case fil copy - nasa · mately 1000 (octal) words. ... nbro number of trajectories already...
TRANSCRIPT
NASA TECHNICAL
MEMORANDUM
CMw»CO
NASA TM X-3529
CASE FILCOPY
THREE-DIMENSIONAL RANDOM EARTH ATMOSPHERES
FOR MONTE CARLO TRAJECTORY ANALYSES
Janet W. Campbell
Langley Research Center
Hampton, Va. 23665
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON D. C. • AUGUST 1977
https://ntrs.nasa.gov/search.jsp?R=19770023097 2018-07-17T06:55:27+00:00Z
1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.
NASA TM X-35294. Title and Subtitle
THREE-DIMENSIONAL RANDOM EARTH ATMOSPHERESMONTE CARLO TRAJECTORY ANALYSES
7. Author(s)
Janet W. Campbell
9. Performing Organization Name and AddressNASA Langley Research CenterHampton, VA 23665
12. Sponsoring Agency Name and AddressNational Aeronautics and Space AdministratWashington, DC 205^6
5. Report DateFOR August 1977
6. Performing Organization Code
8. Performing Organization Report No.
L-1062310. Work Unit No.
506-26-30-08
11. Contract or Grant No.
13. Type of Report and Period CoveredTechnical Memorandum
ion14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
A set of four computer tapes containing random three-dimensional Earth atmo-spheres is available for Monte Carlo trajectory analyses. The four tapes - one foreach season - contain sufficient atmospheric tables to allow over 1400 replicationsof any trajectory below an altitude of 99 km. The atmospheres were provided by anempirical model designed to generate random atmospheres whose distributions matchthose in a data base of over 6000 sounding-rocket measurements. A readily imple-mentable means of linking the tapes to any existing trajectory simulation computerprogram is. described. It involves the addition of three subroutines which arelisted in an appendix.
17. Key Words (Suggested by Author(s))
Three-dimensional atmosphereRandom atmospheresMonte CarloTrajectory simulation
/•
19. Security Qassif. (of this report) 20. Security Classif. (of this
Unclassified Unclassified
18. Distribution Statement
Unclassified - Unlimited
Subject Category 88
page) 21. No. of Pages 22. Price*
45 $4.00
* For sale by (he National Technical Information Service, Springfield. Virginia 22161
THREE-DIMENSIONAL RANDOM EARTH ATMOSPHERES FOR
MONTE CARLO TRAJECTORY ANALYSES
Janet W. CampbellLangley Research Center
SUMMARY
A set of four magnetic computer tapes containing random global Earth atmo-spheres is available for Monte Carlo trajectory analyses below an altitude of99 km. The four tapes - each representing a different season - contain atmo-spheres in sufficient quantity to permit over 1400 independent replications ofany trajectory.
The atmospheres were generated by a statistical atmosphere model based onover 6000 rocket and high-altitude soundings. The model was constructed empiri-cally to generate temperatures, densities, and pressures whose distributionsmatch those in the data, and whose vertical gradients are likewise statisticallysimilar to gradients in the data.
A readily implementable means of interfacing the tapes with an existingtrajectory simulation program is described. The method involves the additionof three subroutines, which are linked to the trajectory "program through simplecalling statements. The core storage required for the subroutines is approxi-mately 1000 (octal) words.
INTRODUCTION
There is a recognized need among aircraft and spacecraft designers for ameans of estimating the impact of atmospheric variability on a vehicle's perform-ance. Standard Atmospheres (refs. 1 to 3) are commonly used to calculate trajec-tories, but the Earth's atmosphere is variable and always differs from any stan-dard. For example, high-altitude densities at a fixed location can vary by over100 percent within the same season. Densities vary by at least 10 percent atall altitudes.
The Random Earth Atmosphere Computer Tapes (REACT) are a set of four mag-netic tapes containing typical nonstandard densities and temperatures for athree-dimensional global atmosphere. The tapes are intended to be used alongwith a trajectory simulation program to generate samples of trajectories passingthrough different random atmospheres. Such samples can help define the variabil-ity in a vehicle's performance parameters resulting from atmospheric variations.Any vehicle (spacecraft or aircraft) can be studied by using the tapes in theflight regime where the altitude is below 99 kilometers.
The REACT atmospheres are profiles of temperatures and densities generatedby the empirical random atmosphere (ERA) statistical model (ref. 4) which was
developed empirically from over 6000 meteorological rocket and high-altitudesoundings. Pressure and other atmospheric properties (e.g., viscosity, speed ofsound, etc.) can be calculated from the temperatures and densities. The atmo-spheres on the tapes are described as typical in that their statistical distri-butions match those of the sounding data at corresponding altitudes, latitudes,and seasons.
The tapes represent four seasons as follows:
Tape
12
. 34
Months
March to MayJune to August
September to NovemberDecember to February
Tape identification
SPRINGSUMMERAUTUMNWINTER
(The season designation refers to that of the northern hemisphere. For example,the."SPRING" tape actually contains autumn atmospheres for southern hemispherelocations.)
Each tape contains sufficient atmospheric data to permit up to 361 indepen-dent replications of the same trajectory, the actual number depending on therange of longitudes covered by .that trajectory. The trajectory's performanceparameters (for example, range, surface heating rates, dynamic pressures, struc-tural loads, etc.), which are of interest to the vehicle designer, will varyfrom one replication to the next because of the atmosphere's variability. Thecollection of parameter values resulting from the simulations forms a randomstatistical sample which can be used to estimate that parameter's underlyingprobability distribution. Such distributions are needed in estimating the prob-ability of exceeding existing design values, in establishing new design values,and in designing adequate guidance and control systems. Sources of error otherthan the atmosphere can be incorporated and a thorough error analysis performed.
Because the three-dimensional REACT atmospheres introduce certain densitygradient effects which are not provided by the standard atmospheric models, adiscussion of horizontal and vertical density gradients is included. Densitygradients on the tapes are compared with those in the data and with proposed"design" gradients in reference 5. Vertical density gradients are consistentwith those found in the data. Since the data base consists of isolated verticalprofiles, inferences about horizontal density gradients cannot be made directlyfrom the data. Horizontal gradients were controlled by assuming a minimum dis-tance between uncorrelated profiles of 600 nautical miles (ref. 6). A fairlyhigh percentage of the REACT density gradients exceeded the "design" gradientsof reference 5, even though the latter were based on this same assumption. Itis believed, however, that the REACT density gradients are more representativeof realistic gradients.
Three subroutines which can be used to link REACT to a trajectory programare described and a listing of the subroutines is given in appendix A. Appen-
dix B presents a brief description of the empirical atmosphere model (ref.used to generate the tapes.
