The synthetical reference outdoor climate

Energy and Buildings, 2 (1979) 151 - 161 © Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

151

The Synthetical Reference Outdoor Climate A. H. C. VAN PAASSEN and A. G. DE JONG

Department of Mechanical Engineering of the Delft University of Technology, Delft (The Netherlands) (Received

A mathematical model o f the Dutch climate is derived from the meteorological.edata of 10 years by statistical analysis. It is called the 'synthetic reference outdoor climate' and can be considered as a replacement of the reference year. The model can be used as input for computing programs for calculating the yearly energy consumption as well as for programs for sizing the components o f heating and cooling installations. Also, it reduces the required number o f hourly weather data to 17.5% of that o f a complete reference year.

and also their probabilities and autocorrelations are formulated. This model and the weather data generated b y it are called the 'synthetical reference outd o o r climate'. To get a simplified picture of the model it can be considered initially as a sort of computerised polygonal die. Every possible combination of the weather variables is represented on one surface of that die. The die is 'falsified' in such a w a y that the frequencies and sequences of the hourly combinations are in agreement with the real climate (Fig. 1). DATA MENTIONED ON EACH SURFACE -

INTRODUCTION UNSHINE

: 31

/ TEMPERATURE : ~ 0 \

The indoor climate and energy consumption of buildings and dwellings are highly influenced b y the o u t d o o r climate. Consequently there is a need to have adequate information a b o u t the o u t d o o r climate, which can be used as input for various types of calculations. Until n o w calculations have been carried o u t by means of the s o , a i l e d reference year, which consists o f a complete year of weather data, derived from meteorological data over a long period b y means of some selection procedure. In this paper a synthetical reference o u t d o o r climate is described which has some advantages. The approach used for the development of this synthetical o u t d o o r climate is quite different from the one used for establishing the reference year. Instead of selecting representative parts from the hourly measured meteorological data, new hourly data are generated which have the same statistical properties as the measured data. The synthetical data are generated by a mathematical model of the o u t d o o r climate, in which all the linear and nonlinear correlations b e t w e e n the weather variables

\

/

/

HUMIDITY

:5 \

WINOSPEED

:3 \

Fig. 1. A polygonal die as a generator for meteorological data.

This approach was chosen for various reasons; 1, a mathematical characterisation of the o u t d o o r climate w o u l d be obtained which leads to a better understanding of its behaviour; 2, such a model would be a tool for reducing the n u m b e r of weather data required for energy consumption calculations. To develop the mathematical model the hourly meteorological data of the Dutch weather bureau for the period 1960 to 1970 was analysed in detail. This analysis was partly carried o u t by means of spectral analysis of the variations of the weather variables [ 1]. A similar spectral analysis was carried o u t by Cumali [ 2 ] . In his paper the use ,of spectral analysis as a tool in identifying correlations, annual and diurnal patterns of weather data of

152 Fresna in California was presented. But no successful use of the results of this analysis was given. By spectral analysis only linear relations between the variables can be found. However, existing relations are in most cases highly nonlinear. Therefore an additional approach is necessary. It appears that average daily variations of weather parameters are suitable to be used for tracing these nonlinearities. These variations are obtained by taking the average of all the measured values of a specific weather variable, which have taken place at the same time of the day. Such variations could be described by empirical equations. Both different approaches are used simultaneous for constructing the model. Successively the following items will be discussed in the paper: the framework of the mathematical model; the properties of the weather variables such a global radiation; air temperature; absolute humidity; wind speed and wind direction and the model itself. Also an indication of the accuracy of the model will be given.

