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3 1822 04429 6655
FEB 1988
ATMOSPHERIC VISIBILITY
TECHNICAL NOTE NO. 208
SA
Offsite
(Annex-JO
rinals)
QC
974.5
. T45
no. 208
SKY COVER MODELING CONCEPTS:
AN ANALYSIS
Wayne S. Hering

UNIVERSITY
CALIFORNIA
SAN DIEGO
The material contained in this note is to be considered
proprietary in nature and is not authorized for distribution
without the prior consent of the Marine Physical Laboratory
and the Air Force Geophysics Laboratory
.
NIVS
Contract Monitor, Dr. Joseph Snow
Atmospheric Sciences Division
the CC www
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hVER
*1868
Prepared for
Air Force Geophysics Laboratory, Air Force Systems Command
SCRIPPS
INSTITUTION
OCEANOGRAPHY
MARINE PHYSICAL LAB San Diego, CA 92152-6400
UNIVERSITY OF CALIFORNIA, SAN DIEGO
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3 1822 04429 6655
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January 20, 1988
MPL File: 88.06
TECHNICAL MEMORANDUM
To:
Richard W. Johnson
From:
Wayne S. Hering
1
Subject:
Sky Cover Modeling Concepts: An Analysis
1. Introduction
An AFGL/UCSD field program to provide minute by minute measurements of sky
cover will be carried out during the next few years. Solid-state, whole-sky imagery
(WSI) systems are to be installed in a network configuration consisting of six sites in
the western United States. The primary goal is to develop a data base that is
required to evaluate and extend cloud models. The models are to be used for
estimating the impact of clouds on ground based systems that depend upon
unobscured paths of sight to satellites in space. As a prelude to the planning of the
WSI data acquisition and processing procedures, some elements of existing cloud
modeling procedures have been reviewed.
In particular, the following reports set forth advanced methods that have direct
application to the modeling of sky cover and cloud-free-line-of sight, CFLOS:
(i) Gringorten, Irving I., "Conditional Probability for an Exact, Noncategorized
Initial Condition", Monthly Weather Review, Vol. 100, No. 11, 796-798, 1972.
Gringorten presents a model for estimating the conditional probability of an
event given an exact noncategorized initial condition. The model assumes a form
of Markov process known as the "Ornstein Uhlenbeck" process to give the
probability estimates. For immediate reference, let us repeat the basic expression
here. For a given cumulative probability distribution of variable, Y, there is a
corresponding equivalent normal distribution, y. Thus there is an equivalent
normal deviate (END), y, that corresponds to an initial value Y and a later
END value, yy, that corresponds to Y. The expression for the stochastic process
is
Yt =Pty. +(1 - 2227
where p, is the correlation coefficient between y and y. over time interval t, and
p is the normalized value corresponding to the conditional probability of y..
.
For a Markov process,
p=po
Where is the correlation over a unit time interval.
.. (2)
over a
Burger, Charles F. and Irving I. Gringorten, "Two-Dimensional Modeling for
Lineal and Areal Probabilities of Weather Conditions," AFGL-TR-84-0126,
Environmental Research Papers No. 875, 1984.
Using simulation methods, Burger and Gringorten have derived analytical
expressions for estimating the probability that a given weather condition will
cover a given area or length or cover a given fraction of the area or length. The
input data are the unconditional point probability of the event and a
representative measure of the spatial correlation coefficient.
Boehm, Albert, Irving I. Gringorten and Charles F. Burger, "Simulated
Duration of CFLOS from Multiple Sites to a Satellite", private
communication of AFGL Technical Memorandum No. 125, 1986.
Glas
.
The report by Boehm et al. summarizes a wide range of important and
directly relevant research, and in particular describes the development and theory
of the CFLOS4D and CFARC models. These complex models simulate the
probability of CFLOS from a network of ground sites to a configuration of
satellites. The models of necessity consist of a lengthy series of model
approximations and assumptions as determined from data analysis procedures.
The simulation model has not been tested extensively, although many of the
component techniques have been applied successfully during the past few years.
(iv) Lund, Iver A. and Donald D. Grantham, "Estimating the Joint
Probability of a Weather Event at More Than Two Locations, J. of
Appl. Meteor., 19, 1091-1100, 1980.
