note that the resolution requirements may depend on the cloud size. Internal structure estimates for larger clouds can be made with lower spatial/angular resolution while keeping the value of N s h p admissible. This means that the spatial resolution of a given sensor determines the minimal size of a cloud it can be used to study.

To make some quantitative estimates of the tomographic retrieval accuracy, let us assume that the sensor flies with the speed v at the altitude H above ground while making instantaneous multi-angle along-track measurements (scans) separated by the time intervals Δ t. Thus, the spatial distance between every two subsequent scans is Δ x = v Δ t. We denote by Δ θ the angular spacing between the measurements within a single scan. Here θ is the viewing angle (θ = 0 corresponds to nadir view). The angular range is assumed to be ± θ m a x from nadir. For example, in Part I the RSP data were simulated with H = 2400 m, Δ θ = 0.8 ° = 0.014 rad, θ m a x = 60 °, and Δ x = 100 m. Given that the real RSP makes scans every 0.8 s, this spatial resolution corresponds to the aircraft speed of 125 m/s, which is consistent with the typical speeds of NASA’s low-altitude aircraft (P3-B, B-200, UC-12, and C-130). For the real RSP Δ θ coincides with the instrument’s field of view (FOV) of 14 mrad, while the FOV was not considered in our RT simulations. Generally, FOV has a smoothening effect on the measured reflectances and should be much smaller than the angular size of the observed cloud.

Let us consider, for simplicity, a round cloud with the radius R (corresponding to the lowest brightness threshold) and the center at the altitude h c (see Figure 3). In this plot, the darker disk with radius r inside the cloud represents the cloud shape corresponding to the next brightness threshold. It is convenient to introduce the difference between the sensor and the cloud center altitudesbecause it will enter the expressions below. We assume that h c > R (so the cloud is above the ground) and also h > R (thus, the sensor flies above the cloud, not through it). The cloud top height (CTH) is then h c + R, and the distance between the sensor and cloud top is h − R.

h = H − h c (1)

The parameter Δ r = R − r characterizes the ability to separate cloud shapes corresponding to different reflectance brightness thresholds and also the accuracy of determination of the cloud shapes themselves in the radial dimension. For our measurement geometry, the actual Δ r can be estimated aswhere l is the horizontal distance between the sensor and the cloud center. This R-independent estimate corresponds to measurements near the cloud’s center, to which cloud shapes converge with an increase in the reflectance threshold. The spatial (radial) accuracy Δ r reaches its maximal value Δ r m a x at θ = θ m a x.

Δ r ≈ Δ θ l 2 + h 2 1 / 2 = Δ θ h tan 2 ⁡ θ + 1 1 / 2 , (2)

For the RSP specifications listed above, tan 2 θ m a x = tan 2 60 ° = 3 and Δ θ = 0.8 ° = 0.014 rad, so Equation 2 reduces to

Δ r m a x ( R S P ) = 0.028 h . (3)

The required spatial accuracy that allows resolving the internal cloud structure iswhere N s h p is the required number of shapes (at least three).

Δ r r e q = R N s h p , (4)

In the angular dimension, the cloud shape accuracy is determined by the difference in the Δ φ angle between the selected tangent view lines from subsequent scans (it should not be confused with the angular resolution Δ θ within each scan). In the case of a round cloud, Δ φ coincides with the angular resolution of the resulting cloud shape polygon in the polar coordinate system associated with the cloud center. Δ φ has a simple relationship with Δ x:because the distance between the sensor and the cloud center is h / cos ⁡ θ, the projection of Δ x on the line perpendicular to the direction to the cloud center is Δ x ⁡ cos ⁡ θ, and Δ φ is the ratio of these two. The maximal valueis achieved at the nadir view where θ = 0.