FRAMEWORK OF THE MATHEMATICAL MODEL The first step is the determination of the linear and nonlinear correlations between the various weather variables b y means of spectral analysis and average daily variations. Next the independent variations o f each variable are defined by subtracting the dependent variations from the original data {Fig. 2). As the last step, these independent variations are analysed statistically and their statistical properties used to generate the independent variations o f each parameter by means of data generators. These generators deliver hourly values with the same probability and autocorrelation as the independent variations OTHER VARIABLES

1~ TOTAL VARIATION OF WEATHER VARIABLE

"Jr-

I

DEPENDENT PART -~ INDEPENDENT PART

Fig. 2. Splitting weather variables into dependent and independent parts.

of the variable itself. The way these generators are realised is discussed in Appendix 1. Global radiation has been assumed to be completely independent and other variables are considered as more or less dependent on global radiation. However, in some cases (e.g. the relation between global radiation and wind direction), global radiation is n o t always the leading variable. Thus in such cases the influence between t w o variables is interpreted wrongly in a physical way, b u t it will not have any effect on the calculation results. For this type of analysis it was convenient to discern daily and hourly dependencies. Global radiation has an hourly as well as a daffy influence on air temperatures and on absolute humidity. On wind speed and wind direction, only the daffy influence is relevant. As well as global radiation, air temperature t o o has a daily influence on absolute humidity. In Fig. 3 the most important dependencies are shown schematically. Adding the independent parts delivered by the random generators to the dependent parts results in the weather variables. The model of the o u t d o o r climate is based on the frame of Fig. 3. Later correlations become more complicated because of the many refinements.

STATISTICAL ANALYSIS OF WEATHER VARIABLES In the following paragraphs the statistical properties and the mathematical approximations are discussed for each variable.

Global solar radiation Figure 4 shows that the average daily variation of global solar radiation can be approximated b y a half-sine function: q(t)=Aqsin

b--a

a<~ t < . b

(1)

a and b are respectively m o m e n t s of sunrise and sundown. Daily variation of global radiation on a specific day can n o w be characterised by the amplitude Aq. The complete series of 10 × 365 values of Aq is derived from the recorded hourly values of qw(t) b y equalizing the daffy total a m o u n t of radiation to that correspondhag to the sine function. Integrating the same function leads to:

153 GLOBAL I SUNSHINE RADIATION .

OF AIR

I

FI~

TEMPERATURE

I

I

OF ABSOLUTE

I

OF WIND

OF WIND

,UM, DITY

j

VELOC,TY

O,RECTION

I

INDEPENDENT

d = DALLY h : HOURLY

=EPENOENT

] I N D E P E N D E N T ~"1"

I,NOEPENDENT

PART AIR TEMPERATURE

ABSOLUTE HUMIDITY

J- T

WIND DIRECTION

WIND VELOCITY

Fig. 3. C o r r e l a t i o n s b e t w e e n w e a t h e r variab]es.

I

700 [kcatlh.m21 JULY q

/ ~ /'~

6OO

i

~d :

measureddl approxlmati¢ s.nsh;neau~at~o.

,1

SAMPLE CORRELATION FUNCTION

,0.5 "0

5OO

~ - rd >90%

•-0.5 TIME (days)

400

,-1

300 COMMON LOGARITHMOF NORMALIZED AMPLITUDE SPECTRUM ESTIMATE

200

¢/

100

2

4

,

•

6 8 10 12 14 16 18 20 22 m

time

24 [h]

Fig. 4. Average dally variation of global radiation for various types of weather.

lit 0 t14

[

i

cycles/day

-1'.5 -1 -0.5 (t0 LOG) FREQUENCY

Fig. 5. Autocorrelation function of the amplitude Aq

of global radiation during January.

b

Aq --~z ~ / ( b - - a )

~, qw(t) t ffi a

In the mathematical model the amplitude Aq will be the parameter from which all the influences of global radiation on the other variables will be derived. An impression of the accuracy of the approximation of the average daily variation by equation (1) is given by Fig. 4. In Fig. 4, the approximation of average daily variation is compared with the variation derived from all measured data. This comparison is also carried o u t for bright and cloudy days, characterised by sunshine duration. Sunshine duration (rd) is defined as the percentage fraction of the integrated duration of all the bright periods during the day with regard to the m a x i m u m possible period of sunshine.