W
(3)
The basic expression for multisite joint occurrence probability of a given
weather conditon as set forth by Lund and Grantham is
Pijk =PjPk + (Pij - PjPk)exp - az(dz/25)
where P.:1, is the multisite occurrence probability (3 sites in this case), P., is
the 2-site occurrence probability, a, and b are constants and the coefficient a
d, is a function of the unconditionål event probability and the effective
spatial correlation coefficient pe (for 3 stations). Expressions of the form Eq.
3 were applied to independent Sky-cover data for as many as 6 of 8
midwestern stations with excellent results.
An attempt is made in this review report to examine further the problem of
estimating the joint occurrence probabilities of cloud obstruction as it applies to
ground to space line of site communication from a number of sites. The approach is to
reassemble some of the techniques presented in the papers cited above into somewhat
.
different analytical forms. The objectives are to achieve a format which will enable a
more direct evaluation of the individual component techniques inherent in the Boehm
simulation cloud models and to propose supplementary techniques for estimating the
impact of clouds on the ground based systems.
2. General Considerations
As emphasized by Boehm, the simulation models do not provide direct estimates
of cloud free arc probability. The CFLOS statistics are generated on a point by point
basis at specified time intervals from multiple ground sites to a fixed point in space
(geostationary satellites) or for a prescribed sequence of viewing angles (for å
configuration of orbiting satellites). A "down" condition at a single site occurs if
cloud obscures the line of sight to a particular point in time and space. System down
time duration at a single site is calculated as a sustained recurrence of the cloud
obscured condition. Multiple station down time is determined in an analogous
manner. Answers to basic questions are required from the user for the most effective
definition of down time duration. For example, what criteria are appropriate for a
termination of down time duration at a single site or for multiple sites? Will an
isolated open or CFLOS condition at one or two 1-minutes intervals at a single site
enable sufficient ground-space communication so as to constitute a down time duration
termination? In other words, the time and space sequence of possible cloud
observation along a satellite track in relation to the operational system requirements
must be a factor in down time definition for the application of simulation LOS models.
For some multisite ground-based system operations an alternate scheme might be
used that provides the operator with estimates of the joint occurrence probability of a
given fraction of sky cover over specified regions of the sky dome or along typical arcs
corresponding to satellite traverses. The technology for estimating the unconditional
probability that a given area or arc segment will be covered by clouds, or will be
obscured by a prescribed cloud fraction is available in the form of the Burger Area
Algorithms BAA (Burger and Gringorten, 1984). The problem of estimating the
persistence probability of a given fractional sky cover over a subarea of the sky or
along a sky arc in time and space presumably can be handled with modeling
techniques similar to those applicable to the persistence of sky cover over the entire
sky-dome.
With this option to provide cloud obscuration probability information, the user is
challenged to define his "down" or "non-operable" condition in terms of what fraction
of arc or subarea must be covered by clouds to critically limit the quality of
communication. To the extent that such operational definition of a down condition
can be specified, meaningful down time duration probability statistics can be derived
through extension and refinement of existing techniques. As is the case with
simulation models, reliable information on mean cloud cover and representative time
and space correlation parameters are required input to the analytical modeling
scheme. :
As the WSI data acquisition program progresses, unique information will become
available to define more accurately the time and space persistence of cloud free arcs
and cloud free areas. Meanwhile, we may be able to prescribe to a good first.
approximation the scaling factors for time and space correlation from conventional sky
cover data.
rekom
Provisional techniques for estimating the joint occurrence (multisite) probability
and the persistence of cloud free arc, CFA, and cloud free fields of view CFFOV, over
time and space are discussed in the following paragraphs. The BAA (Burger and
Gringorten, 1984) for lineal and area probabilities are to be used in combination with
conditional probability estimates of the time in space behavior assuming a Markov
process as given by Equation 1 (Gringorten, 1972) to yield estimates of down time
duration statistics for ground-satellite communication systems requiring CFLOS. The
analytical techniques proposed for dealing with temporal and spatial variation of
limited-area sky-cover or arc-segment sky-cover are similar to the techniques used by
Boehm, et al. (1986) to define the input variables to the CFARC and CFLOS4D
simulation models and by Lund and Grantham (1980) to estimate multistation
occurrence probabilities.
3. Cloud obscuration probability estimates
CU
...