Δ φ ≈ Δ x h cos 2 ⁡ θ , (5)

Δ φ m a x = Δ x h (6)

Note that while Δ r m a x increases with increase of the relative sensor altitude h, the angular spacing Δ φ m a x decreases. This happens because the cloud overpass length at a higher altitude is larger, so the number of measurements made during the overpass within the same angular range is also larger:

N m e a s = 2 h Δ x tan θ m a x . (7)

4 Effects of the polygon aspect ratio

Another important parameter influencing the ability to characterize cloud shape is the aspect ratio of the polygon cut out by the tangent view rays (not to be confused with the aspect ratio of the cloud itself, which will be addressed at the end of this section). This single parameter characterizes the combined influences of the cloud horizontal extent and the cloud top height (relative to the sensor altitude). If this polygon has an elongated form (vertically or horizontally, corresponding to a high or low aspect ratio, respectively), the inscription of a cloud shape into it is uncertain and may produce multiple small clouds instead of a single cloud. Our tests on synthetic cloud shapes (Alexandrov et al., 2016a) showed that polygons with A p o l ∈ [ 0.8 , 2 ] can be fitted well by smooth cloud shapes. However, this condition only means that the inscription of a shape into polygon is possible but does not guarantee the accuracy of such inscriptions.

Figure 4 (left) presents a cloud shape corresponding to some brightness threshold (not necessarily the lowest). This is a round cloud: a disk with the radius r. This figure shows that the altitudes of the top and bottom vertices of the polygon (depicted by small red disks) are determined by four view rays closest to the horizontal direction and corresponding to the view angle θ m a x. For round cloud shape with radius r, these altitudes arerespectively. If h p . b o t appears to be negative, the polygon should be truncated by a horizontal line at zero altitude; h p . t o p cannot be larger than H by construction. The horizontal dimension of the cloud shape is constrained by two nadir view rays (dashed lines in Figure 4) and is equal to 2 r for our round cloud.

h p . t o p = h c + r sin θ m a x a n d h p . b o t = h c − r sin θ m a x , (8)

It follows from Equation 8 that when h p . b o t ≥ 0, the polygon’s aspect ratio A p o l depends only on the angular range limit θ m a x:

A p o l = h p . t o p − h p . b o t 2 r = 1 sin θ m a x . (9)

For the RSP θ m a x = 60 °, so we have A p o l = 2 / 3 ≈ 1.15, which fits well within the admissible range. However, in some real measurement cases, the view rays corresponding to large viewing angles appear to be distorted by atmospheric conditions and must be omitted. This reduces the effective θ m a x and increases A p o l. We often experienced such situations in real-life retrievals reported by Alexandrov et al. (2016a). In addition, a good example of this effect can be found in Part II (Figure 4, bottom left), where the forward and backward space-carving rays are nearly parallel, yielding A p o l ≈ 3.5 and causing uncertainties in cloud vertical extent. Note that all three accuracy characteristics (Δ r m a x, Δ φ m a x, and A p o l) are independent from the cloud shape radius r and, therefore, are applicable to all round cloud shapes regardless of their reflectance thresholds.

The polygon aspect ratio can be related to the aspect ratio of the corresponding cloud shape itself. It is easier to demonstrate this relationship for a rectangular (“box”) cloud than for an oval or elliptic one. Let us consider a rectangular cloud with the length L and the thickness H cld (Figure 4, right). The cloud aspect ratio in this case is A c l d = H c l d / L. As in the round cloud case, the polygon is defined by six viewing rays: four correspond to the view angle θ m a x and pass through the vertices of the rectangle, while the other two are vertical rays along the cloud’s sides. The horizontal dimension of the polygon is L, the same as that of the cloud, while the polygon shape in the vertical dimension differs from the shape of the cloud by the two triangles formed by the viewing rays at the top and bottom of the cloud. The height of each of these triangles is L / ( 2 ⁡ tan θ m a x ), so the vertical dimension of the polygon isThus, the polygon aspect ratio in this case is

H p o l = H c l d + L tan θ m a x . (10)

A p o l = H p o l L = H c l d L + 1 tan θ m a x = A c l d + 1 tan θ m a x . (11)

This equation poses constraints on the aspect ratios of clouds admissible for tomographic investigation, given the instrument-dependent value of θ m a x. Note that due to the difference in cloud geometries, Equation 11 does not coincide with Equation 9 in the case of A c l d = 1. Also note that as the cloud shape becomes more stratiform ( A c l d → 0 ), the corresponding polygon aspect ratio saturates to a constant ( A p o l → 1 / tan θ m a x ). For the RSP specifications with θ m a x = 60 °, the last term in Equation 11 is equal to 0.58 ≈ 0.6, so the condition A p o l ∈ [ 0.8 , 2 ] translates into A c l d ∈ [ 0.1 , 1.4 ].