SYMBOLS AND ABBREVIATIONS
ATMOS one of three subroutines used to interface REACT with trajectoryprogram
COMMON/LINK/ common storage area containing block information from TAPE10shared by ATMOS, INITBLK, and READBLK subroutines
D atmospheric density,
DL increment used to convert longitude to "longitude" on TAPE10, deg
g acceleration due to gravity, m/sec^
INITBLK one of three subroutines used to interface REACT with trajectoryprogram
k refers to a season in northern hemisphere; k = 1 (spring), k = 2(summer), k = 3 (autumn), and k = 4 (winter)
N in subroutine READBLK, refers to next block to be read from TAPE10
NBR simulation or replication number
NBRO number of trajectories already simulated from TAPE10 on previous runof program
NMAX maximum number of independent trajectories which can be simulated byusing TAPE10
NO number of TAPE10 block currently stored in COMMON/LINK/
P atmospheric pressure, N/m^
Pg2 1962 Standard Atmosphere pressure, N/m^
R gas constant used in equation of state, 8314.34 J-kmol~^-K~^
REACT Random Earth Atmosphere Computer Tapes
READBLK one of three subroutines used to interface REACT with trajectoryprogram
RHO three-dimensional array containing random atmospheric densities fromTAPE10, kg/m3
S speed of sound, m/sec
SEASON alphanumeric word used to identify REACT tape (first record on eachtape)
s standard deviation of atmospheric temperature estimated from soundingdata, K
T atmospheric temperature, K
f mean atmospheric temperature estimated from sounding data, K
TAPE 10 REACT tape linked to a trajectory program at any given time
TAU three-dimensional array containing random atmospheric temperaturesfrom TAPE10, K
W mean molecular weight of air, 28.964 kg-kmol~^
XLAT latitude, deg
XLATI initial latitude of trajectory, deg
XLATO latitude locating RHO and TAU arrays, deg
XLOMAX maximum longitude of trajectory, deg
XLOMIN minimum longitude of trajectory, deg
XLONG "longitude" on TAPE10, deg
XLONGI initial longitude of trajectory, deg
XLONGO "longitude" locating RHO and TAU arrays, deg
Z altitude, m
z altitude, km
X longitude, deg
p atmospheric density, kg/m^
P62 1962 Standard Atmosphere density, kg/m^
<t> latitude, deg
TAPE SPECIFICATIONS
The tapes are 1.27-cm (0.5-in.) nine-track magnetic tapes written in binary(odd parity) mode, using a density of 629.9 characters per cm (1600 characters perin.), and labeled with ANSI standard labels. They were generated on a CONTROLDATA CYBER 175 computer which uses a physical record size of 512 60-bit words.
If it is necessary to read the tapes in double precision (for example,where the computer being used has a word size smaller than 60 bits), the informa-tion should be converted to single precision and transferred to a tape compati-ble with the computer to be used. Since the densities and temperatures are ran-dom, the degree of accuracy lost in reducing word size would not be significant.
GENERAL CONTENTS OF EACH TAPE
The Standard Atmosphere tables (refs. 1 to 3) are one-dimensional tablesgiving atmospheric properties as functions of altitude. The REACT atmospheresare three-dimensional tables which give densities and temperatures as functionsof altitude, latitude, and longitude. REACT properties at a given altitude andlatitude are modeled after data taken at the same altitude and latitude. Thus,these two coordinates, altitude, and latitude, correspond exactly to the samecoordinates of the trajectory. The "longitude" coordinate used in the REACTtables does not correspond to physical Earth longitudes but, instead, representsa longitudinal measure, in degrees, from the midnight meridian; that is, thezero "longitude" is interpreted as midnight and is not the Greenwich meridian.Moving eastward, each increase of 15° in "longitude" represents the passage of1 hour in local time as determined by the Sun angle. At a "longitude" of 180°,for example, the atmospheres are characteristic of those at noon. The termreferring to a coordinate on the REACT tapes will henceforth be placed in quota-tion marks to distinguish it from the physical longitude of-the trajectory.
It is assumed that only one of the four tapes is linked to the trajectorysimulation program at a given time, and this tape will be called TAPE10. Sincethe four tapes have identical formats, the sequence of statements used to readTAPE10 is appropriate for reading any of the four tapes.
Each tape contains one file which is composed of 2161 logical binaryrecords. That is, in order to read the entire tape, 2161 unformatted READ state-ments must be executed. The first record contains one Hollerith word SEASONwhich is one of the following:
SEASON = <
TOHSPRINGbbbb
lOHSUMMERbbbb(b = blank) (1)
lOHAUTUMNbbbb
lOHWINTERbbbbv.
depending on the tape.
The remaining 2160 records contain blocks of density and temperaturetables, each extending to an altitude of 99 km and covering a specific geographi-cal region, which is 30° x 30° in latitude and "longitude." The 2160 blocksform 30 complete global atmospheres when pieced together. In general, many rep-licates of a given trajectory can be made by using the same global atmosphere bytranslating the trajectory 30° eastward (2 hours in time). This procedure isexplained in more detail in a later section.
Each of the blocks must be read with a statement of the form
READ (10) NO,XLATO,XLONGO,RHO,TAU
where NO is the block number (NO = 1, . . ., 2160); XLATO and XLONGO are a ref-erence latitude and longitude which locate the block geographically; and RHO isan array of densities and TAU is an array of temperatures, both dimensioned34 x 4 x 4. The elements RHO(I,J,K) and TAU(I,J,K) are, respectively, atmo-spheric density, in kg/m3, and atmospheric temperature, in K, at
Altitude, km: z = 3(1 - 1) ( 1 = 1 , . . . ,
Latitude, deg: <|> = XLATO + 10(J - 1) (J = 1, . . ., 4)\ (2)
"Longitude," deg: X = XLONGO + 10(K - 1) (K = 1, . . ., 4)
Within each array the values are located every 3 km in altitude and every 10° inlatitude and "longitude." From the dimensions of the RHO and TAU arrays, it fol-lows that these tables cover the three-dimensional volume defined by
0 z z £ 99 km ^|
XLATO g <|> g XLATO + 30° \ (3)
XLONGO ^ X g XLONGO + -30°J
Figure 1 shows how the 2160 blocks fit together to cover the "longitude"-latitude plane. The number appearing in each block indicates the block numberNO. The reference point (XLONGO,XLATO) for a block is the grid point in thelower left-hand corner of that block. Blocks were stored on the tapes beginningat XLATO = -90° and XLONGO = 0°, and proceeding northward, keeping XLONGOfixed and increasing XLATO in increments of 30° until XLATO = 60°. Then XLONGOis increased by 30° and the same six values of XLATO are repeated. As theblocks encompass the Earth 30 times, the value of XLONGO goes from 0° to 10 770°(= 30 x 360° - 30°). It was more expedient from a programing standpoint toallow XLONGO to increase monotonically rather than set it back to 0° after eachrevolution. Near the equator the blocks cover an area approximately 3300 km by3200 km and near the poles they cover an area approximately 3300 km by 860 km.
The computer core storage required to read each block is 1091 words orslightly more than 2000 octal words. It should only be necessary to read andstore one block at a time. As the trajectory enters the three-dimensionalregion associated with a particular block, a subroutine should be called toread the appropriate block from TAPE10. The trajectory simulation can then con-tinue through this region until a boundary is reached. As the trajectory passesinto a new region, the subroutine should be called again to read the next appro-priate set of RHO and TAU values.
REPLICATION OF TRAJECTORIES
It is assumed that the user will want to maximize the number of independentreplicates of a trajectory which can be simulated by using TAPE10. That is, theuser will have a specific trajectory, such as a space shuttle entry trajectory,and will want to replicate this trajectory as many times as possible by usingdifferent random atmospheres from TAPE10.