3.1 Climatological probability of CFLOS at a single site
Lund and Shanklin (1973) derived methods for estimating CFLOS probability
using whole-sky photographs made hourly over a 3-year period at Columbia, Missouri.
The resultant graphs yield CFLOS probability for various cloud types as a function of
total sky cover and the zenith angle of the path of sight. General empirical
expressions representing the composite behavior for the distribution of cloud types in
the Lund Shanklin data base are given by Allen and Mahlick (1983).
After further examination of the applicability and accuracy of these expressions,
Boehm, et al. (1986) utilized the Allen and Mahlick CFLOS equations in their
computations of CFLOS site climatology as part of the CFLOS4D simulation process.
A step-wise procedure for calculating the climatological probability of CFLOS from the
conventional meterological data is given in the report.
3.2 Climatological probability of cloud free fields of view, CFFOV, and cloud free arc,
CFA, at a single site
Lund, Grantham and Davis (1 )) extended earlier studies of the Lund-Shanklin
data base from Columbia, Missouri, to develop models for estimating cloud free fields
of view, CFFOV. As is the case for CFLOS, the probability estimates for CFFOV are
given as a function of total sky cover and sky dome position for zenith centered areas
of increasing size as well as pilot results for areas centered other than overhead.
As suggested in Section 2, a more general technique for estimating the probability
that a prescribed region of the sky-dome or segment of a satellite track as viewed from
the ground will be totally or partially obscured by clouds is the BAA (Burger and
Gringorten, 1984). In particular, examples of BAA application to the problem of areal
and lineal sky cover are illustrated in the report. Important considerations with
respect to the effective use of BAA for limited area sky covering estimates center on
specification of the input parameters of mean sky, cover and scale distance and also
the designation of area size or arc length.
For BAA estimates of total sky cover distribution, it was assumed that the sky
dome as viewed by the ground observer has radius of 27.8 km or equivalent floor space
area of 2424 km". Probability estimates of sky cover for restricted portions of the sky
dome require specification of the floor area that is equivalent to the sky area fraction
of interest. In effect, a representative cloud altitude must be prescribed in order to
approximate the fractional floor area corresponding to a given subarea of the sky
hemisphere or the lineal floor distance corresponding to a segment of a satellite track
obscured by clouds as viewed from a ground site. For cloud simulation purposes,
Boehm, et al (1986) selected a median cloud height of 15,000 ft for the sawtooth model
determinations of CFLOS probability. They observed that the simulation statistics at
single site were not very sensitive to the assumed average cloud level, except for
elevation angles within 10 degrees of the horizon.
The scale distance input parameter to the BAA for the determination of limited
area or arc length sky-cover statistics can be handled in the same way as for the
determination of whole-sky statistics. The appropriate parametrization is to use "sky-
dome" scale distance (Burger and Gringorten, 1984) that reflects the structure and
spacing of the individual cloud elements as opposed to the "inter-site" scale distance
that refers to the broad cloud system features. The determinations of sky-dome scale
distance given by Burger and Gringorten (1984) and Boehm, et al (1986) show that the
values usually range from about 0.3 for tropical cumulus conditions to 2 or greater for
winter mid-latitude cloud conditions.
The other decisive input parameter to the BAA is the unconditional single point
probability of cloud cover, accounting for its variations with geographical location,
season and time of day. For whole-sky dome determinations, Burger and Gringorten
(1984) assumed that the mean sky-cover represented the probability of a cloud
presence vertically overhead. For the modeling of the fractional sky cover (from clear
to 10/10) as viewed from the ground or limited satellite arc segments, it is appropriate
to use the climatological probability of CFLOS for the particular portion of the sky
dome of interest as the BAA input variable. In this way the effect of the systematic
variations of CFLOS probability with the zenith angle of the viewing path can be
introduced into modeling process.
In summary, it is reasonable to expect that the BAA coupled with modeling
techniques similar to those which are basic to the development of the CFLOS
simulation models will yield reliable analytical models for estimating the climatological
probability that a specified fraction (0 to 10/10) of an arc or subarea of the sky dome
will be obscured by clouds. The accuracy of the estimates is dependent primarily upon
the reliability and representatives of the BAA input data as derived from the
conventional climatological data base. The WSI program will provide important
insight into CFA and CFFOV model performance and its sensitivity to uncertainties in
the input variables.