In order to detect the way various types of weather are succeeding each other, the autocorrelation function of the time series of the amplitude Aq is used. In Fig. 5, this function is given for January. It shows that only for time shifts lower than one day this function is greater than 0.35. Thus the mean coherence between daily values is weak, or in other words a bright day predicts hardly anything about the brightness of the following days. Air temperature, absolute humidity, wind speed and direction The autocorrelation functions and amplitude spectra of these variables show a strong periodical character. Figure 6 showing these functions for the air temperature. The ampli-

154

SAMPLE CORRELATION FUNCTION

1

SAMPLE CORRELATION FUNCTION

1

J

0.5 0

0.5 0

-0.5

-0.5

TIME (hours)

-&) -~ -I

-,.; -2; ~

o I A

2~

TIME (days)

~

6~

8~

.-1

COMMONLOeARITHMOF NORMALIZEDAMPLITUDE

-1o

-~2 -~ -1'6 -~

6

~

,~

2'4

3'2

i 40

--I

-! COMMON LOGARITHM OF NORMALIZED AMPLITUDE SPECTRUM ESTIMATE

-2

-3

o -J - 4.

-4

8 -5

,1/24I/,1,~ -6. _~ -~.s -i

-,.s

-1

~y~l°slho~ -;.~

;

-6

(10 LOG)FREQUENCY a) HOURLY VALUES ANALYSED F i g . 6.

-'2

-,'s

l/ll0

1~4,

-,

-us

cycIeslday

G

(10 LOG) FREQUENCY b) MEAN DALLY VALUES ANALYSED

Autocorrelation function and amplitude of air temperature.

tude frequency spectrum shows peak values at frequencies 1/12 and 1/24 (cycles/hour). This points to a daily variation, which can be described mathematically, as could be done before for global radiation. When these functions are determined for the daily mean values they turn out to be smooth, due to the elimination of periodical daily variations (Fig. 6b). For air temperature and absolute humidity, the coherence between succeeding daily values is much stronger than for global radiation. Generally, the mean values, of for example air temperatures occurring several days ago, still have an effect on the momentary values.

SQUARED COHERENCY SPECTRUM ESTIMATE

0.5 cyctes/hour

0 -2.5

-1.5

-I -0.5 6 (10 LOG) FREQUENCY

-I COMMON LOGARITHM OF NORMALIZED AMPLITUDE SPECTRUM ESTIMATE

-2 -3 -4 -5

cycles/hour

=

-2.5 t80

Correlation between global solar radiation and air temperature The crosscorrelation between two variables can be defined by means of the amplitude and phase spectrum of the spectral density function. Their mathematical definitions are given in Appendix 2. The coherency function indicates the correlations between both variables for each frequency; its value is 0 where no correlation exists, and 1 where the variables are completely linked to each other. The amplitude and phase spectrum give numerical values for respectively the linear proportional factor and the phase lag between the leading

-2

/

-2

-1.5

-1 -0.5 0 (10 LOG) FREQUENCY PHASE SPECTRUM ESTIMATE

90 0 -90

-180

FOR MONTH APRIL

F i g . 7. Squared coherency and amplitude spectrum of global radiation and air temperature.

variable and its effect on the other variable. The coherency spectrum in Fig. 7 shows that peak values occur at the frequencies 1/24 and 1/12 cycles/hour, showing a strong correlation

