ncar/tn-121+str the delta-eddington approximation for a...
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NCAR/TN-121 +STRNCAR TECHNICAL NOTE
July 1977
The Delta-Eddington Approximation fora Vertically Inhomogeneous Atmosphere
W. J. Wiscombe
ATMOSPHERIC ANALYSI ANAND PREDICTION DIVISION
NATIONAL CENTER FOR ATMOSPHERIC RESEARCHBOULDER, COLORADO
4
-~P---P·-·~---~~I----I --I~-~-YU ~ ~eger 1~-~ II~- -·IC-- ~ e - -~ - I I I I -- C a
3zpzti
iii
ABSTRACT
The delta-Eddington approximation of Joseph, Wiscombe, and
Weinman (1976) is extended to an atmosphere divided up by internal
levels into homogeneous layers. Flux continuity is enforced at each
level, leading, as the mathematical essence of the problem, to a
penta-diagonal system of linear equations for certain unknown con-
stants. Fluxes (up, direct down, diffuse down, and net) are then pre-
dicted at each level. Unphysical results of the model are examined in
detail. Potential numerical instabilities in the solution are noted
and corrected, and an extremely fast, well-documented computer code
resulting from this analysis is described and listed. Actual computed
fluxes are given for several test problems.
v
PREFACE
It has always been painfully obvious, in my multi-spectral-
interval, multi-layer, and multi-angle computer models of solar and
IR radiative transfer, that the angular part of the calculation was
the pacing item. Since my interest has primarily been in fluxes, it
always seemed particularly unfortunate to spend the lion's share of
computing time on obtaining angular information, which was then used
only to compute fluxes. It would obviously have been preferable to
calculate fluxes directly, but none of the existing approximations for
so doing (variants of two-stream and Eddington) seemed sufficiently
flexible - for in marching through the solar and IR spectrums, and
vertically upward through an atmosphere, one encounters huge varia-
tions in optical depth and single-scattering albedo, and the existing
approximations were only valid for restricted ranges of these para-
meters. Even worse, they seemed not to be able to handle the asym-
metric phase functions typical of clouds and aerosols very well.
Then J. H. Joseph, J. A. Weinman and I (1976) discovered the
excellent accuracy of the delta-Eddington approximation (which cal-
culates flux directly) for all phase functions, no matter how asym-
metric, and for all optical depths and single-scattering albedos; but
we investigated only homogeneous layers. It was therefore only natural
for me to work up a multi-layer version, which I used to replace
adding-doubling in my spectrally-detailed radiation models for a
cloudy atmosphere. The solar flux changes from so doing were
25-5-15 watts/m (on the order of present uncertainty in the solar
vi
constant), and could equally well be caused by almost undetectable
variations in the effective radius of the cloud drops. [These results
were presented in August, 1976, at the Symposium on Radiation in the
Atmosphere in Garmisch-Partenkirchen, West Germany.] I concluded from
these results that the multi-level delta-Eddington approximation was
potentially of great utility to a wide variety of users who wanted
radiative fluxes of roughly 1% accuracy. This was the motivation for
publishing the present document.
I would like to thank Dr. V, Ramanathan for his careful review
of and suggestions for improving this publication.
Warren J. WiscombeMay 1977
vii
TABLE OF CONTENTS
ONE-LAYER FORMULAS ..........
ALBEDO OF A SEMI-INFINITE LAYER . .
MULTI-LAYER FORMULATION . . .....
THE COMPUTER CODE ........
4o1 ELIMINATION OF w=1 BRANCH .. . . .......
4.2 MULTIPLE DIRECTIONS OF INCIDENCE .....
4.3 SOLVING THE LINEAR SYSTEM ......
4.4 ILL-CONDITIONING .. . . . . . . . . I
4.5 SPURIOUS AND NEGATIVE FLUXES IN HIGHLY ABSORBINGCASES . . .. . . .. . . . ..
4.6 FAILURE TESTING .. .................
4.7 SPEEDING UP THE CODE . ................
5. ACCURACY . . . . . . . . . . . . . .
REFERENCES .... ... ....................
APPENDIX A: LISTING OF COMPUTER CODE FOR THE MULTI-LAYERDELTA-EDDINGTON APPROXIMATION ........ ...
APPENDIX B: TEST PROBLEMS 1-11 .... ...... ......
APPENDIX C: COMPARISON OF DELTA-EDDINGTON AND ADDING-DOUBLINGFLUX COMPUTATIONS .................
Page
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18
18
20
22
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30
31
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1.
2.
3.
4.
0 0
0 a
0
0 0
0 0 0 0 0
0 *I 0 0 0 0 0
0 0 0 0 0
1
1. ONE-LAYER FORMULAS
The delta-Eddington formulas for a single homogeneous layer will
be required in the multi-layer formalism. Let us therefore derive
them, using the notation:
w = scaled single scattering albedo of layer
g = scaled asymmetry factor of layer
T = scaled optical depth within layer
AT = total scaled optical depth of layer
\P = cosine of zenith angle of monodirectional beam incidentupon top of layer
S = incident-beam flux
A = surface albedo.
The scaling relationships are
= * 2 fl-e* )0*2g T ( w*g* )* T = (1 -*g*2)T* , )
-*- +l~ g W*~~g1 - w*g*
where g*, T*, a* are the actual values of the layer's optical parameters.
If we follow the procedure outlined in the paragraph following
Eq. (16) of Joseph et al. (1976), we arrive at two differential equa-
tions, which are:
dG 3 3 3ST/g 0
dT + 2 (l-g)H = wSgpe (la)
dH 1 -T/P0d + 2(1-w)G = wSe (lb)
where G, H are proportional to the i0, i1 of Joseph et al.,
G -E i0 , H -E 2 ri (2)
2
The solution of these equations is straightforward when w and g do not
vary with T. There are two cases which arise:
(1) w < 1
G = cle + c2 ae O (3a)
X- -XT T/1O iH = -Pcle + Pc2e -/ e (3b)
2
where cl, c2 are arbitrary constants and
X = /3(1-o)(l-wg) (3c)
I ..... (3d)p 2 X = X | - , (3d)
3 1-cg ,
3 1 + g(l-) (3e)4 2 2(l/po)-X
3g(l-w)pO0 + (1/p0)=3 2 WS 2 2 - (3f)
2 (1/) X
(2) = 1
G = c1 - (l-g)c2 T - 4 Se (4a)
2 1 S-/ 0H = c 2 - Spe 0 (4b)
where cl, c2 again are arbitrary constants.
The solution (3a-f) is invalid when p0 = 1/X, for then the
particular and homogeneous solutions of the differential equations
(la-b) coincide. Eqs. (3a-b) do approach a finite limit as p0 - 1/X,
but computational overflows will occur in practice, since the limit
3
is of the form 0/0. However, D0 can only equal 1/% if X > 1, or if
g+l - - g + 1
2g
Since dol/dg < 0, and since wo1 equals 2/3 at g = 0, it is clear that
we need never concern ourselves with this case if w > 2/3. Even
should it arise, it can easily be avoided by changing pO slightly.
Only upon attempting an analytic integration over pO might it be
necessary to bear this case in mind (numerical integration could
simply avoid picking pO = 1/X as a quadrature point).
Diffuse up, diffuse down, and net fluxes for the layer are given
by the following expressions:
w - l =
-1F = 27 r
0pi(T,11) dp
-1
= 27 10
(Di0 + i12i1 ) dl
2= 7(i 0- i1 ) = G- H
+, .1F = 27 I pli(T,v) dp
0
2= Tr(i 0 + il) = G + H
-T/PO + F + -T/pOF = p Se + F - F = p Se + 2H
Note from Eqs. (5c) and (lb) that
d = -Se + dH +/ 0^ = - - - 2 = -(1-c) (Se 0 + 4G 1
This shows that when there is no absorption, w = 1, the approximation
(5a)
(5b)
(5c)
(6)
4
conserves flux. Furthermore, when absorption is present, w < 1, the
approximation guarantees positive absorptivities, dF/dT < 0, provided
G > 0. [Clearly G, being proportional to mean intensity, should be
positive on physical grounds.] The differential equation satisfied
by G, from Eqs. (la-b), is
2 3 -T/P0G" - X G = - w [l+g(l-o)]Se
Because the right-hand side is negative, G cannot have a relative
minimum at which it is negative or zero; for at such a minimum, G" > 0
and X2G < 0, which is impossible. Hence
G(O) > 0 and G(AT) > ==0 G(T) 0 for O < T < AT
because otherwise G would, upon leaving G(O) 2 0 and becoming negative,
have to 'turn back,' constituting thereby a relative minimum, to reach
G(AT) 2 0.
One can argue for G(O) 2 0 as follows. The top boundary condi-
+tion, prescribing a diffuse down-flux F0 at T = 0, is
G(0) + H(O) = F0 . (7)
But physically the net flux at the top of the layer cannot exceed the
incident flux 10S + F0 , which, from Eq. (5c), implies H(O) < - F .
Therefore G(O) > 0 from Eq. (7). [This is not, of course, a rigorous
proof, in that it draws on an assumption not properly part of the
equations.]
Given G(0) > 0, one can argue rigorously for G(AT) > 0. Let
+there be a prescribed up-flux F at T = AT, as well as Lambert0
5
reflection with albedo A(p ):
G(AT) - H(AT) = F0 t + A(D 0 ) G(AT) + H(AT) + p 0Se i (8)
[The reason for a p0 dependence in A is explained later.] Eq. (la)
allows this to be reduced to a boundary condition involving G alone,
G'(AT) + Y1G(AT) =
where all we need to know about y1 and y2 is that they are positive.
Suppose G(AT) < 0. Then G'(AT) > 0 from the last equation. It follows
that G(T) must have a negative relative minimum in order to turn up and
reach G(0) 2 0. Since this is impossible, G(AT) > 0 is proved by con-
tradiction. In sum, positivity of G(0) establishes positivity of G,
and therefore of absorptivities, at all optical depths.
6
2. ALBEDO OF A SEMI-INFINITE LAYER
We examine the albedo of semi-infinite layers in the delta-
Eddington approximation, both out of intrinsic interest, and because
it relates to ill-conditioning in the code and to possible unphysical
cases which can arise.
Consider a semi-infinite layer with single-scattering albedo less
than one (the case where it equals one being trivial). If we take the
delta-Eddington solution, Eq. (3), with boundary conditions (7) and
+(8), setting F0 = 0, then the layer albedo,
a ( -o) G(O) - H(O)a( )+
P0S + F0
becomes, in the limit XAT + oo,
(l-P)F0 + 2(6-Pa)ao (p) 9= 0 (9)
(l+P)(GoS+Fo )
This limit is actually reached, for all practical purposes, when
e << 1 .(10)
Eq. (10) is precisely the situation in which ill-conditioning arises
in the multi-layer code, as described in Section 4.4.
Note that the surface albedo, A(p0 ), does not appear in Eq. (9);
from this we conclude that levels above a layer satisfying condition
(10) are completely unaffected by levels below it. Radiative
7
+communication is effectively blocked. [Of course, a large enough F
could make itself felt even through such a damping layer, but we have
+considered that F = 0 in Eq. (9).]
Consider the numerator of Eq. (9) (the denominator is always
positive because P > 0). The coefficient of F0 may be negative in
the following circumstance:
1 - P < O w < 4- 3g
which is easily deduced from Eq. 3(d). The second inequality holds
for both scaled (w,g) and unscaled (w*,g*) variables, on account of
similarity relation (17b) of Joseph et al. (1976). For g* = 0.85,
this inequality becomes o* < 0.69, and as g* -+ 1, it tends to w* < 1.
Thus, the more asymmetric the phase function, the larger the range of
w* over which 1 - P < 0. Clearly, if only a diffuse flux F is inci-
dent on the semi-infinite layer, the delta-Eddington albedo (9) will
be negative when the single-scattering albedo is sufficiently below
unity. Its largest possible negative value, assumed at w = 0, is -7.2%,
i.e.
min = 1- = -0.0720<W<1 P 1
For the second term in the numerator of Eq. (9), we have the
following expression:
1
B - P = +? -( Pg0 )0s1 + XP0 2 0 0~g~O~O
8
which is always positive because P i /3, g < 1/2, and p0 ' 1. Hence,
if only a direct flux poS is incident on the semi-infinite layer, the
delta-Eddington albedo (9) will always be positive.
Negative albedos can arise at any level within a multi-layer
delta-Eddington calculation where (a) at least some diffuse down-flux
exists, and (b) the total of all layers below that level is fairly
highly absorbing and "semi-infinite" in the sense of Eq. (10). Test
Problems 7-9, Appendix B, illustrate negative up-fluxes at internal
levels arising from simultaneous satisfaction of conditions (a) and
(b). Fortunately, in many problems of practical interest, including
the earth's cloudy atmosphere, condition (b) is seldom if ever met.
The above considerations clearly point up an inconsistency in the
delta-Eddington approximation for semi-infinite layers. If we regard
the diffuse flux F as made up of numerous direct fluxes, then the
semi-infinite albedo with only diffuse flux incident should be an inte-
gral over p of Eq. (9) with F = 0. But the latter integral is always
positive, while the former quantity is sometimes predicted to be nega-
tive by Eq. (9) with S = 0.
While one should be aware of the problems associated with Eq. (9),
we nevertheless suggest that, in two circumstances, it may be useful in
specifying surface albedo, i.e. A(p0) = ao(P):
(1) when the "surface" is just an internal level (e.g., a cloud
top);
(2) when the surface can be modeled as a multiple scattering
medium, as, e.g., snow (cf. Bohren and Barkstrom, 1974) or
loose dust.
9
In both cases one must estimate w* and g* for the medium beneath the
"surface," and, to avoid negative albedos, these should satisfy
w* > 1/(4 - 3g*).
10
3. MULTI-LAYER FORMULATION
Let us assume an arbitrary inhomogeneous layer to be subdivided
in such a way that W and g are constant, or at least sensibly so, within
each sublayer. Then the situation will be as in Fig. 1, i.e., the
complete layer is modeled as a concatenation of N-1 homogeneous sub-
layers with single-scattering albedos i., optical depths (AT)i, and
asymmetry factors gi (all of which are scaled, remember). By imposing
top and bottom boundary conditions, Eqs. (7) and (8), and requiring
flux continuity across interior levels, we shall arrive at a set of
penta-diagonal linear equations for the unknown constants cl, c2
[Eqs. (3a-b)] in each layer.
S
level 1 -.. T1 = 0
wl, gl (AT)1
level 2 - T
2 ,g2, (AT)2
level 3 -- ----- 3
level N-l -. ------ ---- T 1
N-' gN-' ()N- 1
level N / / / / / / / / N
Fig. 1
11
The solution for layer i (i = 1, .,N-l), valid for Ti < Tis s oi i ii+ 1
where Ti is scaled optical depth from the top to level i, is
ilG i = 2i-le
H. = -P2- ei ix2i-1
-. T
+ 2ie - ie
X.T -X.T+ Pix 2 ie
(lla)
(llb)- Bie
where the unknown constants for layer i are written in the notation
x c (i)2i-l = 1 X =2i c (i),2i - 2
in order that the usual matrix-vector notation (Cx=d) for a linear
system might subsequently be applicable. All told, there are 2N-2 of
these unknown constants.