Figure 2 shows the ground track of an entry trajectory from a 28.5°inclined orbit. The initial longitude and latitude, when z = 99 km, are -175°and -5°, respectively, and the final longitude and latitude are -119° and 36°,respectively. Figure 3 shows a set of independent replicates of this trajectoryrelative to the atmospheres on TAPE10. Notice that the first trajectory beginsat a "longitude" of 0°. The second begins at 30°, the third at 60°, and soforth. In simulating trajectories using different atmospheres from TAPE10, rep-licates will be independent provided that no two simulations use the same ele-ment of a RHO or TAU array to compute atmospheric properties. A 30° translationin "longitude" should result in trajectory replicates which are essentially inde-pendent. If ground tracks for different replicates approach or intersect oneanother when plotted on a latitude-"longitude" plane such as that in figure 3,the two replicates are not strictly independent. However, if their altitudes atpoints of intersection are substantially different, the dependence should beweak and can be ignored.
Since "longitudes" on TAPE10 actually represent time lines, the first tra-jectory in figure 3 starts at midnight, the second at 2 a.m., the third atk a.m., and so forth. If a large sample of replicates is generated or if thenumber of replicates is a multiple of 12, then the effect of local time willaverage out.
INCORPORATING REACT INTO A TRAJECTORY SIMULATION PROGRAM
In order to interface a trajectory program with REACT, it is suggested thatthree subroutines, such as those listed in appendix A, be added. A flow chartutilizing these subroutines is shown in figure *J. The first INITBLK is calledonce at the beginning of each trajectory simulation to initialize the block. Itdetermines the trajectory's initial latitude and "longitude" relative to theblocks on TAPE10, and then calls the second subroutine READBLK which reads thecorrect block of information from TAPE10.
A third subroutine ATMOS is called at each point along a trajectory when-ever atmospheric properties (i.e., density, temperature, pressure, and speed ofsound) are needed. Presumably, ATMOS will replace any atmospheric model alreadyin the trajectory program and, as it will be called frequently, it should be asefficient as possible. ATMOS interpolates in the RHO and TAU arrays to give anatmospheric density and temperature at the altitude, latitude, and longitudespecified in its calling statement. Atmospheric pressure is then obtained fromthe equation of state, and the speed of sound is calculated from the temperature.(See formulation of ATMOS.) Before interpolating, a test is made to determinewhether the trajectory's altitude, longitude, and latitude are still within therange covered by the block currently stored. If it is outside the longitude-
latitude range, ATMOS calls READBLK to locate and read the correct block. Ifthe altitude is above 99 km, ATMOS calls an alternate atmosphere model which theuser must supply. In most instances it is sufficient to let this be the nonran-dom atmosphere previously used to calculate trajectories.
The subroutines are written so as to be readily incorporated into an exist-ing trajectory program. The only link between the trajectory program and thesubroutines is through the calling statements to the subroutines INITBLK andATMOS whose arguments are in terms of true longitudes. Conversion to "longi-tudes" relative to the coordinates on TAPE10 is handled automatically by thesubroutines. Replication of trajectories is also handled automatically in thatINITBLK calculates the initial "longitude" of each successive replicate, trans-lating it 30° eastward from its previous initial "longitude." A more detaileddescription of the formulation of each subroutine follows.
SUBROUTINE INITBLK - FORMULATION
INITBLK must be called once prior to each replication of a trajectory. Thebasic purpose of INITBLK is to initialize the COMMON storage area
COMMON/LINK/ DL.NO.XLATO.XLONGO.RHOGM,1*) ,TAU(3M,4)
which is shared by all three subroutines. (See appendix A.) During a simula-tion, NO, XLATO, XLONGO, RHO, and TAU will contain the most recent block ofinformation read from TAPE10, and DL is an increment used to convert longitudesto "longitudes" on TAPE10 by the translation
"Longitude" = Longitude + DL (4)
The calling statement for INITBLK is
CALL INITBLK (NBR, NMAX, XLATI, XLONGI, XLOMIN, XLOMAX)
where NBR = 1, 2, 3, • • • is the number of the replication about to be simu-lated; NMAX is an output variable which gives the maximum number of independentreplications which can be simulated using TAPE10; XLATI and XLONGI are the ini-tial latitude and longitude, respectively, of the trajectory; and XLOMIN andXLOMAX are the minimum and maximum longitudes, respectively, of the trajectory.All arguments except NBR should remain constant throughout the simulations.
The first time INITBLK is called, the calling program must set NBR = 1.At this time, DL is initialized so that the minimum longitude XLOMIN correspondsto a "longitude" on TAPE10 of 0.001°. (This value is effectively zero but asmall positive increment is used to prevent a negative zero "longitude" fromoccurring because of machine round-off error.) Thus, the increment DL is calcu-lated as
DL = 0.001 - XLOMIN (5)
The initial longitude XLONGI becomes the "longitude"
XLONG = XLONGI + DL (6)
on TAPE10, and the maximum number of independent replicates is calculated as
NMAX = HO 80° - XLOMAX - DL] + 1 (7)L 30 J
where here and in later formulas the brackets indicate the "greatest integer"function defined by
^greatest integer £ x if x £ 0M = ' (8)
( smallest integer ^ x if x < 0
INITBLK then sets NO = 0 and calls READBLK to read the block covering the point(XLATI, XLONG), where XLONG is defined in equation (6).
On subsequent calls to INITBLK, NBR must be greater than 1. At this timeDL is increased by 30° and the new XLONG is calculated by equation (6). Aftereach replication is simulated and before calling INITBLK for the next replica-tion, NBR should be tested to make sure it does not exceed NMAX.
As an example, consider the trajectory shown in figure 2. The first callto INITBLK would be
CALL INITBLKd,NMAX,-5.0,-175.0,-175.0,-119.0)
When INITBLK returns control to the calling program, the value of NMAX will be359 and will remain constant throughout the run. Thus, a sample of 359 indepen-dent replications of this trajectory can be generated from each tape or 1H36 rep-lications by using all four tapes. Of course, the user can always make fewerreplications.
As another example, figure 5 shows the groundtrack of an entry trajectoryfrom a 10^o inclined orbit. The first call to-INITBLK for this trajectory wouldbe
CALL INITBLKd,NMAX,73-0,4.0,-123.0,1.0)
Note that the minimum longitude (-123°) is neither the initial nor the final lon-gitude in this case. Here INITBLK returns with NMAX = 356, and the possiblereplications are illustrated in figure 6.
SUBROUTINE READBLK - FORMULATION
READBLK is called both from INITBLK, to initialize RHO and TAU at the begin-ning of each trajectory replication, and from ATMOS whenever the trajectoryenters a region requiring new RHO and TAU arrays. The calling statement toREADBLK is
CALL READBLK(XLAT,XLONG)
where -90° < XLAT $ 90° and 0° 1 XLONG $ 10 800° are the present latitudeand "longitude" relative to the blocks on TAPE10.
At the time READBLK is called, NO is the number of the previous block readfrom TAPE10. READBLK uses XLAT and XLONG to determine N, the number of the nextblock to be read, by the formula
N = 6rXLONG] + [XLAT + 90] + T (9)[30 J I 30 J
For example, if- XLAT = -18.6° and XLONG = 239-5°, then
N = 6[?.98] + [2.38] + 1 = 42 + 2 + 1 = 45 (10)
READBLK now compares N to NO + 1. If N < NO + 1, READBLK backspaces overblocks NO to N, and then reads block N. If N = NO + 1, then it simply readsthe next block on TAPE10. If N > NO + 1, READBLK skips blocks NO + 1 to N - 1and then reads block N. In reading block N, a new set of values for NO, XLATO,XLONGO, RHO, and TAU is thereby stored in COMMON/LINK/.
