Analytical Approximation of the Definite Chapman Integral for Arbitrary Zenith Angle

Yue, Dongxiao

doi:https://doi.org/10.5194/egusphere-2023-3112

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https://doi.org/10.5194/egusphere-2023-3112

Preprints

15 Jan 2024

| 15 Jan 2024

Analytical Approximation of the Definite Chapman Integral for Arbitrary Zenith Angle

Dongxiao Yue

Abstract. This study presents an analytical approximation of the definite Chapman integral, applicable to any zenith angle and finite integration limits. We also present the asymptotic expression for the definite Chapman integral, which enables an accurate and efficient implementation free of numerical overflows. The maximum relative error in our analytical solution is below 0.5 %.

Received: 22 Dec 2023 – Discussion started: 15 Jan 2024

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Dongxiao Yue

Status: closed

RC1:
'Comment on egusphere-2023-3112', Anonymous Referee #1, 23 Jan 2024

This interesting paper derives a possibly better and simpler approximation for integrating the optical depth and attenuation of (presumably) sunlight through an atmosphere. The mathematics look clean, but it is still an approximation, it has errors, it is not a closed-form analytic solution.
The real problem is that the method relies on too many approximations (g = constant, H = constant, no refraction!). In real-use situations, the model atmosphere has many distinct layers with highly varying composition and opacity (think clouds). Current methods of calculating spherical atmospheres are decades beyond this research – see references below.
I do not see how this paper can be published here. It does not contribute to the current science of calculating the attenuation of sunlight in atmospheres. Revisions would not improve this rating.
K.E. Trenberth, L. Smith (2005) The mass of the atmosphere: A constraint on global analyses, J. Clim. 18, 864–875.
M.J. Prather, J.C. Hsu (2019) A round Earth for climate models, Proc Natl Acad Sci, 116 (39) 19330-19335.

Citation: https://doi.org/10.5194/egusphere-2023-3112-RC1
- AC1:
  'Reply on RC1', Dongxiao Yue, 24 Jan 2024
  
  1. The solution to the definite Chapman integral was in a closed form. It was an approximation, as described in the title, and it was a very good one. It made significant advancements from previous research on this problem. Please see the references cited in the paper, particularly the 2021 paper by Vasylev, which provided an easy-to-read table listing previous results, in addition to his own contributions.
  2. For an atmosphere much thinner than the planet's radius, a constant 'g' is a valid approximation. The exponential drop in density is a result of Boltzmann's distribution. H can indeed be different for different components of the atmosphere; the solution reported in the paper can be applied separately to them. It's true that cloud densities don't necessarily vary exponentially with height; the paper does not pretend to solve all problems but touches only a small yet unsolved problem of the definite Chapman integral.
  3. A search for "Chapman function" (the improper integral with a limit of infinity) in the current research literature shows that it is widely used. For textbook examples, one may refer to:
  a. "Ionospheres: Physics, Plasma Physics, and Chemistry (Cambridge Atmospheric and Space Science Series)" (2009) by Robert W. Schunk, et al. On page 258, it presents the Chapman function. The book states, "A great deal of effort used to be devoted to obtaining good analytic expressions for this Chapman function. However, with the availability of high-speed computers, an exact evaluation of the optical depth is relatively easy..."
  b. "Aeronomy of the Middle Atmosphere" by Brasseur GP, Solomon S (2005), Springer, Berlin. On pages 174-175, the Chapman function is presented, and the approximations by Smith and Smith are shown.
  c. "Planetary Aeronomy Atmosphere Environments in Planetary Systems" by Siegfried Bauer, Helmut Lammer (2013). Page 17 shows the Chapman function, with the statement "A number of analytical approximations to the Chapman function have been developed...", showing some previous approximations. (Google Books provides free access to that page - see attached).
  d. "An Introduction to Atmospheric Radiation" by K. N. Liou (2002). Page 111 of this book even includes an exercise problem asking the reader to derive the Chapman function. (Google Books provides free access to that page).
  e. "GPU Pro 360 Guide to Rendering" by Wolfgang Engel (2018). Page 180 shows the Chapman function and its use in atmospheric scattering.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3112-AC1
  - RC3: 'Reply on AC1', Anonymous Referee #1, 08 Feb 2024
    
    In light of the careful review by referee #2, I withdraw my objection to publication.
    I do believe the author should include a more thoughtful discussion of how this analytical formulation may not be useful in real, inhomogeneous, refracting atmospheres.
    Could it be applied layer-by-layer in a more realistic atmosphere? How can refraction be included?
    