155

between daily variations of global radiation and air temperature. Amplitude and phase can be derived from the amplitude and phase spectrum of the same figure. For example for the month April it can be derived that the crosscorrelation can be characterised by a linear factor of 0.016 °C h m 2 kcal-1 and a phase lag of 30 degrees, which is equivalent to 2 hours. Also it is shown by the same figure that there is no crosscorrelation for the slower variations. This information is used in a qualitative way to find the mathematical expressions for the correlations. Numerical values are obtained almost entirely by observing the average daily variation of the parameters considered. Studying this variation on bright and cloudy days shows that the effect of global radiation on air temperature should be considered as caused by direct and diffuse radiation separately. Direct radiation can be obtained by subtracting the diffuse radiation from the global radiation. However, the diffuse radiation is unknown, so that the following approximation is necessary. It is assumed that the average daily variation of diffuse radiation is the same each day and equal to the average daily variation on fully clouded days. Again this daily variation is simulated by a half-sine function, simular to equation (1), with the a m p l i t u d e Aq.di f. By this assumption the direct radiation can now be written as: qdir(t) = Aq.dir sin

other terms in the above equation. Around this constant value, independent random variations 0r can be observed. They will be discussed later. The other terms A0d and AO(t) depend on the brightness of the day, which is expressed by the amplitude Aq. These two terms are also dependent on time; the first is changing daily and the latter hourly. A comparison of the mean daily variation on bright and cloudy days indicates that in winter the mean daily level of the air temperature is much lower on bright days (Fig. 8a, b). This is expressed by thefollowing equation (2a)

AOd = --14Aq.dir

The proportional factor 14 is defined empirically for each month by means of the dally variation. The hourly variation A0 (t) is approximated by equations, valid in three different succeeding time periods; one o'clock to sunrise, sunrise to sundown and evening to midnight. During the period from one o'clock to sunrise, the temperature drops to its minimum value (Fig. 8). The variation in that period can be approximated by two (negative) sine functions with amplitude Ao,. The second sine function starts at tl, the moment the minimum value occurs, and it is necessary to get the right connection to the function used in the following period. These functions for the first period are:

~ ( t - a)

AO(t)=Ao, sin ~fl t - - t l +

b--a

with 1 < t < t~

(3)

fora< t< b =0

for t < a and t > b

° Ao sm

l, (,l +

1

)l]

with Aq.dir

----

Aq

--

Aq.di

f

(2)

It is emphasised that the definition for the diffuse radiation, as used here, is not equivalent to the definition commonly used. It has to be considered as a calculation variable, without physical meaning. The mean daily variation of the air temperature can be split into parts:

o(t)=O-+o~ +~Od + AO(t) [°C] The first part ~ is a monthly constant value. It is equivalent to the real monthly mean temperature decreased by the mean values of the

withtl
1

2f2

(4)

Variation during this period of the day is caused by sky radiation and consequently amplitude A0, depends on the clearness of the sky, which can be expressed by a linear function of the amplitude of the direct radiation occurring during that day.

AO, = ll Aq.dir Again the parameters 11, fl and f2 are defined expirically for each month. During the period sunrise to sundown, the strongest correlation

156 [*C] 26 0 {'C]

JULY

/ ~ \ /

I

e

ANUARY 22

~at

/%

day~ L

/

~

measure( data approxirr rd = sunshine

- - - --

-2

,,

i/

./

L rd<5% -4

>90%

%..

-6 -7

t4

0

4

8

12

16

20

"

24 measure

['cl t6 APRIL

e

10.5

/

rd >90% t2

t0

rd < 5%

['el, s.s

e

\\ -&X.

/i. /;/ ///,,

all days

4

~

I,.

8

12

16

data

appfoxin rd = sunshinl duration l i 20

24

i OCTOBER

I

,,-;-%,

12

all dais - ~

~"

rd >90%

0

t2

4

a) Fig. 8.

16 ~

20

4 4

24

I))

time [h]

8

12

16

20

24

time [h]

Mean daily variations o f air temperature and their approximations.

between global radiation and air temperature occurs. The variation caused by the solar radiation can be split into two parts, caused by diffuse and direct radiations. The diffuse radiation effect can be expressed by AO(t) = Ao, sin {nfs(t -- t2)} t<

t2 + - -

f8

with A02 = 12Aq.di f

A0 (in) = {/3 q d i r ( t ) + TpA0 (tn -- 1))/(rp + I)

(6)

1 a<

radiation data from which the diffuse radiation is subtracted. It results in the following first order differential equation correlating the direct radiation to the air temperature.