SUBROUTINE ATMOS - FORMULATION
ATMOS is called at each step along a trajectory when atmospheric propertiesare needed. Its calling statement is
CALL ATMOS(Z,XLAT,XLONG,D,T,P,S)
where Z is the altitude in meters, XLAT and XLONG are the true latitude and lon-gitude in degrees, and D, T, P, and S are output variables which upon returningto the calling program will contain, respectively, atmospheric density in kg/nP,temperature in K, pressure in N/m^, and speed of sound in m/sec for the location(Z,XLAT,XLONG).
ATMOS first tests to see whether Z is in the range 0 g Z 99 000. Ifnot, ATMOS calls a subroutine (which the user must supply) to get atmosphericproperties outside this range. In the version of ATMOS in appendix A, the sub-routine called is AT62 which is a subroutine used to get 1962 Standard Atmo-sphere properties.
If Z is in the correct range, then ATMOS tests XLAT and XLONG to determinewhether
XLATO g XLAT 1 XLATO +30 (11)
and
XLONGO <. XLONG + DL < XLONGO + 30 (12)
10
If either of these conditions is not satisfied, READBLK is called and the test isrepeated.
Once Z, XLAT, and XLONG are found to be in the correct range, D and T areobtained by interpolation by using RHO and TAU as three-way tables. All inter-polations are linear except in vertical planes of the RHO array. In this direc-tion linear interpolation is performed on the logarithm of density.
Once D and T are obtained, P is computed from the equation of state
P = 521 (13)W
where R is the universal gas constant (8314.34 J-kmol~1-K~1) and W is the meanmolecular weight of air. Although W begins to vary above about 90 km, for thepurposes of REACT it was assumed constant (28.964 kg-kmol~1).
The speed of sound is computed as
S = (1.4 RTV/2 (14)
where the constant 1.4 is the ratio of the specific heat of air at constant pres-sure to that at constant volume. Equations (13) and (14) are the same as thoseused in the U.S. Standard Atmosphere (refs. 1 to 3).
RESUMING AN INTERRUPTED SET OF SIMULATIONS
.Suppose a user wishes to run 350 replications of a trajectory where350 g NMAX, and for some reason only 100 replications were run the first timehis program was executed. If the original program is resubmitted to run theremaining 250 replications, there will have to be some assurance that none ofthe previously run simulations are duplicated. To effect this, the 101st repli-cation should begin on TAPE10 at a "longitude" 30° east of where the 100th repli-cation began.
In general, let NBRO be the total number of replicates already run fromTAPE10, and let XLATI, XLONGI, XLOMIN, and XLOMAX be the arguments of INITBLK asdefined earlier. Before calling INITBLK for the first time, subtract 30 x NBROfrom XLOMIN (that is, XLOMIN = XLOMIN - 30*NBRO). Then call INITBLK as usualwith NBR = 1. Upon returning from INITBLK, the correct block from TAPE10 willhave been read in order to resume simulations, and NMAX will be the maximum num-ber of independent replications which can be simulated by using the remainder ofTAPE10. That is, if NMAX were, for example, 359 when the first 100 trajectorieswere run, then this time NMAX will be 259. The user may wish to convert back tothe original count after INITBLK is called .the first time. This can be accom-plished by setting NMAX = NMAX + NBRO, and numbering the first simulation ofthe present run NBR = NBRO + 1.
A general programing sequence which would work whether one is resuming simu-lations or starting a new set would be the following:
11
READ 400, NBRO,XLATI,XLONGI,XLOMIN,XLOMAX400 FORMAT(I5,4F10.3)
REWIND 10READ(10) SEASONXLOMIN = XLOMIN - 30.«NBROCALL INITBLK(1,NMAX,XLATI,XLONGI,XLOMIN,XLOMAX)NBR = NBRO + 1NMAX = NMAX + NBRO
10 WRITE 401, NBR,SEASON401 FORMAT(20H1 SIMULATION NUMBER,13,9H SEASON =, A10)
(complete trajectory simulation)
NBR = NBR + 1IF(NBR - NMAX) 20,20,30
20 CALL INITBLK(NBR,NMAX,XLATI,XLONGI,XLOMIN,XLOMAX)GO TO 10
30 .
(wind up)
STOPEND
Note that when NBRO = 0, this procedure is equivalent to starting a new set ofsimulations.
DENSITY GRADIENTS
Density gradients along a vehicle's flight path can be as important as theactual density. For example, an extreme positive density gradient during entrycan create higher surface heating loads than either a uniformly high or low den-sity atmosphere. A set of "design" gradients for horizontal and verticalchanges in density has been published in reference 5 for use in studying theeffects on a vehicle's performance of extreme gradients in atmospheric densityalong a flight path. The horizontal and vertical gradients in density found onthe REACT tapes are compared with these design gradients and with measured gradi-ents in the atmospheric data base described in appendix B.
The ERA (Empirical Random Atmosphere) model used to generate the REACTatmospheres was designed to provide atmospheric properties with realistic verti-
12
cal gradients. That is, one of the objectives in developing the model was toimitate vertical density gradients in the data base. Thus, vertical densitygradients in the REACT atmospheres are inherently similar to those found in thedata. Figures 7(a) and 7(b) compare the mean and standard deviation of verticaldensity gradients in the data with corresponding properties of modeled gradi-ents. Table I shows the percentage of vertical gradients in both the data andthe model which exceed the "design" gradients of reference 5.
In the case of horizontal density gradients, the data base was of no usesince it consists of isolated one-dimensional soundings. In creating the REACTtapes, control of horizontal gradients was exercised by the choice of longitude-latitude grid size and by correlating density profiles which are relativelyclose together. A telephone contact with one of the contributors^ to refer-ence 5 revealed that horizontal design gradients were based on the conclusionin reference 6 that hot and cold atmospheric regimes can occur within 1111.2 km(600 n. mi. or 10° of latitude) of one another. For the construction of thethree-dimensional REACT atmospheres, this statement was interpreted as meaningthat the minimum distance between two statisticlly independent profiles is1111.2 km.
A spacing of 10° was chosen for the longitude-latitude grid. Accordingly,constant-latitude lines are 1111.2 km apart and, therefore, profiles on differ-ent latitude lines could be treated as independent. The longitude lines, on theother hand, are 1111.2 km apart at the equator but converge to a single pointat each pole. Thus, profiles at some adjacent grid points were closer than1111.2 km and these had to be correlated. This was accomplished by first gen-erating independent profiles spaced 1111.2 km apart along a constant latitudeline, and interpolating linearly between these to obtain profiles at the gridpoints. This procedure for correlating adjacent profiles is consistent with theoverall scheme of interpolating between horizontal grid points to get a densityor temperature at any arbitrary location.
A comparison was made between the horizontal design density gradients ofreference 5 and the horizontal density gradients in the REACT atmospheres.Table II lists the percentage of horizontal density gradients on the REACT tapeswhich exceed the design gradients. A significant disparity exists because thereis a greater difference between hot and cold atmospheres in the REACT model,even within the same latitude-season category, than in the model used to con-struct the design gradients in reference 5. Since the REACT model is based onan extensive amount of data, it is believed to be more accurate in its represen-tation of these differences.