5 Effects of overpass length and time

The overpass length l o p is the distance between the first and the last scan points at which the view rays with the view angle θ m a x are tangent to the cloud’s surface (apparently at its bottom half). In our measurement geometry (see Figure 5), it has the following expression:

l o p = 2 h + R sin θ m a x tan θ m a x = 2 h sin θ m a x + R cos θ m a x . (12)

It expectedly diverges as θ m a x → 90 °, while l o p = 2 R when θ m a x = 0 ° (i.e., the overpass length equals the cloud diameter when only the nadir view is available). In the case of the RSP when θ m a x = 60 °, sin θ m a x = 3 / 2 and cos θ m a x = 1 / 2, so Equation 12 reduces to

l o p ( R S P ) = 2 3 h + 2 R . (13)

The corresponding overpass time iswhere v is the sensor’s platform speed. While t o p does not play any role in retrievals from simulated data, it is important for analysis of real-life measurements because the observed cloud can change and/or move during the overpass. While general changes in cloud structure are difficult to quantify, for our estimates, we simply focus on the cloud’s advection by the wind with, say, w = 10 m/s speed. Then, the cloud’s displacement d o p = w t o p during the overpass can be compared with such parameters as R, Δ r m a x, and Δ x. More detailed analysis of cloud advection effects on measurements is presented in Section 6.

t o p = l o p v , (14)

6 Effects of cloud advection

Cloud advection, if not properly taken into account, leads to certain biases in the derived cloud dimensions and height. Figure 6 shows the schematic geometry of measurements and retrievals for the case in which, during the overpass, the cloud is moving along the sensor’s track with the speed w in the same direction as the sensor (left panel) and in the opposite direction (right panel).

In the measurement geometry outlined in Figure 6 (left), the sensor flies from left to right with the speed v at the altitude h relative to the cloud’s center. The actual cloud (blue) with radius R is advected in the same direction by tail wind with the speed w. The plot shows the cloud and the sensor in two positions: first, when the cloud is observed at the first half of the overpass (from the left) at the viewing angle θ (of the view ray passing through the cloud’s center), and, second, when the cloud observed from the right (at the same viewing angle) after the sensor has passed over it. Here, we assume that both view rays bounding the 1D cloud mask are available for each scan.

We select the midpoint between these two measurement positions when the sensor is directly above the cloud center. The cloud and the sensor arrive at this midpoint at the same time t (from the moment of the first measurement). The distance traveled by the sensor from the first measurement point to the midpoint (and from the midpoint to the second measurement point) is x = v t. The corresponding distance traveled by the cloud at the same time is y = w t.

We see from Figure 6 (left) that due to the cloud’s movement (while the retrieval algorithm assumes that it is stationary), the view-ray polygon constraining the cloud’s shape and position appears to be consistent with a larger “fake” cloud (shown in red) with radius R ′ > R located at a lower altitude (so h ′ > h). The similarity of two right triangles with the vertices at the centers of the real and the fake clouds respectively implies that

R R ′ = h h ′ = x − y x = v − w v = 1 − w v . (15)

Thus,where

R ′ = α R a n d h ′ = α h , (16)

α = v v − w = 1 − w v − 1 . (17)

If w < v, that is likely to be the case, α > 1; thus, R ′ > R and h ′ > h.

The geometry corresponding to the head wind, when the cloud is advected in the direction opposite to that of the sensor’s flight, is presented in Figure 6 (right). In this case, the “false” cloud appears to be smaller and higher than the real one: R ′ < R and h ′ < h. The expressions for these parameters differ from those in the tail wind case only by the sign of w. Thus, the cloud’s size and position retrieval biases in both cases can be described asrespectively. Here,where the “+” and “–” subscripts correspond to the cases of head and tail wind, respectively. Note that α ± depends only on the ratio w / v of the wind and platform speeds and is independent from the viewing angle θ (and, thus, from the sensor’s position relative to the cloud).