The interior flux continuity conditions, supplemented by top and
bottom boundary conditions, Eqs. (7) and (8), are
G1(0) + H1(0) F
Gl (T2 ) - G2(T2) = 0
Hl(T 2 ) - H2 (T2 ) = 0
(12)
GN_2 (N- 1)GN- N- 1 =
HN_2 (,TN_ 1 )-HNl1 (TN 1) =
0
0
[^(P~o^^N-1( N = F + A(W )Se N O
[1-A(p0)I]GN (TN)-[l+A(li0)1]- (TN) = F0 + A(p0)p0 Se
12
This is the system of 2N-2 linear equations one must solve for the
unknown constants x.. Note that we have written the flux continuity
conditions in Eq. (12) in terms of G and H, rather than F (=G-H) and
F (=G+H). This is because, by so doing, the coefficients have a
simpler form, and, more importantly, numerical ill-conditioning is
avoided.
Note that, by properly specifying F0, in the first equation in
(12), level 1l may be taken somewhere below the top of a scattering
atmosphere. Similarly, by properly specifying A(pO) in the last
equation in (12), level N may be taken above the actual surface, for
+example at a cloud top. F 0 can be used to account for a genuine
source of radiation (e.g., thermal) below or at level N.
Our first attempts to solve Eq. (12) directly for the x. led to
insurmountable ill-conditioning problems for layers, however subdivided,
with total optical depths T such that X.T exceeded about 14. This was
XiT -XiTdirectly traceable to the exponentials e and e in Eqs. (lla-b),
which become very large and very small, respectively, in such circum-
stances. However, we observed that the flux formulas at each level
required, not the bare quantities xi, but rather
- iT i+lx i = e Xi- (13a)X2i-l E x2i-
i- ii+lx2i = e x (13b)
x2i
13
in terms of which the requisite flux formulas are
1-P=X +
l+Px +
e 1
(l+P1)elx2
(1AP e(1-P1) e1x2
F+1 = (1-Pi)x2i-l + (l+Pi)x 2i
+l = (l+Pi) + (1-Pi) 2i
- (oi+3i)
-Ti+l/PO
- (a i+6i)e
- (oi-Si) e
for i=l, *,N-l, and where
.i(AT) ie E e (15)
Eqs. (14) proceed straightforwardly from Eqs. (5), (11), and (13). The
only ambiguity is that flux at level i+l can be calculated from either
Gi+l, Hi+l or Gi, Hi; we arbitrarily choose the latter way, since
Eq. (12) guarantees it to be equivalent to the former.
In terms of the new unknowns x. [Eq. (13)], the linear system (12)
may be written out explicitly as follows:
+F1
F1
(14a)
(14b)
(14c)
(14d)
1-P1--- - x +
el 1
A
x 1 +
-Pl +
- e2
- 4-P 2e2x4
^ 1 ^x2 - xi+l2i ei+l 2i+
Ai+l2ei+lX2i+2
-T i+v /
= (a -a+)e0
-Pix2i-+
PA .jH. ^Aix2i + i+l ^
Pix2i ei+ X2i+l i+lei+lX2i+2( B ,-T +l /
= (Si- )e± i+1
(16)
[1-A+(l+A)PN 1 ]x 2 N_3 + [1-A-(l+A) PN 1 ]x2N 2
-ATN/ p 0[N- N-i+A(0 S-ONl Ni)]eN + F0
The coefficient matrix,which we shall call C, has the following penta-
diagonal form:
14
X21e ̂
- -xe3"2
x33^ 2
P1X2 +- ee2
xi- +x2i-1
= a + + +(1+P 1) eix 2
-T2/P 0(al-ot )
= U --B )e
15
lC11NCl2N 13N N= 5 C12 | 17C C C Cc 21 c22 °23 24
31 32 33 34
C~ C C c C|i42| 43 44 45 46
C= 5 3 C54 C55 5 6
0M1-2..M- 2 M-2,M-2 M-2,M-1 M-2,M
CM-1,M-3 M-1,M-2 M-1,M-1 M-1,M
where 2N st ian.M-2 M-
where M = 2N-2 is the size of the matrix, and all the boxed elements
vanish.
16
4. THE COMPUTER CODE
We present below the special features of the computer code
(Appendix A) which solves the 6-Eddington equations of Section 3. Test
problems (Appendix B), and cases of numerical ill-conditioning and un-
physical results, are described. Note that, while most of the test
problems consider layers all having identical values of w, g, and AT,
the formulation and computer code are in no way restricted to such cases.
4.1 ELIMINATION OF w=l BRANCH
For layers with w*===l, the solution has an entirely different
functional form, Eq. (4), than it does when o<W*<1, Eq. (3). The entire
presentation of Section 3 is based on the Eq. (3) solution. This is
because we discovered that it is feasible to replace w=1, in any layer
in which it occurs, by w=l-£, where E is a suitably small number. This
eliminates branching and makes for a simpler faster code.
Clearly this procedure introduces error, in particular spurious
absorption. The only question is, is it tolerable? Test Problems 1
and 2, Appendix B, address this point; they are for non-absorbing homo-
geneous layers of optical depth 1000 and 100, respectively. [Layers of
lesser optical depth than these cases will obviously incur correspond-
-13ingly smaller errors.] £ is taken to be 10 , or roughly 10 times our
CDC 7600 precision. In Test Problem 1, with a zero surface albedo, com-
puted fluxes agree to all the five figures edited with the correct w=l
solution from Eq. (4). Net flux is constant to at least five significant
figures. In Test Problem 2, with a surface albedo of 100%, net flux is
practically zero at all levels, as it should be; it fluctuates randomly
17
at a level corresponding to imprecision in the LEQT1B solution (Sec-
tion 4.3) and/or round-off error. Test Problem 2 also furnishes other
tests of the code, namely: (a) the albedo at every level should be 100%,
which it is; and (b) a static situation, in which fluxes do not vary
from level to level, should develop in the interior, which it does.
Note that sometimes total downward flux can exceed incident flux, a
realistic physical effect which the model is able to reproduce in both
Test Problems 1 and 2 for the 0 = 1 case.
In all cases, the errors introduced by the w=l + w=l-E replacement
are considerably below what one normally incurs from using 6-Eddington
rather than a more exact radiative transfer method, as discussed in
Section 5.
To understand why using w=l- in Eqs. (3a-f) does not lead to
serious ill-conditioning or round-off problems, we must have a closer
look at the structure of the differential equations (la-b). If these
equations are written in matrix form,
y' = By + b
then it turns out that the eigenvalues of B are +X:
+ +By = +Xy
where X is defined in Eq. (3c) and y are the eigenvectors. The solution
(3a-b) becomes undefined precisely when these eigenvalues go to zero, but
note that the rate at which they go to zero as w+1 is only as /71--. Thus
the ill-conditioning of Eqs. (3a-b) as 0w+1 is strongly damped.
N.B. £ (code variable PREC) must be set properly for the user's com-puter, or the code WILL NOT WORK. E.g., one would take - 10- 7
for IBM single precision.
18
4.2 MULTIPLE DIRECTIONS OF INCIDENCE
Note that the coefficient matrix C, Eq. (17), depends on p0 only
when A(p0) does. If A is independent of p0, the incident direction p0
occurs only on the right-hand side, d, of the linear equations Cx = d.
Then, once C is inverted, little extra computation is required to get
solutions for many p0tS, over what would be required for just one. The
code takes advantage of this fact when A(P0) is independent of pO; it
pre-inverts C, then solves Cx = d in a loop over pO (i.e., over d) in
which the inversion is presupposed.
4.3 SOLVING THE LINEAR SYSTEM
The penta-diagonal linear system Cx = d [Eq. (16)] is solved by
the algorithm of Martin and Wilkinson (1967), as implemented in sub-
routine LEQT1B from the IMSL Library (1975). LEQT1B makes the standard
L-U (lower triangular, upper triangular) decomposition of a row-wise
permutation of C, arrived at by row equilibration and partial pivoting,
and/or solves Cx = d. The "or" feature is useful in the situation des-
cribed in Section 4.2 above. The "and" feature is used when A(p0) does
depend on pO, since then the last row of C is a function of pO and
pre-inversion of C before looping over P0 is not possible.
Note that LEQT1B allows all the right-hand sides d to be entered
at once, which would avoid calling LEQT1B repeatedly as we do in the
present code. But this entails adding a second dimension, for %O, to
several arrays, and we felt that the speed advantage was more than
cancelled by the extra storage requirements.
19
Note also that LEQT1B destroys the matrix C. Therefore the code
takes precautions to save C, in a separate array, before calling
LEQT1B. Other idiosyncrasies of LEQT1B are described in its opening
comment cards.
Comment cards in the code indicate how a second IMSL sub-
routine, LEQT2B, may be used in place of LEQT1B. LEQT2B performs
iterative improvement on the solution generated by LEQT1B. On our
computer (CDC 7600 with "14 significant digits) this improvement never
changed the first four significant digits of any flux for any test
problem. The fifth significant digit was sometimes changed by ±1 when
w was near or equal to unity. LEQT2B called LEQT1B anywhere from 2 to
4 times in test problems, with more iterations being required when
w=l than when w<l. Thus there was a rather severe computational
penalty for using LEQT2B (since calls to LEQT1B are a pacing item in
the code), with no useful increase in accuracy. We mention all this
only for the benefit of users with lower-precision computers than ours,
who may at least want to try LEQT2B to make sure that they don't need
it. If LEQT2B is used, the second dimension of the WORK array must
be increased from 3 to 10. [LEQT2B helped one user with a lower-
precision computer, in that, by taking its error exit, it showed him
that he had not properly set the variable PREC.]
LEQT2B and LEQT1B require the matrix C, Eq. (17), to be sub-
mitted in "band storage mode," which simply means that the diagonals
of C are submitted as successive columns:
20
0
0
C31
C53
CM-2. M-41
CM-1,M-3
M.M-2
0
C2 1
C32
C43
C5 4
CM-2,M-3
M-1 ,M-2
CM,M-1
C11
C22
C33
C44
C55
M-2,M-2
CM-1,M-1
CMM
C12
C23
C34
C45
C5 6
M-2 ,M-1
CM-1,M
0
C24
C46
0
CM-2,M
O0
0
The elements of all but the first and last rows of this matrix are
set in a DO-loop over row pairs, with the first row of each pair having
a leading zero, the second a trailing zero.
Vis a vis the error exits in LEQT1B and LEQT2B, we may note that
numerical (algorithmic) singularity simply means failure on account of
a small pivot element. The matrix C may or may not be mathematically
singular in such a case.
4.4 ILL-CONDITIONING
One advantage of using LEQT1B and LEQT2B is that they check for
ill-conditioning in the linear system Cx = d. That this is a wise safe-
guard, is shown by the fact that these routines have detected three
instances of ill-conditioning already. The first two were mentioned
in Section 3, namely (1) formulating continuity conditions, Eq. (12),
in terms of F and F rather than G and H, and (2) trying to solve
�b
21
for the actual unknown constants x rather than the exponentially-
scaled ones x.
The third instance of ill-conditioning arises from the simultan-
eous presence of exp[Xi(Ar)i ] and exp[-Xi(Ar)i] in the coefficient
matrix C of Eq. (16). If iX(AT)i > A0 (A0 ~ 14 for our computer) for
any i, these two exponentials differ by many orders of magnitude, and
C is regularly flagged as ill-conditioned. Such a layer of course has
a large absorption optical depth, and is "semi-infinite" in the sense
of Eq. (10).
Initially we dealt with such layers by making the upper boundary
of the uppermost one the new "surface," and zeroing fluxes at all lower
levels. This, however, raised the problem of specifying the albedo
A(po) of the new "surface." The obvious choice was A(p0 ) = a(p 0 )
[Eq. (9)]. But a ('0 ) depends on the mix of diffuse and direct radia-
tion striking the new "surface." That mix in turn depends on ao(P0).
Thus we were forced to determine A(p0) by an iterative procedure.
While this iteration converged rapidly in several test problems, it
necessitated several solutions of the n-layer problem (albeit with a
reduced number of layers), and furthermore sometimes converged to
negative values of A(p0), for the reasons put forward in Section 2.
Therefore we abandoned the iterative procedure in favor of one
which is both simpler and faster; namely, subdividing each layer with
Xi(AT)i > A 0 into K equal sublayers, such that
(AT)i
i o *
22
This subdivision is invisible to the user in the computer code of
Appendix A; fluxes are returned only at the originally-specified
levels. The only danger is that the added internal layers may cause
array dimensions to become inadequate; if this happens, an internal
check in the code stops execution.
Test Problem 3 shows the effectiveness and stability of the sub-
dividing scheme in a difficult case in which there are 10 layers, each
of optical depth 100 and single-scattering albedo 0.8. The scheme must
divide each layer into 4 sublayers. While all the fluxes below level 1
are practically zero, the important point is that they are calculated
in an entirely well-conditioned way.
N.B. The variable A0 (CUTPT in the code) is machine-dependent.
Computers of lower-precision than ours will require A0 to be propor-
tionately smaller than the value, 14, which we use.
4.5 SPURIOUS AND NEGATIVE FLUXES IN HIGHLY ABSORBING CASES
Consider, in the one-layer 6-Eddington formulas (3), a purely
absorbing w*=w=O case. Then, since there is no scattering, there
should be no diffuse down-flux if there is none incident (F0 0).
But we can only have F 0 in Eq. (5b) when cl = c2 - 0 in Eqs, (3a-b).
Since the 6-Eddington approximation will always predict non-zero values
for cl and c2 when surface albedo A # 0, it is clear that a spurious
diffuse downward flux F+ 0 is predicted when w = 0 and A # 0.
Furthermore, F is negative, and negative values of F can be found
all the way up to 0w0.5 in certain special cases. Because negative
fluxes are unphysical, and because their occurrence tends to shake
23
faith in the whole 6-Eddington approximation, we explore them here in
some depth.
+If we solve for F2 , Eq. (14c), for an w=0 layer of optical
depth AT, we arrive at
+ / 2 2F2 = ( - -)x + ( +-)x
= 43i F0 (F0 + AO Se )sinh(/3 AT) (18)
where
D = (A+7)sinh(/3 AT) + 4/3 cosh(/3 AT)
Now D > 0, so F < 0 when
0 < 0(F + Ap0Se (19)(F<± 0 , (19)
+ +4- -AT/1 0Thus F < 0 when F =0, and F > 0 when F ASe = 0 (in2 2
particular if F = 0 and only diffuse flux is incident on the layer).
Other cases depend on inequality (19), Loosely speaking, the greater
the proportion of diffuse flux incident on top of a layer, the less
likely that F2 < 0. Note that the right-hand side of (19) grows
exponentially with AT if p0 > l//3 or F0 #0 and decays exponentially
with AT if p < 1//3 and F0 = 0. Thus (19) is much less likely to be
satisfied in the latter case; furthermore the p0 = 0.2 < 1//3 and
P0 = 1.0 > 1//3 cases in the test problems are sharply distinguished.
24
+When F0 = 0, which is true for all our test problems, it follows
from Eq. (18) that
+ -AT/p0F 0F2 + loSe > 0
so that total downward flux is always positive - only its diffuse part
may become negative. [This seems to be true for all single-scattering
albedos, not just w=0.]
Test Problem 4 shows negative values of F predicted for a 10-
layer case in which each layer has w=0 and AT=0.1. The surface albedo
is 100%. This problem has the largest (in magnitude) negative F we
have ever seen, and it is only -0.02, or 2% of the incident flux S=l.
F always becomes less negative as either A falls or W rises. For
example, Test Problem 5 is like Test Problem 4 except w* has been
raised to 0.2, and this clearly and sharply reduces instances of nega-
tive F.
Test Problem 6 shows nearly the largest value of w* (0.4) at which
we have ever observed negative F . Note that this is an optically thin
case. For optically thicker cases, e.g., Test Problem 5, negativity
disappears at considerably smaller values of w*. However, this nega-
tivity threshhold in w* rises as the asymmetry factor rises, so for g*
exceeding the value 0.85 used in our test problems, it would be larger.