CONCLUDING REMARKS
The Random Earth Atmosphere Computer Tapes (REACT) are a set of four magne-tic tapes representing four different seasons which can be used to simulatespacecraft and aircraft trajectories through nonstandard atmospheres character-istic of their respective latitudes and seasons. The atmospheric temperaturesand densities on any one tape (season) at the same latitude and altitude form a
nS. Clark Brown, NASA Marshall Space Flight Center, June 7,
13
random sample whose statistical properties match those of observed temperaturesand densities in a data base of over 6000 sounding-rocket and high-altitudemeasurements.
A method is described whereby, with the addition of three subroutinesinvolving approximately 4000 octal storage locations, the tapes can be linked toany existing trajectory simulation program. Depending on the longitudinal rangeof a particular trajectory, between 349 and 361 independent replications of thattrajectory can be made with each tape. Thus, approximately 1400 replicates canbe generated by using all four tapes.
As a trajectory is simulated, a series of "blocks" is read from one of thetapes. Each "block" contains arrays of atmospheric densities and temperatureswhich span a volume 99 km deep, 30° wide in longitude, and 30° long in latitude.Values in the arrays are located at grid points every 3 km in altitude, andevery 10° in longitude and latitude. To obtain atmospheric properties at anyarbitrary point on a trajectory, the subroutines interpolate by using the arraysas three-way tables. Interpolation is linear in all directions except in verti-cal planes of the density array, where linear interpolation is applied to thelogarithm of density.
Vertical density gradients are consistent with those found in the database. Horizontal density gradients are controlled by assuming a minimum dis-tance between uncorrelated profiles of 1111.2 km. In the case of both horizon-tal and vertical density gradients, a significant percentage of the REACT atmo-spheres exceed the "design" gradients of reference 5. It is believed, however,that the REACT density gradients are more representative of realistic gradients.
Langley Research CenterNational Aeronautics and Space AdministrationHampton, VA 23665May 27, 1977
APPENDIX A
SUBROUTINE LISTINGS
The following listings are suggested versions of the three subroutinesINITBLK, READBLK, and ATMOS:
SUBROUTINE INITBLK(NBR,NMAX,XLATI,XLONGI,XLOMIN,XLOMAX):
COMMON/LINK/ DL,NO,XLATO,XLONGO,RHO(34,M) ,TAU(34,M)IF(NBR-1) 1,1,2
1 DL = .001-XLOMINXLONG=XLONGI+DLNMAX=(10800.-XLOMAX-DL)/30.+1.N0=0CALL READBLK(XLATI,XLONG)RETURN
2 DL=DL+30.XLONG=XLONGI+DLCALL READBLK(XLATI,XLONG)RETURNEND
SUBROUTINE READBLK(XLAT,XLONG):
COMMON/LINK/ DL,NO,XLATO,XLONGO,RHO(34,4,4) ,TAU(34,4,4)NCOLS=XLONG/30.NM1=6.«NCOLS+(XLAT+90.)/30.N=NM1+1IF(NMl-NO) 1,5,3
1 DO 2 I=N,NO2 BACKSPACE 10READ(10) NO,XLATO,XLONGO,RHO,TAURETURN
3 NEXT=NO+1DO 4 I=NEXT,NM1
4 READ(10)5 READ(10) NO,XLATO,XLONGO,RHO,TAURETURNEND
SUBROUTINE ATMOS(Z,XLAT,XLONG,D,T,P,S):
COMMON/LINK/ DL,NO,XLATO,XLONGO,RHO(34,4,4) ,TAU(3M,ll)DIMENSION ANS(4)ZKM=Z/1000.IF(Z) 1,2,2
1 PRINT 401,Z,XLAT,XLONG401 FORMAT(/6(2H »)* ALTITUDE=«F10.1* OUTSIDE RANGE»/6(2H *)«AT LATI
1 TUDE »F6.1* AND LONGITUDE »F7.V)C USE ALTERNATE (1962 STANDARD) ATMOSPHERE MODEL
15
APPENDIX A
ZFT=Z/.3048, CALL AT62(ZFT,ANS)
D=515.379»ANS(1)P=U7.880258«ANS(2)T=ANS(3)S=.30l*8«ANS(4)RETURN
2 IFCZKM-99.) 4,3,13 ZKM=98.9H DXLAT=XLAT-XLATOIF(DXLAT) 11,5,5
5 IF(DXLAT-30.) 7,6,116 DXLAT=29.97 DXLONG=XLONG+DL-XLONGOIF(DXLONG) 11,8,8
8 IF(DXLONG-30.) 10,9,119 DXLONG=29.910 I=ZKM/3.+1.
J=DXLAT/10.+1.JP1=J+1K=DXLONG/10.+1.KP1=K+1R1=(3000.»I-Z)/3000.R2=1.-R1RH01=EXP(R1*ALOG(RHO(I,J,K))+R2*ALOG(RHO(IP1,J,K)))RH02=EXP(R1»ALOG(RHO(I,JP1,K))+R2*ALOG(RHO(IP1,JP1,K)))RH03=EXP(R1«ALOG(RHO(I,J,KP1))+R2»ALOG(RHO(IP1,J,KP1)))RHOi<=EXP(R1*ALOG(RHO(I,JP1,KP1))+R2»ALOG(RHO(IP1.,JP1,KP1)))TAU1rR1«TAU(I,J,K)+R2*TAU(IP1,J,K)TAU2=R1*TAU(I,JP1,K)+R2*TAU(IP1,JP1,K)TAU3=R1*TAU(I,J,KP1)+R2*TAU(IP1,J,KP1)TAU4=R1»TAU(I,JP1,KP1)+R2*TAU(IP1,JP1,KP1)R1=(XLATO+10.»J-XLAT)/10.R2=1.-R1RH012=R1«RH01+R2«RH02RH034=R 1 »RH03+R2«RH04TAU12=R1«TAUUR2*TAU2TAU34=R1*TAU3+R2*TAU4R 1 = ( XLONGO+ 1 0 . *K-XLONG-DL ) / 1 0 .R2=1.-R1D=R1«RH012+R2*RH03U
PrT*D/. 00348362S=.069836«SQRT(T)RETURN
11 RLONG=XLONG+DLCALL READBLK(XLAT,RLONG)GO TO UEND
16
APPENDIX B
EMPIRICAL RANDOM ATMOSPHERE MODEL
To construct the model used in generating the REACT atmospheres, a set ofover 6000 rocket and high-altitude soundings of the atmosphere was modeledempirically. The soundings were divided into categories according to the lati-tude and season of the sounding. The five latitude categories
Band , deg
1530456075
Latitudes covered
0° to 22.5° N or S22.5° to 37.5° N or S37.5° to 52.5° N or S52.5° to 67.5° N or S67.5° to 90.0° N or S
correspond to those used in the 1966 Supplemental Atmospheres (ref. 2). Sound-ings which fell into one of the four nonequatorial latitude categories were alsoclassified by their season. The four season categories are:
Season
SpringSummerAutumnWinter
Northern hemisphere
March to MayJune to August
September to NovemberDecember to February
Southern hemisphere
September to NovemberDecember to February
March to MayJune to August
Soundings in the equatorial (15°) band were not classified as to season sinceseasonal differences at these latitudes are negligible. Table III lists theresultant 17 latitude-season categories and the number of soundings which belongsto each category.