3.3 Persistence and recurrence probability of CFLOS, CFA and CFFOV at a single site
Let us consider the application of existing concepts to the modeling the
persistence of a prescribed fraction of cloud cover along a designated sky arc or a
designated fractional part of the sky dome. The approach is to assume that the
Ornstein Uhlenbeck process as expressed by Eq. 1 will model the temporal recurrence
probability of the sky-cover condition over a limited area or arc segment. In turn, we
must establish a relationship between recurrence (occurrence of a sky condition at
both time t, and later time t) and persistence (continuous presence of the sky
condition from t to t). Since our major interest is in the down time duration
statistics of ground to space communication systems, we must focus on the modeling of
the duration of a sky condition or in other words the persistence probability as a
function of time interval.
The WSI data base will in time provide systematic measurements of sky-cover at
1-minute intervals, so that the persistence probabilities of CFLOS, CFA and CFFOV
can be resolved more closely as a function of location, season and time of day, etc.
Meanwhile let us examine some available evidence of sky-cover variability with time at
a given site.
istence probabilities or CFLos, CFA and OFFok
First, let us assume a provisional relationship between persistence and recurrence
based more on intuition than actual data. Eventually the WSI data base will be of
particular interest and importance for the refinement of such relationships. For the
moment let us define:
P (Y1
rl'ti
conditional probability of
Y, given Y or in other words the recurrence
probabilityº
P(Y) =
unconditional probability of
Y at time =0
P (Y 1 Y) =
persistence probability of Y over
time interval t.
Further let us define y., y, and p, as the END of Y, Y and P(Y | Y) respectively.
We note that Eq. 1 gives us a relationship between the normalized recurrence
probability p and the normalized unconditional probabilities of initial and final values
of sky condition Y. Now we postulate that the expression for persistence probability is
P (Y4 YO) = P,(Y| YO) - P(Y!)
1 - P(Y)
So that the persistence probability becomes an increasingly smaller fraction of the
recurrence probability as the temporal correlation becomes smaller, and becomes zero
for p =0, and the persistence probability approaches the recurrence probability as po
approaches 1.
Before moving ahead to problems of estimating the persistence probability of
cloud free arc and cloud free area, let us examine the applicability of Eqs. 1 and 2 for
estimates of CFLOS persistence probability. An analytical solution to this problem is
in itself important. Data suitable for persistence analysis is very limited. Thus, the
data set published by Lund (1973) and used extensively by Boehm, et al (1986) for
simulation model development is of special interest and importance. Lund determined
both "persistence" and recurrence probabilities of CFLOS from the Columbia,
Missouri, whole-sky photographs obtained at hourly intervals (885 days) and from a
special subset obtained at 5-min intervals (585 hours). So while the data set does not
yield actual persistence (observations of continuous occurrence) The Lund data give
the probability of an uninterrupted sequence of cloudy or clear LOS observations at
both 5-min and hourly observation intervals as a function of the duration interval. In
addition, recurrence probabilities for the 5-min and hourly data for cloudy and clear
LOS were determined.
is
It is important to note that 5-min data base consisted of only observations from
the months of June, July, August and September for the year 1969, whereas the 1-hr
data base consisted of observations from all seasons over a period of approximately 3-
years. The photographs were taken between the hours of 0800 and 1700 (5-min data)
and 0900-1500 (hourly data). For our analysis purposes, attention was confined to the
summer season when both the 5-min and hourly determinations were made. The
lowest of three summer season values is given in the case of the 1-hour data.
Recurrence probability, P, of CFLOS, extracted from Fig. 3 of Lund (1973) are
listed in column (a) of Table 1. The inconsistency between values listed for time
intervals near 55 min and 1-hour are due to the differences in the months and years
comprising the 5-min and hourly data sets as discussed in the previous paragraph.
Listed in column (b) of Table 2 for comparison are the 5-min persistence probability
data, P(5-min), and 1-hour persistence data, P(1-hour), taken from Fig. 2 of Lund
(1973). As by definition, we note and expect that for time intervals equal the sampling
interval (5-min and 1-hr) the recurrence and "persistence" probability values are
identical and the "persistence" probability becomes a smaller fraction of the
recurrence probability with increasing time interval. Furthermore, the actual
persistence probability is even less.