If the layers in Test Problems 4-6 are surmounted by a "flux-
diffusing" layer, i.e., a layer which significantly scatters the inci-
dent beam, then negative fluxes are reduced in extent and magnitude, or
even eliminated, depending on the diffusing power of that layer. In
25
general, our argument following Eq. (19), to the effect that increasing
incident diffuse flux mitigates the negative F problem, is borne out
for all single-scattering albedos, not just L0=0.
In mitigating negative down-fluxes, however, a diffusing layer may
force negative up-fluxes to appear, as in Problem 7. The reasons for
this have been explained in Section 2 (while Section 2 dealt with semi-
infinite layers, in practice fairly small optical depths may trigger the
negativity effects described there). Problem 7 considers a non-absorbing
layer of optical depth 0.1 surmounting 10 purely-absorbing layers, each
of optical depth one. Most of the up-fluxes in Problem 7 are negative,
and one may observe a particularly bizarre case (for P0 = 1) where nega-
tive up- and down-fluxes occur together -- although never at the same
level. Such cases are very rare, since direct radiation causes one
type of negativity, diffuse radiation the other, and only a very narrow
range of the diffuse-direct mix can excite the two types of negativity
simultaneously.
Test Problems 8 and 9 are aimed at showing negative up-fluxes,
and how they can persist up to values of 0.6 or so in single-scattering
albedo. We have 10 layers, each of unit optical depth, and a surface
albedo of zero. The single-scattering albedo is 0.2 in Problem 8,
0.6 in Problem 9. Negative up-fluxes exist at both incident direc-
tions in Problem 8, but only for the more grazing direction in
Problem 9.
No negative fluxes, either up or down, have ever been observed for
> 2 0.8. Thus they will never be seen in a goodly number of cases of
26
practical interest, including clouds and most aerosols in the solar
spectrum.
Negative fluxes represent only a particularly glaring and unplea-
sant aspect of error in the delta-Eddington approximation. Positive
fluxes may also be in error, obviously. One's primary concern should
really be with the size of the error; and on this score, the news is
good. In the most pernicious negative flux cases we have been able to
manufacture, error has never exceeded about 5% of the incident flux S.
Typical errors are much, much smaller. The typical situation is that
the correct flux is positive and very small, while the corresponding
delta-Eddington negative flux is also very small in magnitude. Many
examples of this are furnished by Test Problems 4-9. Indeed, most of
the negative fluxes which one normally encounters are so small that
they are completely and utterly negligible.
A warning: do not yield to the temptation to set negative fluxes
to zero. The price is loss of guaranteed flux conservation and heating
rate positivity. Most fluid dynamic models want fluxes only at the
top and bottom, where negativity is least likely to arise. They want
only heating rates in intermediate layers, and since heating rates
are delta-Eddington's forte, it would be the height of foolishness to
meddle with them by enforcing flux positivity. Indeed, one may turn
the tables and regard the negative fluxes as necessary in order to
attain the more important goal of positive (and accurate) heating
rates.
27
4.6 FAILURE TESTING
We took to heart the maxim of Johnson (1976) that no code should
be accepted as satisfactory until it has been stressed, pushed to the
breaking point if possible -- a procedure he calls "failure testing."
That is how we uncovered some of the quirks of the delta-Eddington
method which were discussed above.
Two other sets of "failure tests" deserve mention. The first set
occurred naturally in the course of doing 80-spectral-interval, cloudy-
atmosphere solar radiation calculations [Wiscombe (1976)]. Here, with-
out ever failing or giving bad results, the delta-Eddington code of
Appendix A handled a wide variety of optical depths (including numerous
thin-thick juxtapositions), asymmetry factors, and solar zenith angles.
Single-scattering albedos were, however, mostly contained within the
range 0.9 to 1.0, except for clear regions of pure water-vapor absorp-
tion for which c=0.
The second set of tests selected random numbers for optical depths
and/or single-scattering albedos, in a wide variety of ways, for a
40-layer atmosphere. Again, no failures could ever be provoked. Thus
one may have confidence that Appendix A furnishes quite a 'robust' code.
4.7 SPEEDING UP THE CODE
For applications in which computer time is critical, there are two
principal ways one may speed up the delta-Eddington code. First, use a
fast exponential, of lower precision, rather than the normal software
"EXP" routine. Second, adapt LEQT1B particularly to penta-diagonal
systems with one right-hand side, transmit its argument list through a
28
common block, and explicitly take account of the alternating zeroes in
the outermost diagonals.
29
5. ACCURACY
Joseph et al. (1976) discuss delta-Eddington errors for single
layers at some length. Errors for multi-layer cases are shown in
Wiscombe (1976) and in Lenoble (1976). Since the author performed the
comparisons for the Lenoble report, and since that report may not be
readily available to many readers, we have taken the liberty of repro-
ducing these comparisons in Appendix C. Details are given there. One
may perhaps summarize by noting that accuracy deteriorates somewhat at
interior levels, as compared to the top and bottom boundaries where
the errors shown by Joseph et al. (1976) and Wiscombe and Joseph (1977)
are more typical. Flux errors at interior levels may be as large as 5%
or so of the incident flux at the top boundary. Heating rates across
interior layers are generally somewhat more accurate than interior
fluxes.
Lenoble observes that, among the many approximate methods com-
pared, delta-Eddington and one other are tied for the best overall
performance. Thus, in the cases where one finds delta-Eddington to
be not terribly accurate, one may rest assured that other methods of
comparable simplicity are probably worse.
30
REFERENCES
Bohren, C. F. and B. R. Barkstrom, 1974: Theory of the optical pro-
perties of snow. J. Geophys. Res. 79, 4527-4535.
International Mathematical and Statistical Libraries, Inc., 1975:
Library 3, Edition 5 Reference Manual (IMSL LIB3-0005). Avail-
able from Sixth Floor, GNB Bldg., 7500 Bellaire Blvd., Houston,
Texas 77036.
Johnson, T. H., 1976: Failure testing: a proposal for increasing
confidence in the results of numerical simulations. J. Comp.
Phys. 21, 245-250.
Joseph, J. H., W. J. Wiscombe and J. A. Weinman, 1976: The delta-
Eddington approximation for radiative flux transfer. J. Atmos.
Sci. 33, 2452-2459.
Lenoble, J., ed., 1976: Standard Procedures to Compute Atmospheric
Radiative Transfer in a Scattering Atmosphere: Numerical Com-
parisons of the Methods. Under auspices of Radiation Commission,
International Assoc. of Meteor. and Atmos. Phys. (I.U.G.G.),
available from Universite des Sciences et Techniques de Lille,
U.E.R. de Physique Fondamentale, Laboratoire d'Optique Atmos-
pherique, B.P. 36-59650, Villeneuve d'Ascq, France.
Martin, R. S. and J. H. Wilkinson, 1967: Solution of symmetric and
unsymmetric band equations and the calculation of eigenvectors
of band matrices. Numer. Math. 9, 279-301.
Wiscombe, W. J., 1976: Radiative fluxes in a cloudy atmosphere using
the delta-Eddington approximation. Abstract Volume, Symposium on
Radiation in the Atmosphere, Garmisch-Partenkirchen, West Germany.
Wiscombe, W. J. and J. H. Joseph, 1977: The range of validity of the
Eddington approximation. Icarus (accepted for publication).
31
APPENDIX A:
LISTING OF COMPUTER CODE FOR THE
MULTI-LAYER DELTA-EDDINGTON APPROXIMATION
32
MAIN PROGRAM (DRIVER)
Set input variables
(common block IN)
Call subroutine DELTED
Edit computed fluxes
(common block OUT)
33
PRoGRAt DPIVFRCC CALL DELTA-EDnINGTON POUTINE AND EDIT COMPUTED FLUXES THEREFROMC
REAL MLUIUNLOGICAL MUDEPTCOMMON/TN/OTAU(5n),*OM cO),,G(S0), MUSUN(20n) ALB(20) SOLFLX*FDINC,
* FFUINCMlJDEPTq7Z,N71M ,NSOLNLFVCOMMON/OUIT/D T R (n ,O) ,FLXU (50 ;on ,FLXD (,n ,20)
C LEVEL DIMENSIONDATA NLFV/50/
C SUN ANGLE DIMFNSION (TN ARRAYS MUSUN, IR*FLXDFLXU, FLXSUNALB )DATA NSUN/0O/DATA NZ/11./DATA G/lnO0.RS/DATA NSOL/?/, MUSlJ/n .,91 .O/DATA MUDFPT/.FALqE./. ALR/O./DATA SOLFLX/../, FOINC/ 0 ,/. FUTNC/O,/
CNZMI = N7-1IF'(NZ.GT.NLPV .OP. NqOL.GT.N;SU\l) STOP OTMERR
CDO 5n00 NT = 1 ,UTA = l0.*(NT-3'Dn 5 I = 1,N7M1
5 DTAU(I) = OTADO 500C NO 1,DO 10 I = 1,NZl1
10 OM(I) , .^.NOUO 50o00 t,14 1,2IFO(!TA.rtT.11, .AhlD. eM(1).LT.0.7Q) GO TO 5000IF(NA, EQ.1 ) aLB(i') = O.IF(NA.EO.?) ALR(1) = 1.0
C WpITE OUT INPl!T VART!ALESwRITE (f.lc()0 N7.SOLFLXgFDINCFltNCNSOL, (MUSUN(NP),NP=1,NSOI.)IF (MIJnDT) .MALB - ISOL1F .NOi.MLDEPT) !Ale = 1W R I TE (6 1 0 ) ( A R I) T =1 NA LB)WRI rF (6 100?) (T .DTAU (I) OM ) ,G I ), I=1 ,NZM1
CCALL OELTFD
CC FDIT qFrSLLTSC
UO 600 \IP = 1,NSnLWRTT F( 6,?nOO) MUI.JI (NP)D) 60 I = 1 .N7
C rDOW!FL TOTAL DOWN-FLU.IX AT LEVEL -I- (DIR+FLXD)C FLJX NFT FLUJX AT LEVEL -I- (DTP+FLXD-FLXU)
DOwNF. n DTR ¢I .NP) FI.XO (I NP)r-LI.X = DOWNFLXFLYUI INP)
600 WRTTF (f,001) IDIR(ItNP)LXDNPFLXD,NP)DOWNFLFLXU(INP)FLUX5000 CONTINUE
C
34
CALL EXTT1000 FORMAT (Il, 55(1Ht ) / ?X, *RADIATIVE FLUXFS FOR VERTICALLY. ,
*INHOMOGENEOUq*/2X,,*LAYEREn ATMOSPHERE BY DELTA-EDDINGTON ,** *APFPOXIMATIONs / 5( 1H*) //* NUMBER OF LEVELS-- NZ r*I4/* * INCIDENT DIPFCTr-EAM FLUX"- SOLFLX*E1 2.4/* ·* INCTOFNIT DIFFUSF DOWN-FLUX-. FDINr=*E2l.4/* t INCIDENT DIF'USE UP-FLUX AT ROTTOM ROUNDARY-- FUINC=*E12.4/
** * NUMBER OF DTrFCT-BEAM ZENITw ANGLES-- NsOL=*I4/* * COSINES OF rIPEC.T-REA4 ZENITH ANGLFS-- MUSUN=*(10F,.4))
1001 FORMAT(* S$UJFACE AI.BFDO(S)-- ALR=* (OFR.4))1002 FORMAT( / * LAYEQ,.4X,*OP DEPTH.4X,*SIlNGSCAT ALB#,4X ,ASYM FAC*/
* (I5 ,F13.4,F17.4.F2. 4 ))?ooo FORMAT( // * MUSUIN =*F8.4/13X,* nTRECT*,cXt*DIFFUSE*97X,*TitOTAL
.SX, *TFFULSE*/* LEVEL*, 8x ,i *Fl X* ,5X O X WN-FLUX4, 4X DOWN-FLUX** 6X. LJUP-FLLJUX*,5X. *i.IET FLUX*)
o001 FOpMAT(It,SE13..END
35
SUBROUTINE DELTED
X AT > A0
Subdivide layer into N sublayers,each of optical depth AT/N, where
X AT/N < A0 , and set X, P,etc. for each sublayer
YES
STOP
RETURN
36
SURROUTINE DELTFr)
CALCULATF UP- ANn nOWN-FLUXES OF RADIATTnN IN A VERTICALLY INHOMO-(E.NEOUS ATMOSPHEPE USING THE DELTA,-EDDINGTON APPROXIMATION
AUTHOr-- WJ1. WTICOMBENATIONAL CENTER FOR ATMOSPHERIC RESEARCHp.O. ROX 300nROULDER, COLORADO 80303
INPUT VARIARLESNZ NUMRER OF LFVF'LS (LEVEL I S THE TO OF THE ATMOSPHERE,
LEVEL NZ IS THE SURFACE)DTAU(I), I=] ,...,N7-1 OPTICAL rEPTH OF LAYER BETWEEN LEVELS I AND 1+1OM(I), I=1,....NZ-1 = SINGLE-SCATTERING ALBEDO OF LAYER BETWEEN
LEVELS I AND 1+1G(I), I_- , ... NZ, 1 - ASYMMETRY FACTOR FOR LAYER BETWEEN. LEVELS I AND 1+1NSOL - NUMREQ OF INCTDENT-BEAM 7?NITH ANGLESMUSUN(I)*I=1 .... NSOL = COSINE(S) OF INCIDENT-BEAM ZENITH ANGLE(S)ALB (I) I=i .... KNSOL = SURFACE AiBEDOMUDEPT = TRUE9 ALB(I) CORRESPONr)S TO MUSIIN(I). FALSE, ALB(1) IS
USED FOR ALL VALUES OF MUSUN(I',SOLFLX = INCIDENT-REAM FLUX (NO.MAL TO REAM) AT LEVEL 1
THE BEAMl) AT THE TOP OF THE ATMOSPHEREFDINC = TNCIDENT DTFFUSE DOVWNr-FLX AT LFVEL 1.FUINC = TNCIDENT DTFFUSE UP-FLUlx AT LEVEL NZNLEV = LFVEL DIMFNSION (OF ARRAYS DTAU9 FTC,)
OUTPUT VARIARLESDTR (T ,NP)
FLXD (T ,NP)
FLXU (T ,NP)
INTERNIALCODE VAPTARLF
GPOMPDTAU lP
PP(I)LMDTAUEX(I)EXSUN(T)AL'PH I)BETA (I)TX(I)Ty(I)TZ(I)
(IN SAME LUNITS AS SOLFLX, FDINC, AND FUINC)DTPFCT FLUX AT LEF\EL -I- FOR SUN ANGLE -NP-(NOTE--IN THE DELTA-EDDINGTON APPROXN, BECAUSF OF THETQUNICATION OF THE FORWARD SCATTERING PEAK. THISQIl.AMTTTY INCLUDES SCATTERED RADIATION TRAVELLING INVFPY NEARLY THE SAME DIRFCTION AS THE ACTIJAL DIRECTFLU X. E.G., IT INICLUDES THE AUREOLE AROUND THE SUN.)rDFFUSE DOWN-FLIXJ AT LEVFL -I- FOR SUN ANGLE -NP.(NOTE--THIS$ WILL nE LESS THAN THE ACTUAL DIFFUSEDOwN-FLUX BY THF SAME AMOUNT THAT THE DIRECT FLUX-DTR- IS AUGMENTFD.)
