优化的Voigt函数与Faddeev函数拟合算法
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收藏 330KB PDF 举报 "本文主要探讨了Voigt线型函数在光谱分析中的应用，以及如何进行快速近似计算。Voigt函数是光谱学中一个重要的线型函数，结合了Gauss函数（高斯函数）和Lorentz函数（洛伦兹函数）的特点，常用于描述光谱线的形状。文章中提出了一个新的算法，该算法优化了Voigt/Faddeev函数的计算效率，特别适合大气层逐行计算的需求。同时，文章还提供了计算Voigt函数及其偏导数的补充算法，这对于理解和模拟光谱线的精细结构至关重要。
Voigt函数在光谱分析中的作用在于，它可以准确地模拟由于碰撞、压力、温度等因素导致的真实光谱线形，这些因素使得光谱线不再单纯表现为理想化的高斯或洛伦兹线型。Voigt函数由两个部分组成：实部K(x; y)和虚部L(x; y)，分别对应于线型的强度分布和展宽。其中，x和y是复变量z=x+iy的实部和虚部，而z与光谱线的频率偏移和自然宽度有关。
Faddeev函数，又称为复杂概率函数或等离子体色散函数，与Voigt函数密切相关。在物理学中，它通常用来描述粒子在等离子体环境中的行为，尤其是在研究光与物质相互作用时。Voigt函数的实部K(x; y)和虚部L(x; y)可以通过Faddeev函数来表示，这使得Voigt函数在处理物理问题时具有更广泛的应用。
文章中提到的新算法旨在提高计算效率，特别是对于大气层逐行计算这种需要大量计算的情况。通常，Voigt函数的精确计算比较耗时，但通过放宽最大相对误差标准，可以实现更快的计算速度，这对于实时监测或大数据处理来说非常重要。
此外，文章还介绍了一种同时计算Voigt函数及其偏导数的方法。偏导数的计算对于理解光谱线形的变化以及参数敏感性分析至关重要。例如，在光谱建模中，了解函数对不同参数的响应可以帮助我们更好地解释实验数据，优化模型参数，并揭示光谱线背后的物理过程。
这篇文章不仅介绍了Voigt线型函数的基本概念，还提出了一种新的高效计算方法，这对于光谱学和相关领域的研究具有重要的实践意义。通过对Voigt函数的快速近似计算和偏导数的求解，科学家们可以更有效地处理和解析复杂的光谱数据，进一步加深对光谱现象的理解。"
Figure 3 shows a contour plot of the relative error in
K
0
,
j
K
0
K
ref
j
=K
ref
, where
K
0
(
x; y
) =
1
p
y
x
2
+
y
2
. The contour drawn with a solid line marks the b oundary of the region where
K
0
has
a relative error less than 10
4
. Other contours are shown by dotted lines. The dashed lines are
x
= 146
:
7
y
and
x
2
= 15100 + 40
y
3
:
6
y
2
which provide readily-calculated approximations to
this boundary. Figure 4 is a similar plot of
j
L
0
L
ref
j
=L
ref
where
L
0
(
x; y
) =
1
p
x
x
2
+
y
2
. The
dashed line is
x
= 146
:
7
y
which can be used as a b oundary to ensure the relative errors on
both
K
(
x; y
) and
L
(
x; y
) are less than 10
4
. Similar plots can be used to demonstrate that for
2
R
5, the region where the relative errors on b oth
K
0
(
x; y
) and
L
0
((
x; y
) are both less than
10
R
is bounded by the line
x
= 1
:
51 exp(1
:
144
R
)
y
.
Likewise, Figure 5 is a plot of
j
K
1
K
ref
j
=K
ref
where
K
1
(
x; y
) =
K
(
x; y
) calculated using the
W4 algorithm for Region I. The short-dashed lines are
x
= 14
:
67
y
and
x
2
= 164
4
:
3
y
1
:
8
y
2
which are suggested revised boundaries. The long-dashed line in b oth Figures 5 and 6 is
x
=
15
y
, the Region I b oundary used by Humlcek. Figure 6 is a plot of
j
L
1
L
ref
j
=L
ref
where
L
1
(
x; y
) =
L
(
x; y
) calculated using the W4 algorithm for Region I. Note the large relative error
for
x
5,
y <
10
6
. (The 10
2
contour in the gure deliberately exagerates the area concerned).
In this section of the domain the use of a computationally-demanding algorithm will have little
or no impact on the eciency of line-by-line applications. The short-dashed line is
x
= 14
:
67
y
as in Figure 5. It can b e demonstrated that for 2
R
5 and
y >
10
6
, the region where
the relative errors on b oth
K
1
(
x; y
) and
L
1
(
x; y
) are less than 10
R
is bounded by the line
x
= 1
:
6 exp(0
:
554
R
)
y
.
Figure 7 is a plot of
j
K
2
K
ref
j
=K
ref
where
K
2
(
x; y
) =
K
(
x; y
) calculated using the W4
algorithm for Region II. The short-dashed line is
x
= 6
:
8
y
and the long-dashed line is again the
Humlcek boundary,
x
= 5
:
5
y
. Both these boundaries include an area near the
y
axis where the
relative error is greater than 10
4
. Figure 8 is a plot of
j
L
2
L
ref
j
=L
ref
where
L
2
(
x; y
) =
L
(
x; y
)
calculated using the W4 algorithm for Region I I with the Humlcek b oundary shown with a long-
dashed line and
x
= 6
:
8
y
with a short dashed line. In this case no attempt has been made to
parameterise the position of the new boundary in terms of the required relative error. Instead
a single b oundary was chosen such that the relative error on b oth
K
2
(
x; y
) and
L
2
(
x; y
) is less
than 10
5
for
y >
10
6
.
Figure 9 is a plot of
j
K
3
K
ref
j
=K
ref
where
K
3
(
x; y
) =
K
(
x; y
) calculated using the W4
algorithm for Region II I. The long-dashed lines are again the Humlcek b oundaries,
x
= 5
:
5
y
and 0
:
195
x
= 0
:
176 +
y
. The short-dashed lines are
x
= 6
:
8
y
,
x
= 2
:
4
y
and
x
= 3
:
097
y
0
:
45,
the rst two of which, together with the
y
axis enclose a region where the relative error of
K
3
is
less than 10
5
. Figure 10 is a plot of
j
L
3
L
ref
j
=L
ref
where
L
3
(
x; y
) =
L
(
x; y
) calculated using
the W4 algorithm for Region I II with the Humlcek boundaries shown by long-dashed lines. The
short-dashed lines,
x
= 6
:
8
y
and
x
= 3
:
097
y
0
:
45 with the
y
axis enclose a region where the
relative error in both
K
3
and
L
3
is less than 10
5
.
The HUMLIK subroutine (with
R
= 4 and the calculation of
L
removed) was used in place
of code based on Schreier for some line-by-line calculations. Execution time was approximately
halved without loss of accuracy in the result.
The co de is deliberately written so that it can easily b e translated without loss of eciency
into other languages by those unfamiliar with Fortran. The '
&
' character is used only as a
statement continuation character. Comments are introduced either by a '
*
' character in column
1 or a '
!
' character in column 73. Upp er and lower case characters are regarded as equivalent.
Lines containing executable statements in lower case can b e removed for greater execution sp eed if
the calculation of
L
is not required. On some systems, extended precision arithmetic is no slower
than single precision. However, it should b e noted that re-co ding in extended precision and
increasing the precision of the constants used in the co de do es not give a signicant improvement
in the accuracy of the results.
4
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