(5)

On bright days the direct radiation causes an additional variation. This is found by applying the crosscorrelation techniques on the global

From evening to midnight a delayed influence of solar radiation, also is described by eqn. (6), is effective. The average daily variations of the air temperature show that the various effects of solar radiation are non linear. On days with an amplitude o f sunshine Aq, exceeding the monthly mean value Aq, the effect is stronger than

157

given by the linear relations. Therefore the corrections on the linear gain factors (li) a r e necessary. The factors Is are replaced by the corrected factor lic, given by the following equation: lic

=

li {1 + k~(Aq -- Aq)}~ when Aq > .4q

with i = 1,3,4.

(6a)

Figure 8 shows an approximation of the daily variations obtained. Correlations b e t w e e n other variables In a simular way the relations between the other remaining variables can be found and described by equations. Therefore only the more important phenomena of these relations are discussed here. More detail is given in reference [4]. Cross-correlations between air temperature (0) and absolute humidity (x) is very strong, particularly for daily mean values. It can be described by a linear equation, which has to be compensated both for completely clear and clouded days. Spectral analysis shows that there is hardly any cross-correlation between global solar radiation and absolute humidity. However, the various average daily variations of humidity, belonging to different classes of brightness, show a noticeable influence. The same is found for the correlation between on the one side the global solar radiation and on the other side the speed and direction of the wind. The correlations, mentioned above, are approximated by the equations (7) to (12), given in reference [4]. Of course, other correlations are possible. But they have shown to be weak and caused mainly by previously mentioned correlations indirectly.

MATHEMATICAL CLIMATE

MODEL

I ~°u*~'°Ns (~)'(=)'(~)'(~) I

OF

OUTDOOR

As stated before, the daffy amplitude Aq of the positive sine curve of global radiation is considered as completely independent. Its probability and autocorrelation function are used to generate daffy values of Aq by means of a number generator. In Appendix 1 it is described how this is realised. All dependent parts of the other variables are derived from the sine variation of global

WEAVHeR BUREAU

Fig. 9. Calculation procedure for temperature 'noise'.

radiation. The independent parts ('noise') are determined by subtracting these dependent parts from the original measured data. Next the probability and autocorrelation functions of the resulting noise data are determined, so that the noise can also be generated according to the method described in Appendix 1. For temperature this procedure is indicated in Fig. 9. Here a differentiation is made between daily and hourly noise variations, 0n,d and 0n,h, because of their different nature. The probability functions of these noise signals are shown in Fig. 10. Mean values of these distributions are respectively 2.9 ° and 0.0 °C. The standard deviation of hourly noise is 1.46 °C. For absolute humidity, wind speed and wind direction, daffy and hourly noise variations are defined in a simular way to air temperature. N o w all the information for the construction of the mathematical model of outdoor climate is available. To get a good overall picture of the main frame of the model a simplified description of it is given in Fig. 11. Each day the "computer die", realised by a data generator, is thrown in order to generate the amplitude of the daily variation of global solar radiation. Next the dependent variations of the other weather variables are derived from the solar radiation and the independent variations, generated by the computer dice, are added to them. The detailed blockdiagram of the model itself is given in Fig. 12. The circles in this figure indicate the data generators of the independent variations and the numbers in the various blocks refer to the equations, describing the correlations. The output of the mathematical model is a series of hourly values of global solar radiation, air temperature, absolute humidity, wind

158 0.14

prot~biiity

density

functions

(PDF)

PDF 0.12

DALLY NOISE

A

0.48

HOURLY NOISE en,h

0.10

:2.9['cl N /

0.08

0.32

=o ['c] o = 1.46

0.06 0.04

0.16

0.02-

0.06

0

r

-12

,

i

-8

-6

a)

i

r

0 -noise

i

~

0

4 8 12 (temperature)

!