DTFFUSE UP-FLUX AT LEVEL -I- FOR SUN ANGLE -NP-
DESCRIPTION (OR NAME IN WRITE-UP)G-PPI:ME (TRANSFORMED ASYMMETRY PARAMETER)ONMFA-PP,,ME (TRANSFORMED SINGLE SCATTERINGDFLTA-TAU-PRIME (TRANSFORMED LAYER OPTICALCUlMuLATIVF OPTICAl DEPTH FROM TOP (I=1) TOLAMBRDA-SUR-IP-S!IB-ILM (! ) *OTA(.UPEXP (LMrTAU)FyP t-TAJ (I)/MULIUN NP) )A L PHA SU B-IR TA -SUR-O,.7 *SOLL.LX*OMP f 1'. ,GP *( .-OMP))On. . SnLFL XOMP3.*GSP ,(1 .-OMP)
ALREDO)DEPTH)LEVEL I
c =CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCccCCCC
C
37
(TX,TYT7 APEISAV
FLXSUN (NP)PREC
CUTPT
NSU9
N L2SS(NL? T)C$ (NL~ ,S )CC (NL? ,;)WORK(NL?,3)X (i L2)
-- NOTE-.
MFRFLY INTERMEDIATE QUANTTTIES FOR COMPUTING ALPH, BETA)ARRAY OF LEVEL INDICES. FLUXESARE CALCULATED ONLYAT THFSE LEVELS.INCIDDFNT FLUX MUSUN(NP)*SOLFLX AT LEVEL 1A NU!MBER SOMEWHAT LARGER THAN THE COMPUTER PRFCISION,SIURTRACTED FROM ANY SINGLE-SCATTERING ALBEDOS WHICHAPE EXACTLY EQI'JAL TO ONE.ANY LAYER FOR WHICH LMDTAIJ.GT.CUTPT IS SUBDIVTDEDINTO EQIUAL SUBLAYFRS, ALL OF WHICH HAVE LMDTAU.LT.CUTPTNiIMREq OF SUBLAYEpS INTO WHICH AN OFFENDING LAYERIS DIVIDFD (THE WHOLE PROCESS BEING TRANSPARENTTn THE USER)2*NLEV-? (INPUT Tn RANDED MATRIX SUBROUTINE)THF PENTA-DIAGONlAI. MATRIX C, IN BAND STORAGE MODESAMF AS -SS- ARRav. USED TO SUBMIT .SS- TO LFQT1B.
. TEMPORARY STOPAGE ARRAY USED RY SUBROUTINE LEQT1BTHE VECTOR (X-HAT). ALSO TEMPORARILY STORESPH.S, ) IN LINEAR SYSTEM C*(X-HAT) = D,
THTS COnE IS NOT PERFECTLY OPTTMIZED, EITHER IN TERMS OFCOiF STOPRGF OR EXECUTTnN SPEEn, BUT IT SHOULD RE NnTED THAT,TN GENEQAL9 THE LIONS SHARE OF COMPUTING TIME Is OCCUPIED INTHE EXPONENTIALS AND THF PENTA-nIAGONAL SOLUTION ROUTINES,SO FLIMINATING A FEW OpERATIONS HERE OR THERE HAS ALMOSTNO IMPACT- ON RUNNING TTME.
REAL MlUSUNLOGICAL MIt.jEPTCOMMON/TN/DrTAU(S.) ,OM (50 ) ,,G (0 ),,MUSUN(2n' ,ALB (P0O) SOLFLX,FDINC,
* FUINC,MIIDEPT . kN7Ml NSOLNLF\ICOMMION/OtIT/DTR (nO ,n ) ,Fr'XJ (5 ;,n ) ,*FLXD (; ,20 )REAL LMCOMMON/TNTFPN/SS'f e9Q) ,CC(9g8,,) .wAORK (98,') ,X(98) ,TAU(50) ,LM(50)
PP (5o), X(O') , ExSUN (50), ALPH (50) RETA (O'), , TX (50, TV0)* TZ(Sn), .SAV(50), FLXSUN(?O)20 NL?REAL L MTAUJDATA Cl/p,6666666666667/
C 0 i*** t0************* n X COMPUTER PRPCIS ION *******************DATA PRFC/T.F-13/
C '4*"*'*"***'*** CUT-OFF POTNT FOR LM (I) *TAUP *s****##***#DATA CUTPT/14,/
CC SET ICIrDENT F, IJw AT TOP OF ATMOSPHERF
DO 5 NP = 1 ,SOl5 FLXSUN(NP) MUSUN(NP)*SOLFLX
CNL2 = 2*NLFV-2
CC SCALE OPTICAL DEPTH, SING-SCAT ai BEDO' AND ASYMMETRY FACTORC AMND CALCULATE VARTOUS FCNS OF THESF VARIaBLES
TAu(1) I 0.DO 10 I = 1.NZMIFF = G (T)**GP = G(T)/( .,+G(T))SCALF = I .- FFOOM (I )
CCCCCCCCCCCCCCCCCCCCCCCCCC
38
UMP (1 .- FF) OM(I )/SCALEIF (OM(I). F0Q .10) OMP = .,-PRECTI = 1-OMP#GPLM(I) - SQRT(3.*(1.-OMP)*T.).PP I) = C *LM (I) T1.
T1 = GP*(1.-OMP)TX(I) = 0,75 SOLFLX ,nMP (1.+Tl)TY(I;) = n .54*SOLFLX* OMPTZ(I) = ..tT1ISAV(I) TUTAUP = SCALF*DTAU(I)LMDTAU = LM(T)*r)TAUP
C TEST FOR A LAyER WHICH IS SO HIGHLY ARSORBING THAT IT WOULDC CAUSE ILL-CON ITInNING IN THE PENTA-DTAGONAL.MATRIX. I F ONEC IS FOI!ND, SUBDIVIDnE IT APPROPRIATELY.
IF('LMDTAlUGT.CUTPT) GO TO 15EX(I) = FXP(LMDTAUtJTAU(I+1) TAU(.lT+rTTA.UP
10 CONTINUEISAV(NZ) - MZ
CC NORMAL CALCUt'ATIONC
IF NZ EQ.2) CALL ONELAY1F(N7.GT.2) CALL NLAYDE(NZ)GO TO 500
CC SIDESTEP POTENTIAL ILL-CONDITIONING BY SUBDIVIDING OFFENDINGC LAYER, ANn ANy nTHERS LIKE IT.C
15 LAYERS = N7M1KTR = I
C20 NSIJB = LMDTA/CIJTPT 4 1.
DTAUP = DTAJP/FLOAT (kS!.J)BEX(I) = ExP(LM (InTA UP)TAu(I+I) = TAU(I) I nTAUPIP1 = I+1IUP = I4NSUPIUPM1 = IUP-1DO 30 I IP] ,IpM1LM(II) I M(I)PP(II) = PP(T)TX(II) . TX(T)TY(II) = TY(T)TZTII) = T7(I)E.X II) = FX (I)
30 'AUL(II+ ) = TAU(TJ) + OTAUPKTR = KTR+¥ISAV(KTP) = ISAV(KTR-1)+NSUBLAYERS = LAYFRS+NS!'B-1IF(LAYERS.GTNLF\') STOP DIMERR
CIF(IUP.rT.LAYERSI (Sn TO 100DO 50 I IlP,LAYFPSFF = G(KTR)**2
39
GP = G(KTP)/(I!,+,(KTP))SCALE = 1.-FF*OM(KTR)OMp = (1 .- FF) *OM (KTR)/SCALE1F(OM(KT) .EO..?e,) OMP - 1.-PRErTi : i,-OMP*GPLM(I) = SORT(3. (1 -OMP) *T1 )PP(I) = Cr*LM(I)/T1T1 = GP*(l. -OMP)TX(I) = 0.75?SOLFLX*OMP( 1.+T!')TY(I) = O).;*SOLFLXOMP-TZ'(I) = 3S.T1DTAUP = SCALE*DTAU(KTR)LMOTAU - LM(T)*DtAIIP
TEST FOP A LAvE WHICH IS SOCAUSE ILL-CONnITIONING IN THFIS FOUND., SULJAIVIDE IT APPROP
IF (LMDTAlJ.GT.CUTPT') O TO 20EX(I) = FXP(LMDTAL.ITAU(I+Il = TAtJ(I'. OTAIJPKTp = KTP+]ISAV(KTP) = TIAV'KTR-1)+1
50 CONTINUE
HIGHLY ARSORBING THAT IT WOULDPENTA-DrAGONAL MATRIX. IF ONE
RIATELY.
]00 CALL NLAYDE(LAYEpSI1)
500 CONTINUERETURNEND
CCC
C
C
40
SUBROUTINE ONELAY
C DO ONE-LAYER CASF SEPARATELY, STNCF MULTT-LAYER FORMALISM BREAKSC uiOWN FOR ONE LAYFR (SYSTEM IS NO LONGER PENTA-DIAGONAL)rC
REAL U$USULOGICAL MIJDEPTCOMMON/IN/DTAU(5n,) ,OQ(50) G(50) ,MUSUN (2noALB (20 ) SOLFLX,FDINC,
* FUI NC, MUDFPT, K7.N7M1 *NSOLNLFVCOMMON/O!T/D IR (5 ,p? ) , LXU (5Q ,On ) ,FLXD (5n ,24) .2.. .REAL LMCOMMON/TNTERN/SS'(9Rq) CC(98.5) ,WORK(98,3) X(98) ,TAU(50) ,LM (.0 ,
* Pp(5n), Ex(xO', ES(N(50) ALPH(50), 9ETA (50 , TX(50), TY(50)TZ(5), ISAV(50), FLXSUN(20), NL2
CPP1 = 1.+PP(I)PM1 = 1-PP(1 )All = Pql1/EX (1)Al? = PP1,EX(1)IAL = 1DO 10 NP = 1,NSOI.RMUO s 1./MUSUN(NJP)T1 = RM.JL I** -LM ( 1 ) 4,PiF(T lEO.U.) STOP SI^JGIALPH( ) TX (1)/'1BETA (1) - TY (1 ) (MIJSUN (NP) 4TZ (1 )RMUO)/T1SOLEXP = FXP(-RMHIO*ITAU(J )IF(MUDEPT) IAL - NPA21 = PP1 - ALB (TAL) *PM1A2? = PM1 - ALB(TAL)*PPlDl = ALPH ) .BETA(1) *FrnINCU? = FUTNC + SOLFXP*(ALPH(l')-BETA(1) + ALB(IAL)*(FLXSUN(NP)
* -ALPH ()-BF¢TA(1)))X(]) = (D144A22-D?,I12)/ AlliA2?.AA?21A12)X(2) = (1-A41 l*X l) )/AlDIR(1,NP) = FLXSIJN NP)DI(P?,NP) = FLXSJN(NPD) *SOLEXPFLXD(1,NP) = FDINCFLXD(2,MP) = PM1,X(l) + PP1iX(?) - (ALPH(1)+BETA(1)) SOLFXPFLXU(1,t IP) = PP1/FX(1)* X(!) + Pm v*EX(1)*X(?2)-(ALPH(1)-BFTA(1))FLXU(2,NP) = PP1*X(1) + PMI)X(P) - (ALPH(1)-BETA(1))*SOLEXP
10 CONTINuFC
RETURNENn
41
SUBROUTINE NLAYDE
Set first row of coefficientmatrix C (Eq. 16) 1
Set middle rows of matrixC in loon over row pairs
dA/dVp + 0 dA/dpo = 0
_________ 00II~ ~ ~ a -
dA/dp0 0
Set last rowof matrix C
I> Save C
Solve[ Solve Cx=d for x (Eq. 13)
ICalculate fluxes (Eq. 14)
End loop over 1O
RETURN
r-I
42
SUBROUTTNE L,.AYr)F(TB')
MULTI-LAYER DELTA-EDDTNGTONIB = NUMBER OF LEVELS
THE TOP AND BOTTOM qOIJNDARY CONIITIONS PLUS THE FLUX CONTINIJITYCONrDITTONS AT EACH TNTLRIOR LEV-L FORM A PENTA-DIAGONAL SYSTEMnF 2?IR-? EQUATIONS FOR THE UNKNOWN CONSTANTS (p FOR EACH LAYER).THE COLIMNS OF THE -SS- ARRAY CONTAIN THE DIAGONALS OF THE COEFFI-CIENT hMATPIX. THE LOWERMOST DIAGONAL IN COLUMN 1, ETC, 'THIS IS THESO-CALLED 9AND sTnRAGE MODE REQIUIRED By IMSL ROUTINE LEQT2B),
REAL MULJUNLOGICAL MIOFPTCOMMON/TN/DTAU (n ) , OM (0 , G (c0 ) ,MUSUN (2n'\ ,ALB (20) ,SOLFLX ,FDINC.
4 FUINC,M.JDEPT, NZ N7Ml NSOL,NL.FVCOMMON/OLIT/D!R (n5o,0) ,FxU (50 ,n ,FLXD (; ,20 )HEAL LMCOMMON/TNTFRN/SS S ,)CC (98 .) ,IAORK (98, ,) ,X (9) .TAU (50) ,LM (0)* PP(50), EX(0;). ExSUN (S'), 4A. H(50), ETA (50), TX(50), TY(50),
* TZ(5n') , ISAV(50', FLXSU (?0), NL2
IBM1 =TP-1LAST ?2*TR-?SS(1,1) = SS(1*, ) = t .S (1,5)=n,SS(1,3) =(1,-PP(T'))/mX(1)SS(1,4) =(1 ,PP (1 )) X (1)LASTM? = LAST-PDO 100 j = ,LASTMP,1 = J/2IPl = I+1SS(J,1) = 0.SS(J,2) = 1.,SS(J,3) = .nSS (J,4) = - ].O/rX (1 1)SS(J,5) = - FX(TP1)5S(J+1,1) = PP(T)SS(J+l,?) = PP(I)SS(J+1 ,') = PP( IPi)/FX(IP )SS(J+1,4) = -PP(TPl) *FX(IP1)SS(J+,1 .) = n
100 CONTINUFSS(LAST,1) : SSS(LAST.4) = SS(LAST,_c) - o.IF (MUDEPT) GO TO 10SS'(LAST,2) 1.+PP(TRMl) - ALR(1)*(1.-PP(IBM1))SS(LAST3.) I= .- PP(TRM1) - AL (1) (1,*.PP (IBM1))
CALCULATE THE L-I DECOMPOSITION OF PENTA-DIAGONAL MATRIX -SS-
LEQT?R CALL FOR TFSTING PU.JPOSESCALL LE(T?R ( SSLAST r ,.2,NL2, X 1 IL2 1 ,WORK,NL?,WORK(1 8))
LEQT1R DESTROS; THE INPUT COFfF MATRIX, SO SINCE WE Ml.STPRESFRPVE -SS-, wE MUST LET IT DESTROY -CC- INSTEAD.