In column (c) of Table 1 are the provisional estimates of persistence probability as
calculated from Eq. 4. Input data for the calculations are the recurrence probability,
P., from column (a), and an average summer CFLOS probability P(Y) of 0.534.
Validation and refinement of these estimates must await appropriate high frequency
observations of CFLOS. For this data set we observe that in the case of the 5-minute
data, the difference in the 5-min "persistence" probability and the estimated pure
persistence is within a few percent after 30 minutes of 5-min observations.
..
Before we can apply some form of Eq. 4 to estimate the persistence probability,
we must derive reliable estimates of recurrence probability. With some modification,
Eq. 1 and Eq. 2 might provide an adequate solution to this problem. Listed in column
(d) of Table 1 are calculated values of temporal correlation coefficient, pu, which
correspond to the observed values of recurrence probability, P., given in column (a).
The determinations were made using Eq. 1. Since the unconditional probability of
CFLOS was not available as a function of time of day, an average value of 0.534 was
assumed for the final event probability, Y , with a corresponding value of 0.08 for y..
A value equal to one half the category frequency was assumed for these initial evenť
probability, Y =0.267 and y. =-0.62.
It is apparent that a simple exponential decay function as prescribed by Eq. 2,
will not handle adequately the indicated temporal variation of CFLOS correlation
coefficient as calculated for the Columbia, Missouri, data sample. This being the case
Boehm et al (1986) assumed a 2-term exponential decay function for the simulation
model development of the form,
po=4%26 + (1 - W43)8€
A similar expression was used by Boehm for modeling the spatial variation for CFLOS.
The approach recognizes that for the modeling the collective behavior of cloud
systems for CFLOS probability one must account for both the broad-scale/long-term
variations in sky cover as represented by Pn and the short term development, motion
and decay of individual clouds as represented by P. The weighting factor W
t , which
siswi
Sweet
dete
ranges from 0 to 1, is large for regions that are dominated by clear sky or extensive
stratus cloud cover regimes and small for tropical cumulus type regimes. An
important consideration is that the concept of sky dome scale distance as discussed by
Boehm et al (1986) can be used effectively to model the weighting factor W.- for both
the temporal and spatial variability of CFLOS correlation. The WSI data base will
enable important refinement of these relationships.
Let us examine some trial determinations which attempt to match the values of
Po given in column (d) of Table 1 with calculations of e. to determine if the form of
Eqs. 1 and 2 are adequate to reproduce the data sample if we are free to select and
adjust the input data. Given that
(6)
Pb = exp(-1/TV)
and
Pc =exp(-1/7)
where Th and T, are the exponential response times for the large-scale and small-scale
variability , respectively. We note that aside from the problems due to differences in
the sampling period for the 5-min and 1-hr data basis, one can achieve a close
correspondence of calculations shown in column (e) with column (d) by proper selection
of the input parameters, 1 TL and W'. The indicated solution to Eq. 5, which may
not be the best, was obtained with 90.1 hours Th=8.5 hours, and W.=0.8.
WA
An alternate set of determinations for the same data sample is shown in
columns (f) and (g) in Table 1. The values of p, in (f) were extracted from Fig. 13 of
Boehm et al (1986). The observed values of recurrence probabilities, P,, determined
by Lund (1973) were transformed into 2x2 contingency tables and the correlations
determined by tetrachoric approximation are shown in column (f). We see that the
tetrachoric correlation coefficients agree reasonably well with the O-U Markov
correlation coefficients in the column (d) for the 1-hour data but are significantly
larger for the 5-minute portion of the data set. Such differences between the
correlation with normalized and unnormalized data are to be expected and can be
managed with appropriate adjustment in the subsequent parameterization of the
Markov and tetrachoric correlation coefficients using Eq. 5. Boehm et al. (1986)
matched the coefficients in (f) using Eq. 5 with input values of T =0.31 hours and tu
=13 hours. The assumed value of W“ was not given in the report but if one uses a
value of 0.71 the results shown in column (8) are obtained, which are in close
agreement with (f).