    Citation: https://doi.org/10.5194/egusphere-2023-3112-RC3
    
    AC3: 'Reply on RC3', Dongxiao Yue, 11 Feb 2024
    
    Thank you for withdrawing the objection to publication. Regarding refraction, it can be accounted for as a minor adjustment in the zenith angle of incoming light, when considered as a first-order approximation. Generally, refraction has a small impact on the density integral. For example, in the Earth's atmosphere, refraction results in a light bending of approximately half a degree at the horizon, with its effect diminishing as the Sun ascends.
    
    Citation: https://doi.org/10.5194/egusphere-2023-3112-AC3
RC2:
'Comment on egusphere-2023-3112', Anonymous Referee #2, 08 Feb 2024

The manuscript "Analytical approximation of the definite Chapman integral for arbitrary zenith angle" by Dongxiao Yue is an interesting analytical approximation of the Chapman mapping function. The Chapman function arises naturally when one studies atmospheres in hydrostatic equilibrium characterized by spherical symmetry. Although both conditions are not met when one is dealing with realistic planetary atmospheres, the Chapman function remains a handy mapping function in several branches of atmospheric physics, such as ionospheric physics, radiative transfer theory, radiation absorption by planetary atmospheres, to name just a few. Compared to the simpler secant mapping function (which is also widely used), the Chapman mapping function is not divergent when the zenith angle argument tends to pi/2.

On the other hand, the Chapman mapping function is represented in the form of the integral, so the exact analytic approximation for it is desirable.

Dongxiao Yue proposes one of the possible approximate expressions in the manuscript. This is actually done for a more general mapping function as given in the form of Eq. (6), with what I would call the incomplete Chapman mapping function. The results obtained are scientifically sound and the derived approximation works quite well. The author discusses in detail the asymptotic behavior of the proposed approximation and proposes the asymptotic expression suitable for computer computations. In summary, the manuscript presents an elegant analytical result that approximates the incomplete Chapman mapping function, and I recommend the manuscript for publication.
Some minor comments are listed below:

1. Line 40. The variable y is not shown in Fig. 1, and its actual introduction does not occur until Eq. (9). The sentence "As illustrated..." is therefore confusing.

2. Line 52. I would move the sentence "Since the thickness..." to line 55. This is logically connected to the next statement "Since $\lambda$ is large...".

3. Line 53. One more explanation of why rewriting (9) in the approximate form of (10) is necessary. Probably the author can add some words why the expression with -sin z term remains under the integral sign and the expression with +sin z term is simplified as given in (10).

4. Line 91. In the sentence "At the same time, the exponential factor in the equation..." it is necessary to specify "At the same time, the exponential factor in the second term inside the brackets of Eq. (11)...".

Citation: https://doi.org/10.5194/egusphere-2023-3112-RC2
- AC2: 'Reply on RC2', Dongxiao Yue, 11 Feb 2024
  
  Thank you very much for your insightful comments and suggestions regarding the manuscript "Analytical approximation of the definite Chapman integral for arbitrary zenith angle". I appreciate the time and care you invested in reviewing the work and providing constructive feedback. Suggestions #1-4 are all well taken.
  In particular, I acknowledge the confusion caused by the premature introduction of variable y without its proper representation in Fig. 1, as well as the logical rearrangement of sentences for enhanced clarity and coherence as you suggested. Furthermore, I understand the necessity of providing additional explanation for the approximation process used in our derivation in Eq.(10), as well as the need for clarification in the description of the exponential factor in Eq. (11).
  I will make all the suggested revisions to address these concerns and enhance the quality of the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3112-AC2