DO 120 TC = 1T ,UO 120 TR = 1 ,LAQT
CCCCCCCCCCC
C
CCCCCccc
43
120 CC(IRIC) = SS(!,TC)CALL LEOTIc (CCLAST,?,2,NL2 ,X,1,NL2, 1 .WORK)
CC FOR EACH SUN ANnLF, CALCULATE THE R.H.S, OF THE BANDED SYSTEM,C SOLVE, AND USE THE SOLUTION TO rONSTRUCT THE FLUXES AT EACH LEVELC
150 DO 400 NP - 1 NS.LRML!O = I1 /MUJUN(NP)T1 = RMLU0I.*-LM (l)**IF(Tl*EO.O.) STOP S.INGUALPH( ) r. TX(1)/T1bETA (1) = TY (I.) (MUSJN (NP) *TZ (1 ) RMUO )/T1X(1) S AL.H (] ) +nrTA (1 ) +DINC
CDO 2o0 J = ?,LASTMPf?I = J/2IPl = I+1T1 = RMI.Jn ? -LM ( TP1 ) *ALPH (IP) TX (IPI)/T1HETA (IPI) , TY(ITP1)*(IMUSUN(NP)*T7(IP1)+RMUO)/T1EXSUN(IPI) = EXPf-PMIJO*TAU(IP ) )X(J) = ALPH(I ) -ALPHIP1 ) * EXq;JN(IPI)X(J+i) = (9ETA(I)-RPETA(IP1)) * vySUIN(IPl
200 CONTTNUFLXsUN(IR) FXP(-.PMUn*TAU(IR))
CIF(MLJOEPT) (O TO ?c()ALPOO = ALR(1)IJOB =GO TO 27;
250 ALPDO = ALR(NP)IJOB = nSS (LAST ,) = 1,+pp(IRPM) - ALBI(!P)(I].-PP(IBM1))5S(LAST,3) = 1.-PP(IlMI) - ALB (!P)*(1i+PP (IBM1) )DO 260 T(C = 1 ,DO 260 TP = 1 ,LAT
260 CC(IRPIC) = SS(IP,TC)C
275 X(LAST) = (ALPH(TRM1)-qETA(IBM]) + ALBDO*(FLySUN(NP)·* -ALPH(I-M1 )-BFKTA (IBM1) ) )*EXSIN\ (IR) + PUINC
CC SOLVE PFNTA-DIAGnNAL SYSTEM JTTH RH.c. -X-. SOLN GOES INTO _X-CC CALL LEOT?R(SS,LASTg,,2,NL2,XlNL2,IJOP.WORKNL,.WORK(1,8))
CALL LEOTI R (CCLAST ?,2,NL2 X, 1 .*lL2 IJOPWORK)C
DIP ( ]NP) = FLXS5lN NP)FLXD(1,qMP) - FDTlCFLXU( 1 KP) (1 .=PP ( 1) )/EX (1) X ( ) + (1 -PP (1) )EX () X (2)
-ALPw (1 ) +RFTA (1 )KTR = ?DO 3Z0 T = 1.IRMI1F(I+1 .NF. ISAVKTRF)) GO TO 3lnDIPR(KTRP.NP) = FLxYSIN NP)* EXSLUN (T* )FLXD(KTP.NP) = (T,-PD( t )*X (2*T-1) *(1.+PP(I) )*X (2T)
44
- (ALPH (I) +RF TA t I) ) *EXSUN(T+ 1 )FLXU (KTP.NP) = (1 .PP (I )) X (21'.I. 1)
*- - (ALPH (I ) -RETA FI) ) *EXSUN (T 4')KTR = KTR+1
350 CONITI-NUF400 CONTTNUF
. (1 .- PP (T.)) 4*X(2*I)
RETURNEND
C
45
SUBROUTINE L E-QTi8(A, NNLC,U ilC, IA,B,M, I8,IJOB, XL)C LFEIB 1C020C-LEQT i3 ------- -S - - -- -- LIBRARY -3 ----------------" LE 18 C CU30CC FUNCTION - MATRIX DECOMPOSITION, LINEAR EQUATION LE180C50C SOLUTI ON - SPACE ECONOM' rS' S-OLU TI ON '- LE-I8BG'C6CC 3AN3 STORAGE MODE LE180070C USAGE - CALL LEQTia (ANNLCNUC',IAB MIIJO3"XL, ta-0C IER) LE 18G09CC PARAMETERS "A ST ' "v - r PC (NUC+NLC+i) SEE PARAMETER IJOB. LEiB311CC N - ORDER OF MATRIX A'AN'J THE NU'M3ER OF'-ROWS IN" LEI'B. 20C 3. (INPUT) LEIB013OC NLC --'NU-.ER OF LO WER COOIAGONALS IN M'AT RIX A.". LEl9I .14C (INPUT) LE1B0150C ...... N UC .- NUMBER OF UPPER CO'D A'AsTO'N AL'S TN T I X .r"'A"' - . .'E' tB C i6'C (INPUT) LE 1B 170C : '' ' IA- - C -*A - POW-'-OIMENSION OF'A AS SPFCIFI'ED IN THE L.E' i ' BO'ia cC CALLING PROGRAM. (INPUT) LE1BC1i3C 9 - 'INOUT/OUTPUT/O T ATRIXOF O IM!SNSION N BY M. LEiB82ZCC ON INPUT, 9 CONTAINS THE M RIGHT-HANO SIOES LE13021CC .. OF T-E EQUATION AX - B. ON OUTPUT, 'T . LEt'ri'22tC SOLUTION MATRIX X REPLACES 3. IF IJOB = 1 LE18 323CC 3 IS NOT USEfC. LEBR-324CC M - NUMBER OF RIGHT HANO SIDES (COLUMNS IN B). LE1t30250C (-INPU T) L E1i3C 26CC S1 - ROW DOIMEMSION OF B AS SPEC.FIEO IN THE L 18 270C . CALLING "-PROGRAM. (INPUT) 'LE1Bn;'28-'C IJO0 - IN:rtT OPTION PARAMETER. IJOB = i ITPLIES WHEN LE1F3C290C I = 3 FACTOR THE MATRIX A AND SOLVE THE LEiBT.C30C EQUATION AX = 3. ON INUT, A CONTAINS THE LE1B0310C COEFFICIENT MATRIX OF THE EQUATION AX := ,LE19Q32GC WH-ER A IS ASSUMED TO QE AN N 3Y N BAND LEB15033C ..... MATRIX. A IS STOR'ED I'N IAN37'STORAGE M.3OOF LE1BC3i40C ANO THERFORr E HAS DIIMENSIONN N Y LEIRG 350C (NLC+NUC+1). ON OUTPUT, A IS REPLACE LE1B036DC 3Y THE U MATRIX OF THE L-tU OECOMPOSITION LE180370C OF A ROWWISE PERMUTATION OF MA TRX A. iJ ISLE'I.J33CC STORED IN 9ANDt STORAGE MODE. LF16C39CC I = 1, FACTOR THE iMATRIX A., A CONTAINS "HE L' t1BC4+CC SAME INPUT/OtJiPUT INFOIMATION AS IF LE1BC41CC IJOB = 0. LF4Bi842tC I = 2, SOLVE THE EOflATION AX = B. THIS LE13043CC OPTION IMPLI'ES THAT L-EQTi' 1AS AS ALREADY LE13440C 3EEN CALLED USING IJO = O 1 SO THAT LE13045CC THE 'MATRIX A HAS ALREADTY f3EFN -FACTODfEn. L ' 13 4E6C 4C IN THIS CASE, OUTPUT MATRICES A AND XL LE13C:470C MUST HAVE BEEN SAVED FOR REUSE IN THE LE13 '48CC CALL TO LEQT19. LE 19'g90C XL - WORK A'.EA OF DIMENSION Ni(NtC+l) , THE FI.ST LE13BC5OgC NLC"1'4 LOCATIONS OF XL CONTAIN COMPCNONTS OF LE13051CC TH " L *IATRTX OF THF 'L-Ut OECOf m OSITION -- O A t1E103520C - OCWJISE PEERMUTATIO, OF A. T-. LAST N LE1i 530IS~~~~~~~~~~~~'H LAS T LE1._ 3
46
C -t -... - ^ ^^ATt0 ^ ^,pSp^^^^^ .O...f ,THZPTOT"' I ES. LEI^-.TC ,~C__________,___________ -_ _-_-,-.--------.-„~----~--,~-LE1B0632
C L ATEST REVISION " -- NOVF1R -27fl9r73 --- tf8E64C L E1 C 06 5 0
oDIENSION -^. - A A--IA-t ^XcTwT-, BTI3I .- ", ̂LE"' It-DATA ZERO/0,/,ONE/I O0/ LE180670
JBEG = N:LC+1 LE18690NLC1 =JBEG ..--- . - -
IF (IJOR .EQ* 2) GO TO 80 LE130710RN N ... .". .... . * ..................... ----- -
C R-ESTRUCTURE THE MATRIX LE180730C ; s ' F t . -. b .w .w . Fi.ss S *S_*_; r t+ RCtR0 I C:t tT4 '' Tr1C ABSOLUTE VALUE IN ROW I LEiB0750
I . . ... 1. ....... . . .... . . . _ 7
NC = J3EG+NUC LE B13 7 CNN NC - .,.JEND = NC LE1B8790IF' - ,1q', ... ';"OR; tc r TT r .E. *.Nt . .. .. ,'Eli, ....01 GTT0U'2TO5' F UD'I..u.S
5 K = 1 LE 19 0810P = ZERO .-- " - - -- - ---- -. -- -- .. .. - L'E '3 8 2 C..LO0 10 J = JFEGJENO LE1 0 830
A(I,K) = A(I,J) - - - - LE190'84CQ ABSS(A(I, K)) LE1885IF - n .GT'. P) =P Q . t L t 8 6 ......... ...K = K-+i LE 19i.8:
10 CONTINUE- - - LEItBO,80 'IF (P ,tQ. ZERO) GO TO 135 LE 13C89GXL(tI NLC1) " ONE/P ' ' L- 091?IF (K .GT, NC) GO TO 20 LF1B091000 15 J = K',NC .... .. . ... . ....... L ' 2' C
A(I,J) = ZERO LE IBC93
15 CONTINUE - LE' 1I3940L20 I =. I+1 L'E1C950
J8EG = J8EG-1 - - -I 1i99650IF (.IEND-JBEG .EQ N) JEN3 = JEND-1 LE109?7CIF. (I .LL . NLC) GO'TO TO 5 .- . ... ,.... .. l'J3EG = I LE 18990NN = JENO - LE IB100
25 JEND = N-NIJC :LtEIBO1DO 40 I = J8EG,N - LEE1S1020
P = ZERO LE1R1C3G90 3 J-:- " 1,.NN -- - U-.... -...... --- ... ... It.-04t
Q = A3S(A(I,J)) LE181050IF (Q .GT. P)P Q P -- I -. LE181060
30 CONTINUE LE181070IF (P .EQ. ZERO) GO -TO 135 LE1R1 - t1E9iO30XL(I,NLC1) = ONE/P LE1B1090IF (I ;,EQ, JENO)- GO TO 3? - -. - .. '' t.tEi81t0
IF (I ,LT. JEN3O) GO TO 40 LE31Q110K = NN+i1 LE1B12600 35 J = KNC LE1 1130
A(I,J) = ZERO LE1911435 CONTItNUE LEi 1115037 NN = NN-1 LE .116040 CONTINUE LE-- 1170
47
L = NLC -iL ...- U E ... ... .. ... ICOM... .POSITION......
L-U DECOMPOSITIONDO 7'5 K = i 'N
P = AS(A (K, 1)) XL (KNLC1)I KIF (L *LT. N) L = L+1
*" K i =. .X - . *** , .. ***,1"* - . - : , . .. - ..... ,,* ._... 1... *._.:,
IF (KI .GT. L) GO TO 50DO 4 5 J = - - - ........ . .. . ... .. ... .......... .. .... ... . . ....
Q = A3S(A(J-1v)).XL(JNLC1)IF (Q .LE. P) -GO T'O' 45P = q
45 CONTINUE50 XL(I, NLC) = -XLK, Nt'LCi)
XL(KNLC1) = ISINGULARITY FOUNO
IF (RN+P .EQ. RN) GO TO 135INTTR'CHANGE ROW' . .'AT
IF (K .EQ. I) GO TO 6000 55 J = i,NC
P = A(K,J)A(K,J) = A(I,J)A(I,PJ) = P
55 .CONTINUE60 IF (Kl .GT. L) GO TO 75
0O 70 I = K1,LP - A(I,1)/A(Kl)IK = I-KXL(K1,IK) = P30 65 "J = 2,NC
A(I ,J -1) A(1,J)-P*A<K,.J)65 CONTINUE
A(I,NC) = ZERO70 CONTINUE75 CONTINUE
IF (IJOB .EQ. i) GO TO 9035FORWARD SUBSTITUTI
30 L - NLC00 105 K = 1,N
I = XL(KNLC1)IF (I .EQ. K) GO TO 9000 85 J = 1"M
P = 3(K,J)3(K,J) = B(IJ)3(I,J) = P
85 CONTINUE90 IF (L ,LT. N) L = L+1
Kl = K+lIF (K1 .GT. L) GO TO 10300 100 r - K1,L
IK = I-KP = XL (K,IK)00 95 J = 1,M
t 3t I,J) -* (I)- tK J)95 CONTINUE
LEI1I319019, 1 i 9 0 2~O
LEiRI2ICLE 1 3 1 2220L E i I B 23LE131230
LE1 iR128C,L.EIM~2TC
LE13129CL F 18 1 29 C
LF181310'" : ' L E 13 13 ~20O
L El 4 i 3 1330LEI B1330
LE1P,35G
LE IB 13 70L - " LE1i3 33CLEIEi 1390L13:.140C
L-El 11~ 14 i C
LE I 1B430
L E R 1 4Qe jLE1~1t4i.Z
L E 1 8 1 4 9 CLE 1i3 145 5 C
LE 515,4%L1l 1 549CLE1t~~31%'
LE13 1510
L E i i s2 7 atE i t- 3 J.
LE IA 1610
LE I B 16 350
'LETiA3IS
L'~ EI 1600CLEVI3 i 64L8 162
L 18 1630
LE 1 71645L El18 4 7 30
C
C
C
. ... ..
0 p:
48
-TO" CONTINUWE L-* -, ^ „tE105 CONTINUE L 1B17 T
C L3AC K 'S U -B- --- S-STITUTO OW - LE7R176-UJBEG = NUC+N4C LE 1177030 - 125 J = i,-' --- -.-.-. -
L = 1 LEII3179.. ..... .. ........ N : .. ,..b.N ;," ..... -w...Y....Y-.. __-- ^ r - .- -....... R T.,.'.'_"'ri^_ ' B". .r.