Each sounding was converted, by interpolation, to a temperature and/ordensity profile with observations every 3 km in altitude over the range of thesounding. Since the soundings did not always cover the same altitude range and,in fact, some did not even overlap, the number of observations at each altitudewas always less than the number of profiles shown in table III. Table IV showsthe actual number of temperature observations at each altitude, and table Vshows the number of density observations at each altitude.
A drastic reduction in available data above 60 km is readily apparent.In categories where temperature data were completely missing, temperaturemeans were estimated by using gradients from the 1966 Supplemental Atmospheres(ref. 2) and variances at the highest altitude where data were available wereextended to 99 km. Missing pressure and density means were then filled in byusing the hydrostatic equation and the equation of state, and their varianceswere also extended from lower altitudes. The modeling procedure is describedin detail in reference U. Only a brief description of the model statisticsrelative to the data is given here.
17
APPENDIX B
For the kth season (k = 1, 2, 3, and 4) and any latitude <J>, the model usesa random-number-generating scheme to produce a temperature-altitude profile. Atany fixed altitude_ z the temperature is assumed to be a Gaussian random vari-able with a mean f(z,k,<|>) and a standard deviation s(z,k,<j>); these values areestimated from the data. Thus, model-generated temperature profiles were pro-vided, within the accuracy of the random number generator, the same means andstandard deviations as the temperature data at corresponding altitudes, seasons,and latitudes. Furthermore, within the same profile, temperatures at differentaltitudes were correlated by using coefficients of correlation estimated fromthe data. This latter covariance property insures that modeled vertical tempera-ture gradients are consistent with those in the data.
If |<t>| g 15°, parameters from the 15° band data are used, and likewise,if |<j)| £ 75°, parameters from one of the 75° bands are used depending on theseason . For 15° < |<l>| < 75°, T(z,k,<|>) and sOz,k,<J>) are obtained by linearinterpolation over <j> by using the tables for T and s in the "category"latitudes 15°, 30°, 45°, 60°, and 75°.
In order to calculate pressure and density profiles consistent with a par-ticular random temperature profile, hydrostatic equilibrium is assumed to exist.Thus, pressures vary according to the hydrostatic equation
dP = -gp dz (B1)
where g and p are gravitational acceleration and density, respectively, ataltitude z. By relating temperature, pressure, and density by the equation ofstate
P = W£ (B2)RT
one can calculate complete pressure and density profiles from a given tempera-ture profile and a base pressure Po. Base pressure Po (at sea level) isselected by using a Gaussian random number generator and is correlated to sea-level temperature. Although it is known that hydrostatic equilibrium is notalways maintained, means and standard deviations of model-generated pressuresand densities (based on hydrostatic equilibrium) agree reasonably well with thedata. (See ref. 4.)
Figure 8, which compares the mean and standard deviation of temperatures inthe model and the data, illustrates the agreement between model and data. Theresults shown here for the 15° latitude-season category are typical of all thecategories. Model statistics in figure 8 and in subsequent figures are based ona sample of 1000 model-generated atmospheres. Figure 9 compares model and datameans and standard deviations of P/?62 where atmospheric pressure P is non-dimensionalized by the 1962 Standard Atmosphere pressure Pfe- Figure 10 com-pares means and standard deviations of nondimensionalized densities P/P62.
18
REFERENCES
1. U.S. Standard Atmosphere, 1962. NASA, U.S. Air Force, and U.S. Weather Bur.,Dec. 1962.
2. U.S. Standard Atmosphere Supplements, 1966. Environ. Sci. Serv. Admin.,NASA, and U.S. Air Force.
3.'U.S. Standard Atmosphere, 1976. NOAA, NASA, and U.S. Air Force, Oct. 1976.
4. Campbell, Janet W.: A Model for Simulating Random Atmospheres as a Functionof Latitude, Season, and Time. NASA TN D-8470, 1977-
5. Daniels, Glenn E., ed.: Terrestrial Environment (Climatic) Criteria Guide-lines for Use in Aerospace Vehicle Development, 1973 Revision. NASATM X-64757, 1973-
6. Cole, Allen E.; and Kantor, Arthur J.: Horizontal and Vertical Distributionsof Atmospheric Density, Up to 90 km. AFCRL-64-483, U.S. Air Force, June1964. (Available from DDC as AD 605196.)
19
TABLE I.- PERCENTAGE OF VERTICAL DENSITY GRADIENTS IN DATA AND
IN MODEL WHICH EXCEED DESIGN GRADIENTS (REF. 5)
Design values for vertical density gradients are expressed asmaximum and minimum percentage increases in density betweenaltitude z and altitude z - 2 km (ref. 5). Exceedancerates are the percentage of times in the data and in themodel that the design value is exceeded.
(a) 15° Latitude (Design values interpolated fromthe July design values)
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
28272423303436
Exceedance rate
Data
0.0.0.2
1.29.125.025.5
Model
0.0.0.3.3*
19. 2»27.918.2
Maximum increase, percent,in 2-km layer
Designvalue
39353231434749
Exceedance rate
Data
2.814.92.03-1
20.015.025.5
Model
1.5*17.5.6**
1.917.913.921.3
"Exceedance rate in the model which is significantly differ-ent from that found in the data at the 0.05 level.
*»Exceedance rate in the model which is significantly dif-ferent from that found in the data at the 0.01 level.
20
TABLE I.- Continued
(b) 30° Latitude
Spring
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
30292325333637
Exceedance rate
Data
0.2.3.3
5.166.723.130.0
Model
0.0.1.0
38.0»»76.036.537.9
Maximum increase, percent,in 2-km layer
Designva 1 UP
39373133414750
Exceedance rate
Data
8.31.34.3.4.0 '
7.710.0
Model
6.31.2.9»».0
6.66.121.2
••Exceedance rate in the model which is significantly different from that found in the data at the 0.01 level.
Summer
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Designva 1 IIP
32"282524303040
Exceedance rate
Data
0.3.0
1.14.0
. 37.510.0
.0
Model
0.0.0
1.63-1
71. 2»10.7.2
Maximum increase, percent,in 2-km layer
Designvalue
40353234454750
Exceedance rate
Data
.0.011.32.22.5.0.0
33-3
Model
0.010.5
.6
.66.25.743.2
•Exceedance rate in the model which is significantly differ-ent from that found in the data at the 0.05 level.
21
TABLE I.- Continued
(b) 30° Latitude - Concluded
Autumn
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
33272226323540
Exceedance rate
Data
1.0.0.3
10.050.025.038.1
Model
0.0.0.0
11.567.017.239.4
Maximum increase, percent,in 2-km layer
Design
42393231363950
Exceedance rate
Data
0.03.32.84.8
30.025.04.8
Model
0.01.8.7*
2.0*19.333.43.6
"Exceedance rate in the model which is significantly differ-ent from that found in the data at the 0.05 level.
