2.11: Gaussian Triple Integral Algorithm
- Page ID
- 8728
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)To approximate the integral
\[I=\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} F(x, y, z) d z d y d x\]
where it is assumed that the roots R and coefficients C are stored in two-dimensional arrays.
BEGIN
h1 = (b – a)/2
h2 = (b + a)/2
I = 0
FOR i = 1, 2,..., m DO
Ix = 0
x = h1*R[m][i] + h2
k1 = (d – c)/2
k2 = (d + c)/2
FOR j = 1, 2,..., n DO
Iy = 0
y = k1*R[n][i] + k2
l1 = (f – e)/2
l2 = (f + e)/2
FOR k = 1, 2,..., p DO
z = l1*R[p][k] + l2
Iy = Iy + C[p][k]*F(x, y, z)
END FOR { k-loop }
Ix = Ix + C[n][j]*l1*Iy
END FOR { j-loop }
I = I + C[m][i]*k1*Ix
END FOR { i-loop }
I = h1*I
PRINT I
END
This algorithm may be generalised further by allowing limits e and f to be functions e(x,y) and f(x,y) and c and d to be functions c(x) and d(x). For our purposes the limits of integration are fixed values.
Applying this algorithm to equation (28) for the Bond albedo and identifying μ with x, we see that
\[\frac{A}{2}=\int_{0}^{1} \int_{0}^{2 \pi} \int_{0}^{1} \times f_{r}(x, \mu, \alpha ; \ldots) \mu d \mu d \phi d x\]
and by further identifying z with μ and y with φ
\[\mathrm{F}(\mathrm{x}, \mathrm{y}, \mathrm{z})=2 \mathrm{x} \mathrm{z} f_{r}(\mathrm{x}, \mathrm{z}, \alpha ; \ldots)\]
where a is itself a function of x,y and z [cf. equation (26)]
\[\alpha=\cos ^{-1}\left[\mathrm{xz}+\sqrt{\left(1-\mathrm{x}^{2}\right)\left(1-\mathrm{z}^{2}\right)} \cos \mathrm{y}\right].\]
For the phase integral, there is no need to invoke the likes of equation (32) since the intensity I(α) is explicitly expressed in terms of α and one stage of the integration is with respect to α. The parameters, … , are, of course, not variables since they retain their values for the duration of the integration.
When applying these integrals it is strongly suggested that A, p and q each be calculated independently in order to verify that the relationship A = p q holds. Taking shortcuts may bury insidious bugs, some possibly as simple as a typo., inside a program and result in at least two undetected erroneous results.