Status: closed

RC1:
'Comment on egusphere-2023-3112', Anonymous Referee #1, 23 Jan 2024

This interesting paper derives a possibly better and simpler approximation for integrating the optical depth and attenuation of (presumably) sunlight through an atmosphere. The mathematics look clean, but it is still an approximation, it has errors, it is not a closed-form analytic solution.
The real problem is that the method relies on too many approximations (g = constant, H = constant, no refraction!). In real-use situations, the model atmosphere has many distinct layers with highly varying composition and opacity (think clouds). Current methods of calculating spherical atmospheres are decades beyond this research – see references below.
I do not see how this paper can be published here. It does not contribute to the current science of calculating the attenuation of sunlight in atmospheres. Revisions would not improve this rating.
K.E. Trenberth, L. Smith (2005) The mass of the atmosphere: A constraint on global analyses, J. Clim. 18, 864–875.
M.J. Prather, J.C. Hsu (2019) A round Earth for climate models, Proc Natl Acad Sci, 116 (39) 19330-19335.

Citation: https://doi.org/10.5194/egusphere-2023-3112-RC1
- AC1:
  'Reply on RC1', Dongxiao Yue, 24 Jan 2024
  
  1. The solution to the definite Chapman integral was in a closed form. It was an approximation, as described in the title, and it was a very good one. It made significant advancements from previous research on this problem. Please see the references cited in the paper, particularly the 2021 paper by Vasylev, which provided an easy-to-read table listing previous results, in addition to his own contributions.
  2. For an atmosphere much thinner than the planet's radius, a constant 'g' is a valid approximation. The exponential drop in density is a result of Boltzmann's distribution. H can indeed be different for different components of the atmosphere; the solution reported in the paper can be applied separately to them. It's true that cloud densities don't necessarily vary exponentially with height; the paper does not pretend to solve all problems but touches only a small yet unsolved problem of the definite Chapman integral.
  3. A search for "Chapman function" (the improper integral with a limit of infinity) in the current research literature shows that it is widely used. For textbook examples, one may refer to:
  a. "Ionospheres: Physics, Plasma Physics, and Chemistry (Cambridge Atmospheric and Space Science Series)" (2009) by Robert W. Schunk, et al. On page 258, it presents the Chapman function. The book states, "A great deal of effort used to be devoted to obtaining good analytic expressions for this Chapman function. However, with the availability of high-speed computers, an exact evaluation of the optical depth is relatively easy..."
  b. "Aeronomy of the Middle Atmosphere" by Brasseur GP, Solomon S (2005), Springer, Berlin. On pages 174-175, the Chapman function is presented, and the approximations by Smith and Smith are shown.
  c. "Planetary Aeronomy Atmosphere Environments in Planetary Systems" by Siegfried Bauer, Helmut Lammer (2013). Page 17 shows the Chapman function, with the statement "A number of analytical approximations to the Chapman function have been developed...", showing some previous approximations. (Google Books provides free access to that page - see attached).
  d. "An Introduction to Atmospheric Radiation" by K. N. Liou (2002). Page 111 of this book even includes an exercise problem asking the reader to derive the Chapman function. (Google Books provides free access to that page).
  e. "GPU Pro 360 Guide to Rendering" by Wolfgang Engel (2018). Page 180 shows the Chapman function and its use in atmospheric scattering.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3112-AC1
  - RC3: 'Reply on AC1', Anonymous Referee #1, 08 Feb 2024
    
    In light of the careful review by referee #2, I withdraw my objection to publication.
    I do believe the author should include a more thoughtful discussion of how this analytical formulation may not be useful in real, inhomogeneous, refracting atmospheres.
    Could it be applied layer-by-layer in a more realistic atmosphere? How can refraction be included?
    