Thus, we are encouraged to believe with evidence from this limited data set that
Po as determined either by Eqs 1 and 2 or by tetrachoric correlation can be
represented analytically using Eq. 5. The accuracy depends primarily on the efficacy
of techniques used to identify the input parameters T, Th, and W“. A sky-cover
relaxation time on the order of 8.5 hours (Markov pa and 13 hours (Tet. p.) for the
mid-day summer season in Missouri is commensurate with other estimates for mid-
western stations (see Boehm et al., 1986, Table 17). Relaxation times on the order of 5
minutes (Markov) and 15 minutes (Tet.) for the influence of individual cloud motion,
development and decay on CFLOS frequency statistics appear reasonable. Key
questions remain as to the importance of seasonal and geographical variations in these
values and the associated modeling requirements. The appropriate approximations for
an
the weighting factor W. - are another consideration. It is important to note that
Boehm et al assumed the same parameterization of W.- for both time and space (their
Eqs. 4.16 and 4.18). Using these expressions, the corresponding values of sky dome
scale distance are 0.6 for the pe data set in column (d) of Table 1 calculated from Egs.
1 & 2 and 0.45 for the tetrachoric pe data set listed in column (f). This is a rather
reasonable range of values for sky-dome scale distance, for summer daytime in the
midwest, however a value of 1.1 (Boehm et al p. 29) was calculated directly from the
Lund (1973) June, July, Aug, Sept 1969 data set.
Another consideration is that Boehm et al. (1986) found it necessary to modify the
relaxation times (cut in half) to accommodate differences in the persistence probability
given by the sawtooth model in contrast with the O-U process. Thus, validation of 1
for the simulation model applications requires analysis of additional factors that affect
the model differences in persistence, probability.
Proposed analytical techniques for the modeling of persistence probability of cloud
free arc, CFA, and cloud free field of view, CFFOV, at a single site are essentially the
same as those described above for CFLOS. Thus the procedure is as follows:
Determine a representative value for the climatological probability of CFLOS as
discussed in Section 3.1. The input variables are the climatological sky cover
distribution and zenith observing angles.
Determine the climatological probability of CFA or CFFOV using techniques
described in Section 3.2. The input variables are the climatological probability of
CFLOS for the portion of the sky dome of interest, the equivalent floor area or
floor distance and the sky-dome scale distance parameter.
(c) Determine the recurrence probability of CFA or CFFOV as in 3.3 above. The
input variables are the initial and final unconditional probability of CFA and
CFFOV, the O-U Markov relaxation times T and I, and the weighting factor
. As the area or arc length increases, the value of Wº would increase from
the point value associated with a single line of sight to unity for the whole sky
dome.
(d) Determine the CFA and CFFOV persistence probability from Eq. 4 or equivalent
method. The input variables are the recurrence probability of CFA or CFFOV
and the unconditional probability of CFA or CFFOV at time t.
W
3.4 Multisite joint occurrence probability
In the absence of significant spatial asymmetry, there is no compelling reason why
the O-U Markov process will not model successfully the recurrence probability of sky
cover in space as well as in time. thus, the joint probability of CFLOS or CFFOV at
sites 1 and 2 is expressed by Eq. 1 except that po becomes P., the intersite or spatial
correlation of the cloud condition,
Yg = psyo +(1-P52) Ps
(8)
where y, is the concurrent END value corresponding to the unconditional probability
of the event at a remote site. A straightforward extension to this process for the
representation of recurrence probability in both space and time is
Yst =Pst yo +(1 - Post?)Pst
.
where y, is the unconditional event probability at site 1 at initial time and yot is the
unconditional event probability at site 2 at later time t, the space-time correlation
between y, and yet is Pat, and Pat is the conditional probability of y, given y . As
before y Ýst, and'pat áře END Values.
Techniques for modeling the intersite correlation coefficient for CFLOS are given
by Boehm et al. (1986). For reasons discussed above, the resultant-form of the
expression for the spatial variation is the same as the Eq. (5) for temporal variation of
the correlation coefficient so that
Ps =W221 +(1 - W 37
(10)
where o is the spatial correlation coefficient for CFLOS. O, is the correlation
corresponding to the sky-dome scale distance and p, is the correlation corresponding to
intersite scale distance. Thus the spatial correlation for CFLOS is given by a
weighted combination of the small and larger scale correlation components. It is
important to note that a simple exponential decay assumption for the large-scale
spatial circulation is a reasonable alternative for site separation distances up to a few
hundred miles where the correlation remains positive. For example, an assumed
exponential relaxation distance of 229 n mi provides a good fit to the spatial
variability of p, for mid-western stations in both January and July for time periods
near noon and midnight. The plotted data are shown in Figs. 3, 4, 5 and 6 of Boehm,
et al. (1986). In any event, the WSI data base will serve to validate and refine these
relationships for both p, and pa.