Winter
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Designvalue
31282323293026
Exceedance rate
Data
0.21.2.3.0
10.04.8.0
Model
0.0.6.5.3
29.3**•8.9
.0
Maximum increase, percent,in 2-km layer
Designvalue
43403435414760
Exceedance rate
Data
2.01.5.0
1.215.0
.0
.0
Model
0.1»«1.1.0.0
11.2.1.0
**Exceedance rate in the model which is significantly different from that found in the data at the 0.01 level.
22
TABLE I.- Continued
(c) 45° Latitude
Spring
Altitude,km
30405060 '708090
Minimum increase, percent,in 2 -km layer
Designva 1 IIP
32312525273538
Exceedance rate
Data
0.012.2
.09.110.055.6
.0
Model
0.013.01.32.78.538.529.6
Maximum increase, percent,in 2-km layer
Design
38342930415358
Exceedance rate
Data
35.726.728.511.7
.022.250.0
Model
35.327.130.64.7.4
11.87.7
Summer
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Designvalue
34302525263038
Exceedance rate
Data
0.0.0
2.35.6.0
30.8.0
Model
0.0.9
6.6»3.014.927-333. 9»
Maximum increase, percent,in 2-km layer
Designvalue
38353130455356
Exceedance rate
Data
6.111.52.34.27.1.0
37.5
Model
4.614.75.04.016.0
.524.5
•Exceedance rate in the model which is significantly differ-ent from that found in the data at the 0.05 level.
23
TABLE I.- Continued
(c) 45° Latitude - Concluded
Autumn
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
35262424283540
Exceedance rate
Data
2.9.0.0
2.042.950.050.0
Model
3.7 ..0.0.1»
27.565.266.0
Maximum increase, percent,in 2-km layer
Design
50403434414456
Exceedance rate
Data
0.01.9.0.0.0.0.0
Model
0.0.1«.0.0.4
1.1.1
"Exceedance rate in the model which is significantly differ-ent from that found in the data at the 0.05 level.
Winter
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
„
Designvalue
35272322191923
Exceedance rate
Data
3.3.0
2.23.8.0.0.0
Model
2.9.3.7.8.8
2.24.5
Maximum increase, percent,in 2-km layer
DesignVfl 1 IIP
474542 .38485562
Exceedance rate
Data
0.0.0.0.0
13.3.0.0
Model
0.0.0.0.2
2.8.9
2.3
24
TABLE I.- Continued
(d) 60° Latitude
Spring
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
323126252636
Exceedance rate
Data
0.01.22.05.3.0
Model
0.03.6*7.6**4.311.2100.085.8
Maximum increase, percent,in 2-km layer
Design
37343030405161
Exceedance rate
Data
50.550.916.48.5.0
Model
50.347.410.88.3.0.0.0
•Exceedance rate in the model which is significantly differ-ent from that found in the data at the 0.05 level.
""Exceedance rate in the model which is significantly dif-ferent from that found in the data at the 0.01 level.
Summer
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
34302525263439
Exceedance rate
Data
1.9.0
2.5.0
Model
8.7**.1
3.91.3
55.270.48.0
Maximum increase, percent,in 2-km layer
Design
37353128395358
Exceedance rate
Data
1.92.4.0
7.1.
Model
4.0.8.2
3.7.0.0.0
**Exceedance rate in the model which is significantly dif-ferent from that found in the data at the 0.01 level.
25
TABLE I.- Continued
(d) 60° Latitude - Concluded
Autumn
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
33282622273340
Exceedance rate
Data
0.0.0.0.0.0
Model
0.5.0
2.0.0
61.363.673.5
Maximum increase, percent,in 2-km layer
Design
51413433404858
Exceedance rate
Data
0.06.711.53.0.0
Model
0.05.812.0
.9
.1
.1
.0
Winter
Altitude ,km
30405060708090
Minimum increase, percent,in 2-km layer
Designvalue
33262324201519
Exceedance rate
Data
1.4.0.0.0.0
Model
5.4»».0.1
1.7.0.0.0
Maximum increase, percent,in 2-km layer
DesignVfl 1 IIP
48444338525963
Exceedance rate
Data
0.7.5.5.0.0
Model
0.01.1.5.2.0.0.0
•'Exceedance rate in the model which is significantly dif-ferent from that found in the data at the 0.01 level.
26
TABLE I.- Continued
(e) 75° Latitude
Spring
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
29282625283738
Exceedance rate
Data
1.11.4
14.332.44.3
Model
0.01.7
18.241.522.0**51.135.4
Maximum increase, percent,in 2-km layer
Designva 1 IIP
3634333037456.1
Exceedance rate
Data
42.525.7
.018.934.8
Model
41.626.22.09.9
21.07.95.2
••Exceedance rate in the model which is significantly dif-ferent from that found in the data at the 0.01 level.
Summer
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
34302524263941
Exceedance rate
Data
23.63-3
24.35.15.4____
Model
19.01.0
16.319.1"29.0"27.935.9
Maximum increase, percent,in 2-km layer
Design
37353327294957
Exceedance rate
Data
3.83.3.0
40.783.8
Model
1.11.2.0
40.042.7««26.426.6
••Exceedance rate in the model which is significantly dif-ferent from that found in the data at the 0.01 level.
27
TABLE I.- Concluded
(e) 75° Latitude - Concluded
Autumn
Altitude,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
30302619272940
Exceedance rate
Data
0.02.93-9.0
9.525.0
Model
0.01.12.6.0
17.227.547.7
Maximum increase, percent,in 2-km layer
Designva T IIP
48413330365157
Exceedance rate
Data
1.319.147.150.028.612.5
'Model
0.018.347.426.3**24.015.411.0
**Exceedance rate in the model which is significantly dif-ferent from that found in the data at the 0.01 level.
Winter
Altitude ,km
30405060708090
Minimum increase, percent,in 2-km layer
Design
29252227261514
Exceedance rate
Data
0.05.5.0
22.93-85.9
Model
0.0.4
1.032.56.2.1.1
Maximum increase, percent,in 2-km layer
Design
36343330376165
Exceedance rate
Data
2.413.716.3
.0
.05.9
Model
5.124.9**9.5.8.7
1.13.2
**Exceedance rate in the model which is significantly dif-ferent from that found in the data at the 0.01 level.
28
TABLE II.- PERCENTAGE OF REACT HORIZONTAL DENSITY GRADIENTS WHICH
EXCEED DESIGN GRADIENTS (REF. 5)-
Horizontal gradients measured along lines of constant latitude.Design values for horizontal density gradients are expressedas maximum change in percent departure from 1962 StandardAtmosphere density per 110 km (ref. 5).
(a) Spring tape
Altitude,km
3033363942
4548515457
6063
• 666972
757881848790
REACT density gradients, percent, at latitude -
10° N
Designvalue
0.30.30.30.30.30
.30
.30
.31
.34
.37
.40
.46
.52
.58
.60
.60
.60
.59
.56
.53
.50
Exceedancerate
1924273339
4141464442
4139485557
636468747679
30° N
Designvalue
0.30.36.42.48.52
.55
.58
.58
.52
.46
.40
.52
.64
.76
.86
.951.041.07.98.89.80
Exceedancerate
2323201515
1619222943
5443363945
485154607078
50° N
Designvalue
0.60.60.60.60.66
.75
.84
.90
.90
.90
.901.021.141.261.32
1.351.381.371.281.191.10
Exceedancerate
1217243030
2931343639
3935373837
434855616469
70° N
Designvalue
1.101.04.98.92.84
.75
.66
.58
.52
.46
.40
.46
.52
.58
.62
.65
.68
.66
.54
.42
.30
Exceedancerate
1824324352
6069717780
8582817981
818183909294
29
TABLE II.- Continued
(b) Summer tape
Altitude,km
3033363942
4548515457
6063666972
757881848790 .