DO 120 I = i,N LE131810
P -= (,KJ) LE £191i83IF (L ,EQ.- I)" GOQ- - -L- -I1I ... -00 110 KK = 2,L LE1 31850
. .. .... ........ : .. ....... .. . ..... .................^^ ..-, I ....-'_... r. e t :: *^e ._ t s S R t.KI~ * "I~ .............:-.' ' ................. .3 L 'P = P-A(KKK)' 8(IK-1iJ) LE 18187
l 11i0 " CONTINUE 'E - - E -- -115 3(K J) = P/A(KI) LE1B1890
* IF (L .LE. JBE 'G)" " .-L .+1 - - .. .. ....... 31 C ... .....120 CONTINUE LE1i1919i25 CO ONTINUE -*
90105 RETURN .. ................ ........
135 CO-NTINUEWRITE ( 6,1J 0)
10 00 FORMAT(//f LtEQTI19 EPPOR- -1A'TTX AL GORITH"M'tCALLY SINT-GtLAR')` ..... .. - -. -STOPENOD- - LET 19RO
49
SU3ROUTINE LEQT238(AN,NLC, NUCIAB M,I 3,IJO0 ,UIU, XL)C .. b0^ .- .. ... ... . . .... .... ... . ... .. . . ........... .. ..... ................ .............._- C * ....................................C-LEQT29 .---.--- S -. LIfBqARY 3 .---- ....-- "----....--------"""". LE28§00.33
C FUNCTION - MATRIX DECOMPOSITION, LINEAR EQUATION LE2BG005C '...' ' ..'''l''ULT ION .- CHIH A-CCU RA"CY S OLUT IN . - .L'E23'0 06C BAN) STORAGE MODE LOE 2B G 070C USAGE ' -CALL LEQT29(ANiNLCNUCIA,8 MBIJO U LE23008C XL, IER) L'2803090C ... PARAMETERS A ..T 4 rIC T TT FAI tC AX = 8, WHERE A IS ASSUMED TO BE AN N BY N LE280110C 3' ' A'NO 'M'ATRIX.- -A I'S ST ')OREO . ..IN 'B"A-ND S'TOR"'AGE .E2' ' ' P 02 tiC MODE ANO THEREFORE HAS OIMENSICN N BY LE 2R013C ' ('NLC + NUJC + I'. (I 9NPU"T ) ' 14C N - OROER OF A AND THE NUMBER OF ROWS IN 8. (INPUTLE2BC.150C ' NLC -. -NUMWE "'OF 'LO O'ER' 'C''TA'ONALS..TLN iATRIX.A.'' .'' ' ...C (I NP!JT) LE2 30170C NUC - NUMBER OF UPPER CODIAGONALS IN MATRIX -A."' LE"2Bi'8'8CC (INP')T) LE2B 190C IA -'ROW OIWENSION OF~ A AS SPECITFI.O IN THE CALLTNGLt2B'C2Q0C PRO ,RA -1. (INPU") LE2B"210C ' ' - INPUT/OUTPU'T' MATR'IX OF DIMENSION N 'Y .....................220C ON INPUT, 1 CONTAINS Tt E RIGHT HAND SInES LE280230C OF T'E EQUATIO! AX = a* OaN OUTPUT, TH;E LE28B0.24.'C SOLUTION MATRIX X REPLACES 3. IF IJOS 1 , LE2BC250C 3 I'S NOT USED-. L'E28260C M - NU1gER OF RIGHT HAND SIDES (COLUMNS IN L). LE2BC270C .. '.. ' .' ""(INPUT) 'L"' ' " '0 ' " ' '- ' '- ." '' ' ''- " LEZ 2280C IB - ROW DI31ENSION OF MATRIX A AS SPEFCIFI--O IN LE 239290C THE 'CALLING PROGRA., (INPUT) LE2B3i3'C IJ03 - INPUT OPTION PARAMETER. IJO. = I IMPLI-S WHEN LE2R 31GC I = 3, FACTOR THE' MATRIX A AND SOLVE THE LFE2B032-C EQUATION AX = 3. LE2B 330C I -' I, ..FACTOR THE- MATRIX A, " LET32334CC I = 2, SOLVE THE EQUATION AX RB. THIS LE2BC35ZC OPTTION IMPLIFS THAT MATRIX A HAS ALREAOY E' EZB3360C 3EEN FACTOREO 3Y LEQ,29 t USING LE23 9C37CC I39 = 0 OR 1. TN THIS CAS- THE OUTPUT LE:23.-38DC 1ATRICES U AND XL MUST HAVE B:EE'N SAVED LE280393C - FO:R ?E$S IN THE CALL TO ............... .TtEZB 40l 'C U - OUTPUT MATRIX OF DIMENSION N ,3Y (N(JC+NLC+3) LE233410C CONTAINING MATRIX U OF THE L-U OE-COMPOSITIONLE2:CN420C OF A ROWWISE PERUITATION OF MATRIX A. U IS LE2B3430C STORED IN 'AND STORAGE MODE OCCUPYING THE 'LE-2BC+44CC FIRST N'(NUC+NLC+I) LOCATIONS. THE R2:AI NItGLE2 L450C ?'2 N LOCATIONS'ARE USE'D AS WORKING S7TOr, AGE. L't-E2'3'460''C IU O- Ow9 OI"3ENSION OF MATRIX U AS SPECIRI'- IN THE LE2 C470C CALLING PROGRAM. (INPUT) LE 2848CC XL - WORK A EA OF DIMENSION NI'(NLC+l). THrE FIRST LRE23490C NL'CN LOCATIONS OF XL CONTAIN COMPOMN't4TS 0'F LF23L5OCCC THE L-U DECOMPOSITION OF A ROWNWISr LE 2C51CC 0F' IR1t1T'ATION OF A, TE. LAST N tOCATIOMS- LE35'C CONTAIN THE PIVOT INDICES. LE2Gu53C
50
C LATEST REVISION - OECEM ER 11,1973 LE28G660
DIMENSION A(IA,1) , U(IU, ) tXL(N ,1) ,B(I9,1i) LE2B068CDOUB'LE PRECISI'ON- '"SUKt ......... -...... L * - S^90DATA ZERO/. 0/,ITMAX/50/ LE28t70 0
NLCPI = NLC+1 LE2BC720NCC NUC+NLCPI -- T""-73""NLCP2 = NLC+2 LE2B074CNM NU C -- -= -N UC 75NU2 = NCC+I LE2B0760NU-3 NCC047' NU'3 ' N C C G"+? .. .. .. ."...4a...-.W r....-)..*< ..- .. .. ..lliG-.T. UO ....*l ....................... .... .......... *.......lm.i >. t~ 'E"Ut7'7a ...IF (IJOB .EQ. 2) GO TO 15 LE2 0780
C ......-... ^ -.. SAV "-^M'ATRI X- A' " .. .-^- 8 ..-- E' B 0£C79g'DO 10 I = 1iN LE2 8800
0 O'5 J 1 ,N'CC" -.
U(IJ) A(I,J) LE2B08205 C ONTI NUE - - -- --- .. -..-.-.... 3.....,,...,.... ,;
10 CONTINUE LE2B084CC - -- FACTOR M-lAT'RIX A -- - LE 2BO0 850
CALL LEQT1 (U, NNLC, NUC, IU, ,M, I,1, XL)IF (IJB ' .*EQ. 1) GO TO 9035
15 00 60 J'= 1, ' ".O0 20 I = 1 N
U(I,NU2) = B(I,J)20 CONTINUE
SAVE THE RIGH9T HAND SIDOS
- t JBTAIN A SOLUTIONCALL LEQT13(UNNLC NUC IU U( 1NU2) ,1, IU,2,XL)
^.........C...... -OMPU'TE rHE 'NOmM .R O F THF. SOLUT-ITOctnXN>ORM = ZEROD0 25 I 1,N
XNORM := AMAXi1IXN3RM, A :S(U(I,NU2) ) )25 CONTINUE
IF (XNORM .EQ. ZERO) GO TO 60-. .... COMPUTE-THE RESIOtJAtS -- .. - - -
O0 45 ITER = 1,ITMAXNC NCCKK ;= 100 35 I = 1,N ...
SUM = (IJ)L = NLCP2-I *IR = 'AXOIL,1)IF (L .LE. 0) KK =-KK+iK KKDO 30 JJ - IRNC
SUM = SUM -ORL (A(I,JJ) ) 3LE U(K, NJ2))= (+ t
30 CONTINUEU 3I,NU3) SUM -IF (I ,GE, NMNUC) NC = NC-1
35 CONJTINUECALL LEQTIB(UNNLCNUC,IU,U(1,NU3),1,IUt,2,XL)
DOXNORM = ZERO s.UPDATE THE SOLUTION
tE 2 q08-C.LE290890
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LE2B1iCO
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LE 28 9730
L 28. 1C40
LE23B1350LtE231S2
LE2B3 090.L£ 29 110 ·
LE 21 1105
L2831120LE2811590LE29 113i
LE 231170L E 2 8 114 C
t LE23 11'
LE 2B 123t_E23 121G
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51
0O 40 = 1,N - ....U(INU2) = U(INU2)+U(I,NU3)3XNORM AMAXi(DOXNO'RM, A S (U'(INU3)))
40 CONTINUEIF (XNORM+OXNORM *E '. X'NOR-} GO TO '50
45 CONTINUEG GO TO '9000
STORE THE SOLUTION50 DO 55 JK = iN
3(JKtJ) = U(JKNU2)55 CONTINUE60 CONTINUE
LE2B122CLE2012 30LE'2824CALE 28125CLE2 1 260LE291270
LE2B9 290.L£23'130i
L' 2B13i2LE2B 1320' 2. ,134C
, ....... .t 5 .. .1...,,^., ... t
9005 RETU RN
9000 WRITE(6,iOO0)1000 FORMAT(//* LEQT23 ERROR--ITERATIVE IMPROVEMENT FA'IL'EC TO CONV-ER- .
*GE SECAUSE BANDEO MATRIX IS TOO ILL-CONDITION£E,)' ST OP ........ .ENO LE2 13390
C
C
C
53
APPENDIX B:
TEST PROBLEMS 1-9
54
.* .* *. * 1' * ' * * , * * * , , , * ' , * , ** * * * 4 * 41,. * * * * * *, , * * * * .. * * 4*¢ 41 * * *
RADIATIVE FLUXES FOR VERTICALL Y INHOMOGENrOUSLAYERED ATMnSPHERE BY DELTA-En)DINGTON APPPOXIMATIONH
NUMRER OF LE'ELS-- NZ = 11INCIDENT DIRFCT-BEAM FLUX-- SoLFLX= 1.0000E+00INtCIDENT DIFFUSE DOWN-FLUX-- FDINC= 0,6INCIDENT DIFFUSE UP-FLIJX AT BOTTOM BOUNDARY-- FUINC= 0,0NUMRER OF D IRECT-BEAM 7ENITH ANGLES-- NSOL- 2COSINES OF DTRECT-BEAM ZENITH ANGLES-- MISUNs .?20o 1 .0000SURFACE ALBEDO(S)-- ALB= 0.OnOO
OP nEPTH100,00001 00 , 0000100,0000100.000010o0.0000
100.0000100.00n00.0000
100.()000100.0000
"S ING-SCAT ALB1.ooo00
1.6ooo1.'00ooi. Roo
1,nOOO
1.nOOO1. noOO1. o000
106000
TEST PROBLEM 1
ASYM FAC
.8500O .50 ()
* 8500
R.500
* 8500.R500.8500
MUSUN = n 00nIFECT
LEVEL FLUX.1 ,OOnnoE-0 12 1.1032F-613 6 ,ns. ;7 1224 3.3570-1825 1.851 8-426 no07 n.08 n09 0,00
10 n.o11 n.n
MUSIJN = l .ononI ECT
LEVEL F LJX1 1.OnnOOO+F2 R.*8733-133 7,8A?4r-254 6.99P?r-375 ^,2132?E- 4 96 5.551 A-617 4.8974F-738 4.34A1F-859 3.86n03r-97
1 0 3.4?73 0 911 3.n049-121
DIFFUSEDOWN-FLUx0.01 ,711E-O11 .0423E-019.1344EE-07.8458E-0?6.5573E-095.2687E-0,3.9802E-0O2.691 6E-0].4n31E-0o1,1454E-n0
DIFFUSEDOWN-FLUX0.01.l?^1E++01, 00?2E+On8.7830E-017.5441E-016,3051E-015. o61E-013.8271E-012.58R1E-011 3491E-011 .1. 13E-o
TOTALDOwN-iLIJX
2 O 0 0 F L n I2.OOOOE-Ol1 1711E- 11.0423E-019. 1344 -?7 8458K-n0?6 .5573E-n?5,2687E-0?
3 9R80E-n?2.6916E-n?1 4031e-0n1 14454E-n3
TOTALDOwN-FLJUX
1 ,0 0 EE+on1 12?6iE+nn1 002?E+on8 7830E-n17 5441E- 16.3o51F-nl5.0661E-o13 .8 676 1 E - Q 13.8271E-n02.5R81E-nl1 3491r-nl1.1 13E-o?
DTFFUSEJUP-FL JX
1.98A5F-nl1.1 597F- 1
1 .0308F-01
7.731 3F-n2A. 44?7Fr-i r5.1542F-n23.866Fr-022.5771F-n21 .2R8eF-n23,63n oF-i?
DTFFUsEUP-FLi X
9. 8899F-11,ll~]F+60q.9119F-nl8,6739F-ll
f. 194QF-nl4, 95s9;F-nl3.7170F-nlP,47tnOF-ml1 .23nF-nl2.91 n4F-1 1
NET Fl UX1 1454F-n31 .14 5 4F-031,1454Fn031.1454Fn31 1454F,031.1454r.n31.1454F-031.1454F-031.1454F-031.1454F-031.1454F-03
NET Fl UXlnl 3F1n
1.1013F-0~1 lollF-0?
1.1 13F-n21 .1013F-0?
1 1013F-02llnlF-n21 .lolF-n2
1 l~l3F-0?1 .l 3F-0?1.1013~-n2
LAYER1234
56789
In0
55
RADIATIVE FLUXES FOR VERTICALLY INHOMOGENEOUSLAYERED ATMOS:HERE BY DELTA-EDDINGTON APPROXI4ATION
NUMRER OF LEVELS--NZ - 1INCIDENT DIRFCT-BEAM FLUX-- SOLFLX= 1.0000E00INCIDENT DIFFUSE DOWN-FLUX-- FDINC= 0.0INCIDENT DIFFUSE UP-FLUX AT BOTTOM BOUNDARY-- FUINC- 0.0NUMBER OF DIYECT-BEA4 7ENYITH ANGcLES-- NSOL- 2COSINES OF DYRECT-BEAM ZENITH ANGLES-- MUSUN!J .2000 1 .0000SURFACE ALBEDO(S)-- ALB= 1.OnOO
OP nEOTH10,000010 000010 000010 000010.000010 000010.000010.000010.000010 .0000
SING-SCAT ALB
1.l ooo
1 .o000
1. 0001 n0001.io000oo
1 .o O O O 01 .6 o 0 o
TEST PROBLEM 2
ASYM FAC.8500,500.9500
.B500
.R500a9 500.9500.B5008500
.8500a 8500's F;0
MUSUN = *.2?nrnIECT
LEVEL FLUX1 .OOOnOE-n02 1.8845E-073 1,77?7F-134 1.6731E-195 1 .5765E-256 1.*4Q 4 E-317 1 .3996-378 1.31PQE- 4 39 1 .24?26-49
10 I. ,1709E-55]1 1,1ln02-61
MUsuN = l.nnoOIRECT
LEVEL FlUX1 i.nnoc+Oo02 2.?349E-023 3.8875E-034 ?9,4?23-045 1.5112r-056 9.4?P5r-077 5.8749E-088 3.66P9F-099 ?.?SlrE-10
1t0 1.4P40F-1111 R.87F 3 E-13
DIFFUSEDOWN-FLUY0.01.3000E-011.3000E-011 .300QE-011 .3000E-011.3000E-011 3000E-011 ,,3000E-01.1.3000E-011.3000E-011 ,3000E-O0
DIFFUSEr)ON-FLUy
0.01.17?1E+001 .2451E.OO1 . 2497E +0 n1.2497EOn1 25)0E+001,2530E+001.2500E*0o1.2500E+001 ,2 500E 01.2500E+001.2500E+00
TOTALDOWN-FLUX
2,0000E F-11 3000E-011 . 3 00E-n]1 3000E-O01 .3000E-o11 3000E-ol1 3000F-n11 3000E-011 ,3000F-n1 3000E-011 30oOo-01
TOTALDOWN-FLilX
1 0000E+n01 2344E+001 2490E+On1 2499E+nn1 2500E+001 2500E+001.2 500F+001 .200E+001.2500E+no1 .2500E+001.2500E+00
DTFFUqEUP-FL IX
?.OOOnF-nl1 .3000E-011,3ooo0F-11.3000F-011.3000F-011 .3000F-01
1 .3000F-nl1,.3000F-011,3000F-011.3onorFn1
DTFFUqE. P - FL ,')X
1 00onF+nO01 2344F+n01 24Q0F+iO
1 .2500F+001 2500F+00
1 2500F+00
1 250OF+001 .25noF+o00] ,2500=+00
NET Fl UX1 .55 4 'F-121.3829F-121 .2301F 1 21.075AF-129.2193F-137.691 AF-1 3.1462F-1 3
4.6185F-133.07 lF.1,31.5365F-1 30.0
NET FlIUX1 4559F' 1 71 33? -lPlT
1 .1823F-111 034Fr.1 18 .86n0F-127.3825F-125,9117F-124.4267F-122.9488F-121 .477F-1?