SWARAN
As pointed out in Section 1, Lund and Grantham (1980) introduced and tested
an analytic technique for the specification of the joint occurrence probabilities of a
weather event at a number of sites. The input variables are the unconditional event
probabilities at each site and a spatial correlation parameter. Their basic expression
as shown for a group consisting of 3 stations is given as Eq. 3 above. Tests with
independent data gave excellent results for estimates of the joint occurrence of sky-
cover conditions at a group of 7 locations in the midwestern United States.
cawan
mail hest fit for all the pairs of statum..
With a view toward using the same technique as Lund and Grantham (1980) but
with equivalent normal variables and Eq. 1, some trial calculations were carried out
using the same midwest data set. Starting first with individual pairs of stations as a
dependent data set, we established by trial and error a single value of spatial
correlation 1. (ore) which gave the overall best fit for all the pairs of stations. With
this value of relaxation distance, I =450 miles, calculations were made with Eq. 1 of
the joint relative frequencies of skỳ cover equal to or greater than 0.8 in winter at the
at the indicated pairs of stations shown in Table 2. The observed joint frequency, P..
as extracted from Table 5 of Lund and Grantham are given in the column labeled
"observed frequency" on the right hand side. The estimates from Eq. 2 using in turn
each station of the pair as the initial site are given in the adjacent columns on the
right. The results are very good, but even so not as good as achieved by Lund and
Grantham with Eq. 3 using the data set as independent rather than dependent test
data. However, even with round off error due to course interpolation of look-up tables,
the results appear to offer encouraging tests of the concept of applying Eq. 1 to the
spatial recurrence estimates.
A graphical presentation of the observed and estimated frequency data for the
pairs of stations is given in Fig. 1. Note the significant underestimates for the station
pairs with the shortest separation distances. Application of the two-parameter
representation the spatial correlation (see Eq. 10), instead of the single coefficient used
here for the trial calculations would enhance the results.
3.5 Determination of the joint occurence probability of sky cover at more than two
sites.
Lund and Grantham (1980) extended existing techniques for the determination of
joint occurence probability for a pair of sites to multiple sites through a stepwise
procedure. Given the probability of joint occurrence at an initial pair of stations,
another station is added to the determination by calculation of the joint occurence
probability of a neighbor station with the closest of the two original stations, and so
on. With some modification let us proceed in a similar way using the Markov
assumption (Eq. 1) as a basis for the determinations.
As a first step let us assume that the conditional probability for the second pair
), is independent of the conditional probability of the first pair, P(j:i).
Thus the joint probability of occurence at the 3 stations, P(ijk), is given
Pijk) =P(i)*(P(j:i)*P(k:j),
(11)
where P(i) is the unconditional probability at the initial station. In a similar manner
the determination can be extended to 4 or more sites.
A preliminary test of the concept was carried out using the data base given by
Lund and Grantham (1980). Data were summarized for the frequency of sky cover
greater than or equal to 8 tenths in the winter months at 7 stations in the central
United States. Determinations of the joint occurrence frequency for combinations of 4
stations are given in Table 3, and for combinations of 6 stations in Table 4.
Shown for comparison in Table 3 are (1) calculations of estimated joint frequency
by Lund and Grantham (1980) using equations of the form given by Eq. 3, (2)
calculations using Eq. 11 above and the Markov joint occurence frequency calculations
for station pairs (Eq. 8), (3) calculations using Eq. 11 with the observed joint event
occurrence frequency for the individual station pairs, and (4) calculations assuming a
Markov process as in (2) except increasing the assumed scale distance for all pairs
after the first to account for the interdependence of station pair correlations.
CCC
Similar results for various combinations of 6 of the 7 pairs are shown in Table 4.
As expected the frequency estimates using Eq. 11 without adjustment tend to be
low due to the fact that the conditional occurence frequency for adjacent pairs of sites
are not independent, but the systematic errors are not large in this case. Lund and
Grantham (1980) introduced an adjustment which modified the spatial correlation
coefficient as a function of the number of sites which was compatible with other
aspects of their modeling procedure. Initial evidence indicates that slight adjustment
in the Markov correlation coefficient or relaxation distance, which is held constant
regardless of the number of sites, will work well for this sky cover data set and is
compatible with the use of alternate Eq. 11 above. For example, if one simply
increases the relaxation distance by 15 percent for all station pair calculations beyond
the initial pair, the results listed in the last column on the right are obtained.