R ′ = α ± R a n d h ′ = α ± h , (18)

α ± = v v ± w = 1 ± w v − 1 , (19)

When the cloud moves along with the sensor α − > 1, and the cloud appears larger and lower in the retrieval, while when it moves in the opposite direction, it appears smaller and higher because α + < 1. In realistic cases when the wind speed is much less than that of the aircraft ( w ≪ v ), we havemeaning that the relative biases in R and h are equal to the wind/platform speed ratio:

α ± ≈ 1 ∓ w v , (20)

δ R ± R = R ′ − R R = ∓ w v a n d δ h ± h = h ′ − h h = ∓ w v . (21)

For example, for typical speeds v = 100 m/s and w = 10 m/s, the retrieval bias in cloud size and relative altitude is 10%.

To correct the retrievals for the biases caused by cloud advection, independent measurements are needed of either the wind speed itself or of cloud top height (made, e.g., by collocated lidar and/or radar). The latter measurements can also be used for estimation of the along-track wind speed (given that the aircraft’s speed is known).

7 Numerical examples

7.1 Simulated clouds

In Part I, we used a tomographic algorithm with N s h p = 5 to analyze the internal structure of a simulated cloud (see Figure 2). The cloud dimensions corresponding to the smallest brightness threshold used were approximately 500 m horizontally and 1,000 m vertically (Figure 1, left). In the notation of this article, these dimensions correspond to the reference radius R ≈ 750 m; thus, the required resolution is Δ r r e q ≈ 150 m according to Equation 4. The sensor altitude H = 2400 m used in this simulation is lower than the typical 7–9 km height at which most of NASA’s aircraft fly during remote-sensing legs. The cloud center altitude h c is approximately 1,300 m corresponding to h = 1100 m. Then, according to Equation 3, the actual Δ r m a x ≈ 31 m, which is better than the required value. The angular resolution Δ φ m a x is 5.2 ° according to Equation 6 with Δ x = 100 m. This is more than sufficient for accurate characterization of the cloud shape.

The polygon’s aspect ratio A p o l from Equation 9 is 1.15 in our case, which is sufficiently close to unity, ensuring the close cloud shape inscriptions shown in Figure 1 (left). The estimated overpass length l o p from Equation 13 is 6,811 m flown in 54.5 s (assuming v = 125 m/s). In the presence of a moderate 10 m/s wind, the cloud’s displacement during the overpass time would be 545 m, exceeding the cloud’s horizontal dimension.

This simulated cloud was first introduced by Alexandrov et al. (2016a) (where it is called Cloud 3) together with two smaller clouds from the same Cu field (called Cloud 1 and Cloud 2, respectively). These simulations were also later used by Alexandrov et al. (2024). The measurements in these tests were made by simulated RSP flying at H = 2400 m and operating with full angular range, so in both cases, A p o l = 1.15.

The dimensions of Cloud 1 were 450 m vertically and 500 m horizontally, with the cloud center altitude h c ≈ 800 m. These parameters correspond to the reference radius R ≈ 475 m, so Δ r r e q ≈ 95 m, assuming N s h p = 5. The altitude difference h between the sensor and the cloud center in this case is 1,600 m; thus, the actual Δ r m a x ≈ 45 m according to Equation 3. This is better than Δ r r e q. The angular resolution Δ φ m a x in this case is 3.6 °. The overpass length l o p estimated using Equation 13 is 7,443 m flown in 59.5 s at the speed 125 m/s. This corresponds to cloud displacement of 595 m by a 10 m/s wind during the overpass.

Cloud 2 had 600 m vertical and 500 m horizontal extents corresponding to R ≈ 550 m and Δ r r e q ≈ 110 m for N s h p = 5. The cloud center was at h c ≈ 900 m; thus, h = 1500 m, and Δ r m a x ≈ 42 m, less than Δ r r e q. The angular resolution Δ φ m a x is 3.8 °. The overpass length l o p = 7396 m corresponds to an overpass time of 59.2 s and a potential displacement of 592 m by a 10 m/s wind during the overpass. This length is almost equal to the horizontal dimension of the cloud.