-7 ln54F-15
LAYER12345.67R9
10
56
RAOIATIVE FLUXES FOR VERTICALLY INHOMOGENEOUSLAYERED ATMnSOHERE BY DELTA-EnDINGTON APPROXIMATIONk
NUMRER OF LFVELS-- NZ = 11INCIDENT DIPFCT-BEAM FLUX-- SnLFLX = 1.OoE+n0INCIDENT DIFFUsE DOWN-FLUX-- F DINC 0.0INCIDENT DIFFUSE UP-FLUX AT BOTTOM BOUNDARY-- FUINC- 0.0NUMRER OF DIPECT-BEAM 7ENITH ANIGLES-- NSOLs ?COSINES OF DTRECT-BEAM ZENITH ANGLES-- MUS[jl= ,20n0 1 .0000SURFACE ALBEDOO(S)-- ALB= O.0o00
OP- nETH100.000010o .0000100.0000100.00001 00 0000100.0000100.0000100. 00010.,000010o.0000
SING-SCAT ALB,~000. 000,p000.R000
.Q000
.9000.TOO
.oOO
TEST PROBLEM 3
ASYM FAC.B500.0500.8500,8500
.8500
.ROO0
.0500,R00. 500.8500
MUSUN = .?*noOnIECT
LEVEL FLUX1 .OOnOE-012 4.6??7F-933 1 .0n5-1844 ?. 466.2765 0.06 0.07 0.08 n.09 0,0
10 0.011 0.0
MUSUN = 1.0nMIRECT
LEVFL FLUX1 1.OOnOc+002 4.7n07?-193 P.?21 E-374 1.0n431F-555 4.91n1E-746 ?.3113F-927 1 .o08-1108 5.117-1299 , 41 A9-14710 1.1349-16511 5.3493-184
DIFFUSEnOWN-FLUX
0 .07,1421E-216.660OE-4n6,2253E-595 .81 1E-7R5 4263E-975.0661-11l4,7298-13c;4.4158-15a4.12?7-1733.8410-19?
DI FFUSEnOWN-FLUJ0.04.4203E-1R2.4935E-3F1.21?3E-545.74>6E-7i2.7n66E-911 .2744-10O5.9992-12?2.8240-1461.3294-1646,2435-183
TOTALDOWN-FLUX
2 .0000 - 17.1421E-P16 6680E-406 2253E-595 8121E-7P5 .463E-975.0661-1 164 7298-1354.4158-1544 1227-1733 8410-192
TOTALDOWN-FLUX1 .OOOE+004 8910E- 82.715lE-31 3166E-546.2336E-732 9377E.Q11 3832-1096.5113-1p83 0651-1461 4428-1646.7777-1q3
DTFFUSEIJP-FL IX
q .981 4F-023.2531F-?23.0372F-41? 8356F-60
.6473F-79?.471.F-982.3075-1172.1544-1362.0113-1551.8778-1747.78Aq-208
DTFFUSEUP-FL'IX
5.0037F-n2?,4R09F-191 .24A6F--76.0417F-562.08614F-741.34A5F-926.3490-1l12,988-1?291.4069-1476 .62?9-1A63.3444-1 8
NET FL UX1 .601lF-nl6.8 168F-216.3641F-405,9418.F595.5474F-795.1791F-974.83s3_1 164,5143.1354.21471 543.9349-1733.8410n.Q2
NET Fi U)X9 49qrAF-l 14 666F-1 8
.59;4F.361 .256?F.545 9474F-732 ,80AF-911 .31971 096 .21-1?82 9244-1461 .3766-1646,7777-183
LAYER1I34
* A789
10
57
* **s * 41 44* 4*4141* 4* *i*tit ***t*i * ***4***t*** 4****4**414*** * ***
RADIATIVE FLUXES FOR VERTICALLY INHOMOGENEOUSLAYERED ATMnSPHERE BY DELTA-EnDINGTON APPROXIMATION
* * i*414*441 i* 4 * ****** *4i*4t* 4*4i*44 *4$1t 4144 ***41*4$* ****4t*41*
NUMBER OF LEVELS-- NZ s 11INCIDENT DIRFCT-BEAM FLUX-- SOLFLX= 1.0000E+00INCIDENT DIFFUSE DOWN-FLUX-- FDINC= 0,0INCIDENT DIFFUSE UP-FLUX AT BOTTOM BOUNDARY-- FUINCa 0.0NUMBER OF DTRECT-BEAM 7ENITH ANGLES-- NSOL= 2COSINES OF DTRECT-BEAM ZENITH ANGLES-- MUSUJN= .2000 1.0000SURFACE ALBEDO(S)--
LAYER OP nEPTH1 .1000
.10003 .1000oo4 .10005 . I o o O
5 .0oo06 .10007 .1000R .10009 .10001 ..lInOO
o.1 oo00
ALB= 1.OnOO
STNG-SCAT ALBo.00oo0.0000o.nooo0.6000O,.OO 0
OnOO0o.aoooO.OO00
0.0000
O.onOOO
TEST PROBLEM 4
ASYM FACF 8500
.8500
.8500
.SO0
.6500
*8500
,R5O0
.m500
MUSIJN = .?norIRECT
LEVEL FLUX1 .Onn0QE-012 1.2131E-013 7.3576E-024 4.46'6E-025 ?.7nOA^-026 1 6417E-027 9,9574E-038 6.035SE-039 3,6631E-03
10 ?.?18E-0311 1 .3476F-03
MUSUN = l.nnnonIRECT
LEVEL FLUX1 1.onnn oo002 9.n04A4-013 .R1873E-014 7.40?E-01l5 6,70n?-016 6.06q3E-017 S.4RR1E-018 4.9659E-019 4.4.Q3F-01
10 4.0A67P-0111 3.67R8E-01
DIFFUSEnOWN-FLUX
0.0-5.57?7E-0h-1.1313E-0O-1.7394E-O0-2.3997E-0-3.1323E-0s-3.9590E-05c-4.9048E-05-5.99R-1E-0-7.2719E-0;-8.7643E-05
DIFFUSEnOWN-FLUy0.0
-1.5213E-03-3.0884E-01-4.74q3E-03-6.551nE-03-8.5508E-01-1.08n8E-0,-1.3390E-0?-1.6374E-07-1 981E-0?-2.3926E-0?
TOTALDOWN-FLUX
2.000UE-011.2130E-nl7.3565E-n24,4609E-0?2.7043E-n?1.6386E-n?9.9178E-n35.9904E-033.6o31-n32.1491E-031 .2599F-03
TOTALDOWN-FLUX10000OE+009 .033?E-o18,1564E-017 3607E-ol6.6377E-n15 9798E-015.300E-n14 8320E-014.3?95E-n13 .872E-0o3,4395E-nl
) TFFUSEUJP-FL IX
?221lOF-n42.64l4Pr-n43. 1443F-n43.741 F-0n44,4517F-045 2956F-n4
.29A7F-n47.4913F-048.909?F-n41 . 5OQE-n31 .5QQF-n3
DTFFUSEUP-FL! IX
A 0549F-0 27,?lnFF-02
9.5817F-n?0.215F-nl
1.4447F-nl11.71QF-nl?.04S1F- 12,43?lF- 1
3.43SF-mnl1
NET Fl UX1.9978F-rT1.21n4F-617.325nF0n?4.4234F-n22.659RF-021.58sAF-n29.2879F-035.24T13Fn032.7122F-031.089^.F031.3878F-1 7
NET Fl UX9.394 5F-nT8 3121 -017.2981F-016.339?F-015.4224F-014 5341F-013 6605F-n12.786QF-0r11 8974F-019,7484F.n?1 7764F-15
58
* * * * * *t 41 41 *4 1 1 i t* **it*4 is i**41** 4t * * 4tit* i 4t 1 4 t * ****1** *4I
RADIATIVE FLUXES FOR VERTICALLY INHOMOGENFOUSLAYERED ATMnSPHERE BY DELTA-EnDINGTON APPPOXIMATION'
NUMRER OF LEVELS-- NZ = 1INCIDENT DIRFCT-BEAM FLUX-- SnLFLX= 1.OOOOE+00INCIDENT DIFFJSE DOWN-FLUX-- FDINC= 0,0INCIDENT DIFFUSE UP-FLUX AT BOTTOM BOUNDARy-- FUINCs 0.0NUMBER OF DITECT-BEAM 7ENTTH A NGLES-- NsOL- 2COSINES OF OTRECT-BEAM ZENITH ANGLES-- MUSUNL ,2On 1.0000SURFACE ALBFDO(S)-- ALB= 1,On00
OP nEPTH.1 000oo.1000. I noo
.1000
.1 000
.1000
.1000
.1000
.1000
SING-SCAT ALB.;:POO.,000,~000.?000,?000.,000.2000
?000.,000
TEST PROBLEMI 5
ASYM FAC.8500.8500. 500.8500.8500.8500.8500.85008R5008500
. nson
. tSOO
MUSIJN = nnIRECT
LEVEL FLUJX1 .? .onE-012 1.3n09E-013 .5n 4E- 024 c,.547F-025 3.6lt'7 -026 2.3560E-027 1 .531F-028 1,onlSE-029 6,5PQ4E-0310 4.2570E-0311 7.77q5E-03
MUSUN = 1 ,nnonIRECT
LEVEL FLUX1 1 .nnOoE+002 q .181E-013 R.4274E-014 7 73A4E-015 7.10?1.-016 6S5197E-017 , 9852E-018 ;* 4944F-019 , 049gE-0110 4,63n4E-0111 4,25n7F-01
DIFFUSEDOWN-FLUY0.02.310 4E-033.501.6E-034.0031E-014.0891E-033.9337E-033.6464E-0';3.2945E-0O2.9189E-0O2.5431E-032,18nl1E-0
DIFFUSEDOWN-FLUY
0o02.3481E-0i3,7747E-0O4 ,3875E-014.2718E-013,49?2E-012.0945E-0q1 0716E-04
-2. 4576E-03-5. 6 5E-0
-9 3497E-0O
TOTALDOWN-FLJX
2 ,000 E-011 ,3?7E-nl8 8516E-n5,9430E-o?4,0226E-n22 7494E-n?1 9 007F-n21 3309E-n29.4483E-n36. 8002E-034.9556E-n3
TOTALDOwN-FLUX1 .00OOE+n9,2036E-nl8.4651E-.17 7803F-n17 1 448E-nl6.5547E-nl6.0061E-nl5 4955E-n15.0194E-014.5743E-n14 1572E-h
DTFFUsEUP-FLfUX
5 08A,4F-n3
3,78q?F-n33 0497F-n3? .6945F-rn32,6014F-n32.69A^F-n32,93?9F-n33,2R^1F-n33.74?8F-o34.29PF-n34.959AF-n3
DTFFUqEUP-FLl!X
1 .34AF-nl1 .184lr-nl1 .3576^F-ol1 .5584F-n11 7906F-012 .0587r-nl
? .7?c; I-n 12,7-1 -n 13 .138Fr- 1,3.618F-014 1,72F-nl1
NET Fl UX1 94qlF.-n1 2892F-ll8.546AF-0n5,673^6F-03,762rF-o?2 4798F-0?1 6074F-021. 003F-n25.7057F-032,5020n-032.7756F-1 7
NET Fl.UX8.96 5 4F-n18.0194F-017.10o76F-l6.2219F-nl5.354pF-n14.496nFo.l3.638nF-012,7704F-n01 8828F-nl9 6355F-n21 .0658F-14
LAYER123456789
10
59
RADIATIVE FLUXES FOR VERTICALLY INHOMOGENEOUSLAYERED ATMOSPHERE 'Y "DELTA-EnDINGTON APPROXIMATION
NUMBER OF LEVELS-- NZ = 11INCIDENT DIRFCT-BEAM FLUX-- SOLFLX= 1.00OOE00INCIDENT DIFFUSE DOWN-FLUX-- FDINC= 0,0INCIDENT DIFFUSE UP-FLUX AT BOTTOM BOUNDARY-- FUINC= 0.0NUMBER OF DIRECT-BEAM 7ENITH ANGLES-- NsOL=COSINES OF 'DIRECT-BEAM ZENITH ANGLES-- MUSUNs .20n) i.0o00SURFACE ALBEDO(S)-- ALB= 1.On00
OP nEPTH.0l00.0100.0)100.0100.0100.0100.0100.0100.0100.0100: 0 1 0
SING-SCAT ALB.4000,4000@4000*4000.4000.4000.4000.4000
4 000,4000
TEST PROBLEM 6
ASYM FAC
Ra500.R500
.8500
.8500
.R500
.RS00.8500.R500
MUSUN = . nof0nIPECT
LEVEL FLUX1 .oo0noE-O12 1 93n01E-13 1.867?E-014 1,7977F-015 1.7349F-016 1.6743F-017 1,618rE-018 1 .59Q4-019 l.50 4 9E-01
10 1. 454E-0111 1.4n16E-01
MUSULN = 1.n00ntRECT
LEVEL FLUX1 1,00OE+002 9,9PQ?-013 Q 85R8E-014 9,7 8qOF-015 Q.71Q6E-016 9 .6n7rEOl7 9. 5R4E-018 9,5145E-019 9 4471E-0110 9.3 nl0E-0111 9,3137E-01
DIFFUSEDOWN-FLUY
0.04,77 1E-049.2803E-041.3514E-O01 7489E-0O2.1216E-032.4704E-O02 7962E-033.0998E-033 3821E-0.3,6438E-03
DIFFUSEDOWN-FLUX
0.05,119 QE-Oc8,5755E-051.0332E-041,0549E-049.0902E-0O6.0 56E-O.01 3366E-05
-4.9359E-09-1.2791E-04-2.2220E-04
TOTALDOWN-FLUX2, 000E-011 9349E-n11 8720E-n11,8112E-011 7524E-n11.6955E-n11 6405E-011 5874E- 11 5359E-011 4862E-ol1 43 8 1E-Ol
TOTALDOWN-FLIUX
1 OOo0E+o09.9?97E-019.8597E-019.70qOE-ol9,7?07E-019.6517E-019.5830E-n19.5146E-019.4466E-n19.3789E-n09.3115E-01
DTFFUSEUP-FL(JX
1 32 70F-011 33AqF-nl1,3470F-r11 3575F-o11 .36A?F-nl1 3792F-nl1 .394F-011 .40?F-nl1 4137F-n11 .4258F- 11 43R1F-n1
DIFFUSEUP-FLIX
R.36T1F-nl8,4518F-018,5432?F-nl8 6395F-nl8.72R9F-n18 8233F-nl8.9!8RF-019.013F-1nl9 1130F-p19.211 7E-n9 31 15F -0
NET Fl UX6.7300F.np5.9804F-n?5,2497F-024.5371F-0n3.84TSFn-023.1633F-022.5009F-n21.854nF-021.2219F-0?6.0416nF-031.77A4F-15
NET FLUX1 6385R-fl11 .4779F-Ol1 3165F-011 1545F-019.9176F-0?8.2833F.026.6417F-024.9927F-02
3336PE-021 6720F-021 * 0658F14
LAYER123456789
10
60
RADIATIVE FIUXES FOR VERTICALLY INHOMOGENEOUSLAYERED ATMOSPHERE BY DELTA-EnDINGTON APPROXIMATION
NUMBER OF LEVELS-- N7 = 12INCIDENT DIRFCT-BEAM FLUX-- SoLFLX= 1.0000E+00INCIDENT DIFFUSE DOWN-FLUX- FDINC= 0 .0INCIDENT DIFFUSE UP-FLUX AT BOTTOM BOUNDARY-- FUINC= 0.0NUMBER OF DIPECT-BEAM 7ENITH- ANGLES-- NSOL 2COSINES OF DTRECT-BEAM ZENITH ANGLES-- MUSuN .2000 1 .0000SURFACE ALBEOO(S)-- ALB = 1.On00
OP nEPTH
.1 0001.00001 00001.00001.00001 .00001.0000
1 00001 00001,.00001.0000
SING-SCAT ALB1.nOOO
0nOOO0.6000
o nooo
0,6oooo.noooo. o00o-io
TEST PROBLEM 7
ASYM FAC
.8500. .n
.8500
.8500* 8500.8500.8500RSO 0
a A 5 00 PF n
MUSUN = .?n00nIRECT
LEVEL FLUX_1 .0onnooF012 1.74n9E-013 1.17 oE-034 7.9n6E-=065 5-3254E-086 3.58R2E-107 . 4 1777E-128 1.6291E-149 1 0 977E-16
10 7. 39q9F'1911 4.98_3E-2112 1,3577E-23
MUSIJN = .OnOoI RECT
LEVEL FLUX1 1.OOnOE+002 9.7263E-013 3.5771E-014 1.31 3E-015 4.84?4E-026 1.7814E-027 6.555E-038 ?.410n9E039 8.8Q,3E-04
10 32.6?PE-0411 1.20n3E-0412 4,41;7E-05