It remains to be seen how well the O-U Markov concept will apply to multisite
estimates with independent data. These pilot studies and the immediate extension of
the technique using the space-time correlation coefficient as given by Eq. 9 are to be
the subjects of a follow-on memorandum.
4. Summary comments
The purpose of this memorandum is to review briefly some aspects of cloud
modeling procedures that are relevant to problems of ground to space, line-of-sight
communication. Our interest is to set the stage for the application of the WSI data
base to the validation and refinement of existing cloud modeling techniques and the
development of alternative modeling procedures.
In summary, the following list is an attempt to isolate some of the individual
components of the cloud modeling procedures that can and should be validated as
soon as practicable with the WSI data. As expected, many of these components are
important for both the simulation models and the purely analytic models. The initial
list of validation items includes:
a. Lund-Shanklin and Allen-Mahlick techniques for CFLOS estimates as a
function of sky cover and zenith angle.
b.
Determination of the probability of a given fraction of sky cover for a limited
area of the sky dome or along a given arc segment as a function of sky-dome
scale distance and CFLOS probability (special application of the BAA).
c.
Determination of pt as a function of location, season and time of day.
d.
Determination of Ps as a function of location, season, and time of day.
e.
Determination of Pst as a function of location, season, and time of day.
Validation and refinement of Eqs. 2, 8 and 9 for recurrence probability of
CFLOS, CFFOV and CFA.
g.
_
Validation and refinement of Eq. 4 for persistence probability of CFLOS,
CFFOV and CFA.
h. Direct comparison of CFARC and CFLOS4D simulation model output with
WSI data for prescribed satellite configurations.
i.
Direct comparison (independent data) of CFA and CFFOV analytical model
output with WSI data.
WA
5. References
Allen, J. H. and J. D. Mahlick, 1983: The frequency of cloud-free viewing
intervals, Preprint 21st Aerospace Science Meeting, 10-13 Jan. 1983, Reno, Nev.
Boehm, Albert, Irving A. Gringorten and Charles F. Burger, 1986:
Simulated duration of CFLOS from Multiple sites to a satellite, private
communication of AFGL Technical Memorandum No. 125.
Burger, Charles F. and Irving I. Gringorten, 1984: Two-dimensional
modeling for lineal and areal probability of weather conditions, AFGL-TR-84-0126,
Environmental Research Papers No. 875, 1984.
Gringorten, Irving I., 1972: Conditional probability for an exact noncategorized
initial condition, Monthly Weather Review, Vol. 100, No. 11, 796-798.
Lund, I. A., and M. D. Shanklin, 1973: Universal methods for estimating
probabilities of cloud-free-line-of-sight through the atmosphere, J. of Appl. Meteor,
12, 28-35.
Lund, I. A., D. D. Grantham, and R. E. Davis, 1980: Estimating probabilities
of cloud-free-fields-of-view from the earth through the atmosphere, J. Appl. Meteor.,
19, 452-463.
Lund, I. A., 1973: Persistence and recurrence probabilities of cloud- free and
cloudy lines-of-slight through the atmosphere, J. Appl. Meteor., 12, 1222-1228.
Lund, I.A., and D. D. Grantham, 1980: Estimating the joint probability of a
weather event at more than two locations, J. Appl. Meteor., 19, 1091-1100.
Xer
lysty..
Table 1. Line of sight recurrence probability statistics based on Columbia, Missouri,
summer data (Lund, 1973)
(a)
(b)
(c)
(e)
(g)
(a)
Pp
(d)
ft
(9)
(b)
P(5-min)
(F)
pt
Time
(min)
Time
P (5-min)
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Tetrachoric
1.00
.87
1.00
.88
1.00
.84
.82
.93
1.00
.87
.85
.82
.93
1.00
.72
.68
.61
.59
.78
.70
.65
.82
.79
. 79
.78
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.60
.77
.74
.73
.76
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.65
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.73
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.72
.75
.68
(hours)
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o
.77
.51
.71
.71
.67
.64
.44
.66
.67
.59
.63
.56
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