REACT density gradients, percent, at latitude -
10° N
Designvalue
0.40.40.40.40.44
• 50• 56.61.64.67
• 70.76.82.88.88
.85
.82
.75
.60
.45
.30
Exceedancerate
912171922
1816161717
1716303740
505259728087
30° N
Designvalue
0.40.40.40.40.44
.50
.56
.61
.64
.67
• 70.76.82.88.88
.85
.82
.75
.60
.45
.30
Exceedancerate
3491313
1313151818
2421232860
494862738086
50° N
Designvalue
0.40.40.40.40.42
.45
.48
.54
.66
.78
.90
.991.081.17
, 1.24
1.301.361.361.241.121.00
Exceedancerate
89141924
3032353127
2223233033
3638466167
- 78
70° N
Designvalue
0.40.37• 34.31• 36
.45
.54
.62
.68' .74
.80
.89
.981.071.10
1.101.101.05.90.75.60
Exceedancerate
2635485959
5757535049
4550586269
-' 697175848893
30
TABLE II.- Continued
(c) Autumn tape
Altitude,km
3033363942
4548515457
6063666972
757881848790
REACT density gradients, percent, at latitude -
10° N
Designvalue
0.30• 30.30• 30-34
.40
.46
.52
.58
.64
.70
.70
.70
.70
.70
.70
.70
.68
.62
.56
.50
Exceedance- rate
1924273333
2723222118
1720364750
575963717578
30° N
Designvalue
0.30• 30• 30• 30.40
.55
.70
.81
.84
.87
• 90.84.78.72.70
.70
.70
.68
.62
.56
.50
Exceedancerate
2531404739
3124212323
2530314662
636464697881
50° N
Designvalue
0.90.90.90.90.96
1.051.141.221.281.34
1.401.461.521.581.64
1.701.761.781.721.661.60
Exceedancerate
2591519
1922232223
2323222627
302632421351
70° N
Designvalue
1.101.071.041.011.00
1.001.00.99.96• 93
.901.051.201.351.42
1.451.481.451.301.151.00
Exceedancerate
2128364548
5155575860
6257
. 514648
495460687480
31
TABLE II.- Concluded
(d) Winter tape
Altitude,km
3033363942
4548515457
606366 •6972
757881848790
REACT density gradients, percent, at latitude -
10° N
Designvalue
0.40.40.40.40.40
.40
.40
.44
.56
.68
.80
.83
.86
.89
.94
1.001.061.07.98-89.80
Exceedancerate
912171926
2730302216
1213273636
433946576269
30° N
Designvalue
0.40.46.52.58.66
.75
.84
.92
.981.04
1.101.251.401.551.64
1.701.761-751.601.451.30
Exceedancerate
1111171310
10991010
9551415
142329364755
50° N
Designvalue
1.601.661.721.781.92
2.102.282.452.602.75
2.902.902.902.902.86
2.802.742.612.342.071.80
Exceedancerate
00111
11111
11125
7814273650
70° N
Designvalue
2.602.722.842.962.92
2.802.682.562.442.32
2.202.232.262.292.28
2.252.222.152.001.851.70
Exceedancerate
00247
914171618
2017161417
192534445259
32
TABLE III.- NUMBER OF PROFILES IN DATA BASE FOR
EACH LATITUDE-SEASON CATEGORY
Latitude band
15°
30°
1450
60°
75°
Profiles in season -
Spring Summer Autumn Winter
'1928
495
184
238
101
468
193
176
128
516
154
217
91
504
147
342
122
Total
1928
1983
678
973
442
6004
33
TABLE IV.- NUMBER OF TEMPERATURE OBSERVATIONS AT EACH ALTITUDE
IN THE 17 LATITUDE-SEASON CATEGORIES
Altitude,km
0369121518
21242730333639
42454851545760
63666972757881
848790939699
Number of temperature observations for season and latitude of -
Annual
15°
1747175917711781177817821781
1772177417681749169016161569
155615601530147613691284986
4861366467696765
665956402625
Spring
30°
375457456457457456456
452451445436419369353
364365368365358321286
183904720141414
13.1311110
45°
165170170170170170171
173173172166158137133
1331331331261089280
5117131110109
662000
60°
234234234234233231233
233230227224221214210
206203199192182155126
8042258000
000000
75°
86888890909089
90909098999787
79746964585044
40332823221914
000000
Summer
30°
319436433435435436436
428429427404395354329
325330332324308295238
129592811121212
11910000
45°
172174175173176176171
175176174168.155147136
1331331331321209676
38171614151313
131210533
60°
173173173173173173172
173171171168166163152
150147144144138120108
8146130000
000000
75°
103107107108111110110
111107108121118113111
102968579776962
53464137342417
100000
Autumn
30°
319454451452451451451
450450451436423380350
366371371366351328284
181884831272827
272221532
45°
142141141142142142143
142142142142140119114
11110810394897653
211287766
655333
60°
214214214214214211212
209207205205200190188
186187179177168152129
9141121100
000000
75°
69747677798081
81817887858278
76696357524944
37282421211816
420000
Winter
30°
363475476477477477476
474470466452427386364
363365362353346326273
154582524242323
201311212
45°
119122122122122122129
129129132129118106102
1021029995887452
2818151513116
641000
60°
338338337337337334327
324320315309300290281
276273265260250229180
9933153000
000000
75°
909496101104104100
1009795108105107102
95847367615345
39363328262117
210000
TABLE V.- NUMBER OF DENSITY OBSERVATIONS AT EACH ALTITUDE
IN THE 17 LATITUDE-SEASON CATEGORIES
Altitude,km
0369121518
21242730333639
1*2454851545760
63666972757881
848790939699
Number of density observations for season and latitude of -
Annual
15°
1744174817531755173417661762
1748172717101680158014871426
141114091377132812571141870
4231216060666260
615451392624
Spring
30°
372452451449453453453
447447438421400335312
322326328327324292261
168854518131313
121210110
45°
163169170167170169168
170166168157154135131
1311321311231069180
5117121010109
662000
60°
232233233233233231232
232227220216204186171
16916315815214512194
6036228000
000000
75°
86868787878787
88868796969179
71666156504337
36312823221914.
000000
Summer
30°
304431429429424431432
424423417390381325292
285289290289272259202
110512511111011
11910000
45°
1721731741731701662
173174170166151139133
1311291301291179375
38171614151313
13109533
60°
168173173173173173169
170166165162158149131
1271241231201129885
6640110000
000000
75°
102105105103106107W7
107105107118114109103
92867770686359
52464037342417
100000
Autumn
30°
316449449445445447449
445435435415404351309
327332331329313294253
160804631272625
262221532
45°
142140141140142141143
141142140142138117112
10910610192877451
201187766
655333
60°
212214212213214211209
203200198196185165153
150148141139129118101
7234101100
000000
75°
68717274757577
78797684807671
70635651474440
34262321211816
410000
Winter
30°
360473475472476475474
471466459445418367342
338340336329322303254
137552422232322
191311212
45°
119122122122122122129
12812712812711210198
99989693867252
2818151513116
641000
60°
335338337337333332326
320313304289272241230
222222215211202185144
8330153000
000000
75°
8891909510010298
95898699938982
73625249434135
29282826252117
210000
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NASA-Langley, 1977 L-10623
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON. D.C. 2O546
OFFICIAL BUSINESS
PENALTY FOR PRIVATE USE S3OO SPECIAL FOURTH-CLASS RATEBOOK
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