DIFFUSEDOWN-FLUx0 01 4706E-OP2,6018E-034 .632E-048,1441E-0;O1 4409E-052 5492E-O04,51OOE-077 9792E-O01.4116E-0O2,4920E-Oq4 11l5E-ln
DIFFUSEnOWN-FLUy0,02,2991E-0O4.0677E-0o7.1965E-041.2732E-042.5?6E-0c3.9848E-067.021 9E-071.0836E-07
-7,05POE-08-5.1943E-072 ,9573E-06
TOTALDOWN-FLUX
2 0000E.0 11 8880E-ol3 7749E-n34 ,623F_-48 1494E-051 4409E-n52 5492En-64,5100E-077,9792E-n8141 16E-n82 4 920F-n94.10olE-1n
TOTALDOWN-FLUX
1 .OOOE+009.9562E-nl3,6188E-1l1, 3 235E-nl4.8552E-021.7837E-n?6.5575E-032,4J16E-038,8703E-043,2621E-n41]1951E-n441200OE-05
DniFFUEUP-FLt X
1 *014F-n2-1 .055QF-3-1 86ROF-04-3 3onrF-n5-5 .847?F-06-1 0345cF-n6-1 83nF-07-3,23a8F-n8-5 72s4F-n9-9 997?F-10-1 0114F-10
,10114F-10
DTFFUCEIIP-FL IX
P 725F-n3-1 6507F-03-2? .9o4F-04-5 .1 6gQ. 5- .1411F-06-1 .61AoF-n6-2 78qq9F-7-1 ,0257F-i82.19?oF-n77.28nFF-n67 .28oF-o^54, 12n n rF-5
NET FLJUX
1 8985F.011 .8985F-'013.9617F-035.0128AF048 7341F-051 5443F-052 7322F-064 8339F-078.5519Fr081.5i1lF-n82.5931F-094.9631 -24
NET Fl..UX9. 9727F.o79.9727F.0n3.6217F-011.3240Fo014.8561F-0?1 7839F-026.557Fn-032, 41T Fn08 8681F.-43 249?Fn041 12?F36 044 33 6 8F-19
LAYER1234567
91011
61
RADIATIVE FLUXES FOR VERTICALLY INHOMOFENFOUSLAYERED ATMOS:HERE BY DELTA-ErDINGTON APPROXIMATIOm
NUMBER OF LEVE.S-- NZ = 11INCIDENT DIPFCT-BEAM FLUX-- S3LFLX=s 1.000E+OINCIDENT DIFFUSE DOWN-FLUX-- FDINC= 0O.0INCIDENT DIFFUSE UP-FLUX AT BOTTOM BOUNDARY-- FUINC- 0.0NUMRER OF DIPECT-BEAM 7ENITH ANGLES-- NSOL= 2COSINES nF ODTrCT-BEAM ZENITH ANGLES-- MUSUtN= ,20n 1.0000SURFACE ALBEDO(S)-- ALB= 0.0000
OP E'rTH:1.00001 .00001.0000
1 . no0001.00001 . 0 0 O1.n000
1.00001.00001. nooo
''STNG-SCAT ALB0?000* 000,?000
0,000.?000.?000.?000
.,000
. ?000
.,w000
TEST PROBLEM 8
ASYM FAC*8500.8500.8500.8500.8o500
.8500
.8500
.8500
~ .~
MUSUN = .?nO0nI ECT
LEVEL FLUX1 ,.OnnOOE-12 2,7755E-033 3.8516e-054 g,345n0-075 7 .4174E-096 1. n03E-107 1 ,4R84-128 9 ?3cE-149 .75n9r16
]0 3.81 75c-1811 5.2977r-2o
MUSUN = l.nnoonI ECT
LEVEL FUX1 1.non0 o ,002 4.?5,n7-013 1.80n9r-014 7.6A04E-025 3.?647E-026 1 .377E-027 B.8988E-038 ?.5074--039 1.06q8e-03
10 4,5305s-0411 1.9?;3r-04
DIFFUSEDOWN-FLUy
0.02.4781E-0O6.3897E-041 5616E-043.8080E-059.?843E-OA2.2636E-065.5187E-071.3455E-073.2796E-0O7.9664E-09
DIFFUSErOWN-FLUy0.01.5140E-0O
.0127E-O?5.2045E-0^2.4317E-031.0871E-0O4.751 E-042.0515 E-048.7978E-003.7579E-OS1.5990E-0
TOTALDOWN-FLLUX
2 .o0OE-on5 2536E-036 .7708F-041 .5670E-n43,8088E-ns9.2844E-062 2636E-065,5187E-n71 3455E-073.2796E-n87 9664E-n9
TOTALDOWN-FLUX1 .00OUE004 4 o 2 1 Enl1 .9081E-lo8 . 2o0E-n?3.So7q9E-n1 ,464F-n?6,3740E-032 7126F-n31 1538E-034 90o63F-n42 O857E-n4
DTFFUqEUJP-FL tX
,.8579F-n3-1 .01ARF-n4-3. 92R4F--5-9.771nF-n6-2,381;OF-n6-5 81 C F-; 7-1 .417RF-n7-1.45c;9M-n8-8,3977F-n9-1 .93F~-n9-3.6644F-?4
DTFFUSEUD-FL.IX
P?50r9F-n31 .1847F-04
-1 .804F-n4-1.334F- n4-7.037RF-n5-3.3?&F.-n5-1 .499AF-n5-6.54Q6F-o6-2.8049F-06-1 .09n4F-n-. 77?F3F-?l
NET FLUX1 96l4F-nl5.355?F-0o37.1634F.n41 6647r-044 0473Fn059.8659F-06? .453Fo65 864rF-071 429 c F-0n73 4739F-n R7 9664F-nQ
NET FLUX9.97 4QF;.n4.40.09~Frl11.9nqqF- 18.2142Fn2?3.5149F-n21.4998F-0?6.3889r-n32,7191F-031 .1566F0o34.9172r-042.0857F0n4
LAYER123456789
10
62
4 1, * *K * * * 4* * * * * 4* * * * * * ¢* ¢* * * * * * * * * 41' * # 41 * * * * ¢* * 4 * 4 * * 4 * 4 * * * ~ 41 ' *
RADIATIVE FLUXES FOR VERTIC ALLY INHOMoG'NE 'OULAYERED ATMOSPHERE 9Y DELTA-EnDINGTON APPROXINMATION
*** * ** * ** ** 0* * * * *4 *4 **4 * *4* *44** 4 *
NUMRER OF LEVELS-- NZ i1 1.INCIDENT DIPFCT-BEAM FLUX-- SoLFLX- 1.0000F00INC DENT DI FFJ'SE .DOWN-FLUX''- F DI NC= 0 .0INCIDENT DIFFUSE UP-FLUX AT BOTTOM BOUNDARy-- 'tJINC= 0,0NUMBER OF DTIRET-BEAM 7EN!'TH ANGLES-- NSOL= _COSINES OF DTRECT-BEAM ZENITH ANGLES-- MUSUN= ,20oO 1.0000SURFACE ALBFEDOS)-- ALB= O.n00
'OP hE'TH1.0000
1.o000
5.* 00001.0000
1 .00001.00001.00001.00oooo
SING-SCAT ALB.AOOO
,. 000.(000.^000.is000
* 6000
* ,o00.... .ooo,000
. ooo( ̂000
TEST PROBLEM :9
ASYM FAC* 8500.8500.850085008500
.8500
.8500* 8500.8500* 8500
MUSUN = nooon I ECT
LEVEL FLUX1 .0 n010E012 J 1773C-023 6,93n3r-044 4.07Q6 -055 ? '.4014.E-066 1.4136e-077 .3214E-098 4.89R4FP109 2 .8835E-11
10 1.6974-1211 9 9*9917r-14
MUStJN = 1.nnonTIECT
LEVEL FLUX1 1,nnnoro002 95671r-013 3.2?n7'-014 1.8277-o015 1 0373E-016 5 88S6E-027 3.34n07--028 1.8959E-029 1, 0 7n9?-02
1.0 l.1n9O-0311 3461;E-03
DIFFUSEDOWN-FLUX.0
1 .8477E-09.6700E-014 5557E-0O2.1198E-039.8487E-044 5748E-042.1 249E-049 .8700E-0c4 5842E-0o2.1286E-0O
DIFFUSEDOWN-FLUtJ0.07 2034E-0?7.4339E-0O5 77?9E-0O3.9981E-0l2 6043E-0p1 .6337E-0O9 9948E-0O6 0082E-03.5659E-0O2.0964E-03
TOTALU)OWN-FLL)X2 ,0) E -013.0250E-n21 0363E-024 ,596SE-032 1222E-n39 850 1E-044 5748F-n42 ,1 4 9E-n49 8700E-054 5 824 ?E-n52, 186E-n5
TOTALO WN-FLUX
1 00 00E+On6.3954E-1n3,9640E-012. 4o5oE-nO1 .3 7 1E- n8 4908E-o?4 9744F-022.8953E-n1 .6767E-n?9 671RE-n35 5616E-03
ODFFUSEUP-FL' X
1 .9 3 q4F-o?7 4966F-o4
-1 37A6F- 4-9 .543F-'5-4 .4665F-65-2?.088F-n5-9 .6685F-06,4 .458R-8P6-1 9934F-0o-7.61snF-o7-? ,69n5F-l1
DTFFUSEUP-FL! X
1 *717?2r-.28 21q7F-033 .95^n0F-n31.91 cSRr-n39 3471F-n44 .58QoF-n42 .?26F?F-n41 .124F-n45.4237F-n52 23 n7F-05F .421 n-PO
NFT Fl UX1 2,8o6?r-n2 95^0oF-?1 0501F-024 6890F_032 1669F-031 .o0058F-034 6715F-042.1695F-n4
4 66i3Fno5?.12R6F-0n
NFT FLUX9 ,28Fn-T16.31] ?F-n3. 9245F-o2 3859Fr-11 42 7 7rF-i8 4449Frn?4.9517F.O?2 8841F-0?1 6713r-029.64QSFn035.561^F-n3
LA'YER-1234567
g9
In
63
APPENDIX C:
COMPARISON OF DELTA-EDDINGTON AND
ADDING-DOUBLING FLUX COMPUTATIONS
We consider five problems, each referring to a homogeneous layer
with a decidedly asymmetric phase function. The characteristics of
these problems are as follows:
Problem Phase function (AT)* a* 0
1 Haze L (X=0.7p, m=1.33) 1 1 1
2 Haze L (X=0.7p, m=1.33) 1 0.9 1
3 Haze L (X=0.7p, m=1.33) 1 0.9 0.5
4 Cloud C1 (X=0.7p, m=1.33) 64 1 1
5 Cloud C1 (X=0.7p, m=1.33) 64 0.9 1
A= F =00
S = T F = 0
X is wavelength, m is aerosol or cloud drop index of refraction, (AT)*
and w* are (unscaled) optical depth and single-scattering albedo, and
the Haze L and Cloud Cl size distributions are given by Deirmendjian.
1Deirmendjian, D., 1969: Electromagnetic Scattering on Spherical
Polydispersions. American Elsevier, New York.
64
The adding-doubling results, which are the 'exact' ones for the
2purposes of this comparison, proceed from the Grant-Hunt algorithm.
3The doubling part follows the development of Wiscombe and uses the
diamond initialization. The phase function is treated using the
4delta-M method [Wiscombe ], an extension of delta-Eddington to arbi-
trarily high orders of angular approximation.
The comparisons are presented in Tables C.1 and C.2. The largest
reflected-flux error in Table C.1 is 18%, for Problem 2, the next
largest is 14%, for Problem 1; the rest are considerably smaller. But
note that these large percent errors occur in relatively small numbers.
As percentages of the incident flux af, they fall to 0.7% and 0.8%,
respectively. The largest transmitted-flux error in Table C.1 is 6.9%,
for Problem 3; the rest are considerably smaller. But again, relative
to the incident flux T/2, this maximum error falls to 3.5%. Among the
net fluxes in Table C.2, the largest significant error is 6.6% (two
very small fluxes in Problem 5 have larger errors) but this error is
only 0.4% of the incident flux.
Grant, I. P., and G. E. Hunt, 1969: Discrete space theory of radia-tive transfer I. Fundamentals. Proc. Roy. Soc. London, A313, 183-197.
Wiscombe, W. J., 1976: On initialization, error, and flux conserva-tion in the doubling method. JQSRT, 16, 637-658.
Wiscombe, W. J., 1977: The delta-M method: Rapid yet accurate radia-tive flux calculations for strongly asymmetric phase functions.J. Atmos. Sci. (submitted for publication).
65
Table C1. Comparison of adding-doubling (parenthesized)and delta-Eddington (unparenthesized) predic-tions of reflected and diffuse transmittedflux for the five test problems.
Problem F [F[T*=0O] F+[T*= (A) *]
1 0.198 1.787(0.173) (1.813)
2 0.147 1.500(0.124) (1.516)
3 0.219 0.858(0.226) (0.803)
4 2.668 0.473(2.662) (0.480)
5 0.354 0.000(0.376) (0.000)
Table C2. Comparison of adding-doubling (parenthesized)and delta-Eddington (unparenthesized) predic-tions of net flux at 7 levels, for the fivetest problems.
Net FluxProblem 1 2 3 4 5
T*=O 2.943 2.995 1.352 0.473 2.787(2.968) (3.018) (1.345) (0.480) (2.766)
(A-)*/20 2.943 2.977 1.334 0.473 1.614(2.968) (3.001) (1.326) (0.480) (1.605)
(AT)*/10 2.943 2.960 1.317 0.473 0.826(2.968) (2.983) (1.306) (0.480) (0.830)
(AT)*/5 2.943 2.926 1.284 0.473 0.183(2.968) (2.948) (1.266) (0.480) (0.196)
(AT)*/2 2.943 2.822 1.193 0.473 0.0013(2.968) (2.842) (1.157) (0.480) (0.0022)
3(AT)*/4 2.943 2.738 1.128 0.473 0.0000(2.968) (2.755) (1.079) (0.480) (0.0001)
(AT)* 2.943 2.656 1.071 0.473 0.0000(2.968) (2.671) (1.016) (0.480) (0.0000)
66
There is abundant evidence in Tables C.1 and C.2 that the delta-
Eddington approximation is particularly accurate for heating rates (net
flux differences). It is somewhat less accurate for up- and down-flux
separately. It should be particularly valuable for radiative energy
budget studies, in which flux errors are required to be small compared
to the incident flux (from the sun, say).