7.2 Clouds from DEVOTE

The real-life RSP measurements used for cloud shape derivation by Alexandrov et al. (2016a) were made from the altitude H = 9000 m during the NASA Development and Evaluation of satellite ValidatiOn Tools by Experimenters (DEVOTE) field campaign conducted in October 2011 north of New Orleans, Louisiana. At that time, the RSP was on board NASA’s UC-12 Huron aircraft. The observed clouds (forming “streets”) then were larger than in the Part I simulation with R ≈ 1000 m. The typical cloud center altitudes h c were approximately 2000 m, and the distances Δ x between the scan points were approximately 96 m. These parameters correspond to Δ r m a x = 196 m, which is consistent with the required radial accuracy Δ r r e q ≈ 200 m corresponding to N s h p = 5. The angular accuracy was Δ φ m a x = 0.8 °, which is better than in the simulation study. The polygon aspect ratio A p o l = 1.15 depends only on θ m a x, the RSP instrument specification, so it is the same in all examples. The full overpass length for this case study exceeds 28 km flown in 235 s (almost 4 min) with v = 120 m/s. A 10-m/s along-track wind in this case can cause a cloud displacement of 2,354 m during the overpass, about the same length as the cloud’s diameter. To determine cloud dimensions with admissible accuracy, Alexandrov et al. (2016a) reduced the angular range to | tan ⁡ θ ≤ 1 |, that is, setting θ m a x = 45 ° instead of 60°. This reduced Δ r m a x to 160 m and increased A p o l to 1.41 (which is still suitable for cloud shape inscription). The overpass length and time were also reduced to 17 km and 140 s (2 min 20 s), respectively, while the potential cloud displacement remained quite large at 1,402 m.

7.3 Clouds from CAMP 2Ex

In Part II of this series, we presented a tomographic analysis of two clouds observed during CAMP 2Ex conducted in the vicinity of the Philippines in August–October 2019. The RSP was deployed on board NASA’s P-3B aircraft. In the first example, we focused on the Cu cloud observed on 25 August (Cloud 1), while in the second example, we analyzed a much larger and optically thicker CuCg (Tcu) cloud observed on 9 September (Cloud 2). Typical aircraft speed during the campaign was approximately 140 m/s, which yields the spatial resolution Δ x = 112 m, given the RSP measurement intervals of 0.8 s. As in all RSP-based examples using the full angular range, A p o l = 1.15 in both cases, which is sufficient for cloud shape inscription.

In the case of Cloud 1, the average aircraft altitude H during the overpass was approximately 4,500 m. The cloud itself was essentially round ( A c l d = 0.93 ≈ 1 ), with the radius R ≈ 1300 m and the cloud center altitude h c ≈ 1700 m (corresponding to h ≈ 2800 m). The number of brightness thresholds and the corresponding cloud shapes was N s h p = 8, so the required radial accuracy was Δ r r e q ≈ 163 m. The actual radial accuracy Δ r m a x ≈ 78 m, computed according to Equation 3, is twofold finer than the required value. The angular accuracy Δ φ m a x = Δ x / h is 2.3 ° (from Equation 5 with θ = 0). The overpass length l o p from Equation 13 is 14.9 km flown in 106 s (1 min 46 s). The cloud’s displacement by 10 m/c wind during this time would be 1,060 m, which is less than the cloud diameter of 2,600 m.