    Citation: https://doi.org/10.5194/egusphere-2023-3112-RC3
    
    AC3: 'Reply on RC3', Dongxiao Yue, 11 Feb 2024
    
    Thank you for withdrawing the objection to publication. Regarding refraction, it can be accounted for as a minor adjustment in the zenith angle of incoming light, when considered as a first-order approximation. Generally, refraction has a small impact on the density integral. For example, in the Earth's atmosphere, refraction results in a light bending of approximately half a degree at the horizon, with its effect diminishing as the Sun ascends.
    
    Citation: https://doi.org/10.5194/egusphere-2023-3112-AC3
RC2:
'Comment on egusphere-2023-3112', Anonymous Referee #2, 08 Feb 2024

The manuscript "Analytical approximation of the definite Chapman integral for arbitrary zenith angle" by Dongxiao Yue is an interesting analytical approximation of the Chapman mapping function. The Chapman function arises naturally when one studies atmospheres in hydrostatic equilibrium characterized by spherical symmetry. Although both conditions are not met when one is dealing with realistic planetary atmospheres, the Chapman function remains a handy mapping function in several branches of atmospheric physics, such as ionospheric physics, radiative transfer theory, radiation absorption by planetary atmospheres, to name just a few. Compared to the simpler secant mapping function (which is also widely used), the Chapman mapping function is not divergent when the zenith angle argument tends to pi/2.

On the other hand, the Chapman mapping function is represented in the form of the integral, so the exact analytic approximation for it is desirable.

Dongxiao Yue proposes one of the possible approximate expressions in the manuscript. This is actually done for a more general mapping function as given in the form of Eq. (6), with what I would call the incomplete Chapman mapping function. The results obtained are scientifically sound and the derived approximation works quite well. The author discusses in detail the asymptotic behavior of the proposed approximation and proposes the asymptotic expression suitable for computer computations. In summary, the manuscript presents an elegant analytical result that approximates the incomplete Chapman mapping function, and I recommend the manuscript for publication.
Some minor comments are listed below:

1. Line 40. The variable y is not shown in Fig. 1, and its actual introduction does not occur until Eq. (9). The sentence "As illustrated..." is therefore confusing.

2. Line 52. I would move the sentence "Since the thickness..." to line 55. This is logically connected to the next statement "Since $\lambda$ is large...".

3. Line 53. One more explanation of why rewriting (9) in the approximate form of (10) is necessary. Probably the author can add some words why the expression with -sin z term remains under the integral sign and the expression with +sin z term is simplified as given in (10).

4. Line 91. In the sentence "At the same time, the exponential factor in the equation..." it is necessary to specify "At the same time, the exponential factor in the second term inside the brackets of Eq. (11)...".

Citation: https://doi.org/10.5194/egusphere-2023-3112-RC2
- AC2: 'Reply on RC2', Dongxiao Yue, 11 Feb 2024
  
  Thank you very much for your insightful comments and suggestions regarding the manuscript "Analytical approximation of the definite Chapman integral for arbitrary zenith angle". I appreciate the time and care you invested in reviewing the work and providing constructive feedback. Suggestions #1-4 are all well taken.
  In particular, I acknowledge the confusion caused by the premature introduction of variable y without its proper representation in Fig. 1, as well as the logical rearrangement of sentences for enhanced clarity and coherence as you suggested. Furthermore, I understand the necessity of providing additional explanation for the approximation process used in our derivation in Eq.(10), as well as the need for clarification in the description of the exponential factor in Eq. (11).
  I will make all the suggested revisions to address these concerns and enhance the quality of the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3112-AC2

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The stunning colors of the sky and clouds result from light scattering in the atmosphere, whose density changes with height. Previously, calculating these colors involves costly, sometimes inaccurate methods. This paper presents a silver bullet: a single elegant formula that simplifies these complex calculations. The result? Faster, more precise predictions of atmospheric colors, from Earth's blue skies and red sunsets to Venus's golden hues.


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