In the case of the larger and thicker Cloud 2, the P-3B’s altitude H was close to 6,000 m, while the cloud center was at h c ≈ 2500 m above ground, thus, h ≈ 3500 m. The cloud aspect ratio varied between 0.90 and 0.99 (depending on the brightness threshold), so we assume A c l d ≈ 1 and approximate the cloud’s shape by a ball with R ≈ 2000 m. (Actually, Cloud 2 slightly deviated from roundness because of a small thin “appendix” visible only on radar, but we disregard it for the purposes of our analysis.) The number of brightness thresholds (and cloud shapes) in this case was N s h p = 4, corresponding to the required radial accuracy Δ r r e q ≈ 500 m. The radial accuracy of the actual measurements from Equation 3 is Δ r m a x ≈ 98 m, which is more than sufficient for our tomographic retrievals. The angular accuracy Δ φ m a x = Δ x / h is 1.8 °, and the overpass length is l o p ≈ 20124 m flown in 144 s (2 min 24 s). The cloud’s displacement by a hypothetical 10 m/s wind during the overpass is 1,440 m, which is substantially smaller than the cloud’s diameter of 7 km.

7.4 Clouds from ER-2 campaigns

When the RSP is deployed on NASA’s high-altitude ER-2 plane (see, e.g., Alexandrov et al., 2015; Alexandrov et al., 2016b) flying at 20-km altitude with 210-m/s speed, its Δ r is 476 m, and Δ φ m a x ≈ 0.57 ° (assuming h c = 3000 m). This large Δ r value indicates that tomographic analysis of the measurements made from this platform can be applied only to clouds with R ≳ 1500 m, while winds can still cause serious problems. For example, for a cloud with R = 2000 m ( N s h p = 4 ) centered at the altitude h c = 3000 m, the overpass length of 67 km is flown in 320 s (5 min 20 s), which may correspond to displacement of 3,200 m comparable with the cloud diameter, and is almost 7 times larger than Δ r.

8 Conclusion

In Part III of this series, we presented a theoretical study of retrieval accuracy and performance of the tomographic algorithm introduced in Part I and applied to real RSP measurements in Part II. We focused on the most challenging part of the algorithm, dealing with the determination of nested cloud shapes corresponding to a range of brightness thresholds. The family of these shapes forms the basis of the algorithm and is later converted into a 2D field of cloud extinction coefficient. In Section 3, we have related the spatial accuracy of the shape determination to the resolution of the multi-angle measurements made by an airborne along-track scanner (or imager). In Section 4, we derived constraints on the aspect ratios of the clouds to which our algorithm can be applied. In Section 5, the expressions were derived for overpass length and time as well as for possible cloud displacement during the overpass. Further study of the effects on the retrievals of cloud movement caused by advection by wind is presented in Section 6. In particular, we demonstrated that the cloud moving in the same direction as the sensor appears larger and lower in the retrievals than it really is, while it appears smaller and higher if it moves in the opposite direction. In Section 7, we applied our accuracy estimation methods to previously published examples of clouds, both simulated and real.

Our plans include further development of our tomographic technique and application of it to more data. The tomographic algorithm presented in Part I was designed for measurements of total reflectance by an airborne along-track scanner with high angular resolution, such as the RSP. However, we plan to extend our accuracy and applicability analysis of it to other airborne (such as AirHARP (McBride et al., 2024)) and satellite instruments (such as HARP-2 (Sienkiewicz et al., 2025), which is currently deployed on board the NASA’s Plankton, Aerosol, Cloud Ocean Ecosystem (PACE) satellite platform).

Applications of a technique requiring multi-angle measurements to satellite data face several problems that must be resolved. Our method is applicable to single-layer isolated clouds, the identification of which is challenging in satellite data (especially when the spatial resolution of the measurements is less than optimal). Measurements at high viewing angles (required to constrain the cloud’s top and bottom) are made at large distances from the cloud, so they can be affected by the long atmospheric path and may be blocked by other clouds. The distance along the orbit between forward- and backward-looking measurement points corresponding to the highest viewing angles is also large, leading to long overpass times. During an overpass that takes more than 10 min, the observed cloud may move, change, or entirely disappear. To address this issue, a multi-satellite constellation concept is being explored (Levis et al., 2020). We will also work on the design of a simplified parametric tomographic algorithm that does not require high-resolution multi-angle measurements and may be more suitable for satellite data analysis.

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Footnotes

Edited by: Xiaoguang Xu, University of Maryland, Baltimore County, United States

Reviewed by: Sean Foley, National Aeronautics and Space Administration, United States

Haixiao Yu, Chinese Academy of Sciences (CAS), China