-8

-6

-4

-2

0

2 noise

6 8 (temperature) 4

Fig. 10. Daily and hourly noise distribution of air temperature for January.

SIMULATIONMODEL OUTDOOR- CLIMATE SIMULATIONMODEL OUTDOOR - CLIMATE (simplified) sun radiation

(~Ag

daily throw

time

l

i.

~ GLOBAL SUN-RADIATION

direction

[ O(q)

~LOl(qt.) [ 02(q,t)

_~O3(q,t)'~ '~

I

time I

I V V v -~

l q -Aq (~

= number generator

J

r"r =.Y ----e..tlme

+ TEMPERATURE

ABSOLUTE HUMIDITY

WIND SPEED

e (t)

x (t)

v (t)

WIND DIRECTION w (t)

7

"

I

j

b v(t)

--o,

-,,-w(t)

Fig. 11. Simplified presentation of mathematical model of outdoor climate.

Fig. 12. Block diagram of mathematical model of outdoor climate.

speed and wind direction. The length of the series of weather data depends on the generating period of the data generators used in the model. Due to the daily number generators a big jump may occur at the changeover of one day to the other. This jump is smoothed by spreading the height of the jump over the variation during the day.

ACCURACY OF MATHEMATICAL OUTDOOR CLIMATE

MODEL

OF

To check the mathematical model, the yearly energy consumption of a specific building was calculated by means of the real weather data, measured over a period of ten years, and those generated by the model. The building

159

considered here has a good sun protection system and is well insulated. For the mathematical model an optimal "generating period" can be selected. The longer the period the better the desired probability functions and auto- and crosscorrelations are approximated by the number generators. Various period lengths are considered, namely 8, 16 and 32 days per " m o n t h " . In Fig. 13(a) the percentage error made in calculating the yearly amount of heating and cooling energy (using the model instead of the real weather data), is demonstrated for various period lengths. It shows, that a good result is obtained when a generating period of 32 days is chosen. However, using a synthetic year of twelve 32-day months would not give any reduction in the number of calculations compared with a natural reference year. In order to reduce the number of calculations the monthly statistical propetries represented by the parameters are averaged over each quarter of the year and the same calculations as before are repeated. Thus 8, 16 and 32 days per quarter. The results are given in Fig. 13(b). Calculations agree also (virtually) with the natural year as before, although the number of calculations has been substantially reduced. Contrary to expectation, the agreement for heating is better for shorter generation periods. Possibly this is caused by the averaging procedure of the monthly parameters. If for heating an error of 3.3% and for cooling 8% is accepted, a period of 16 days is sufficient. Consequently, the generation of the weather data with a generating period of 16 days for each quarter reduces the number of data required for an energy consumption calculation to 17.5% of that of the natural reference year. Instead of the model itself the generated data, recorded on tape, can be used for calculations. This series of data, as well as the model itself, can be considered as a reference climate. In Fig. 14 the probability functions of the heating and cooling loads are drawn for the months of January and June. Again they axe calculated using both the real data and the model with a generating period of 32 days. Comparison shows that the agreement at the extremes is sufficient to consider the mathematical model as a suitable tool not only for energy consumption calculation but also for design purposes.

= heating = cooling

..... 12

["]

error

lO

l'

8 4 2

12.

I I I I i

',

I

i

'

'i

days

[%] ~o. error 8-

4"

I

II

2 I

I

'

I

1

16days J 32days

number of d a y s generated per quarter

n u m b e r of days generated per m o n t h

a)

b)

Fig. 13. Error of mathematical model in yearly heating and cooling for data generated per month (a) and per quarter (b).

20-

= catcu~ted by real data (10 years)

>, 18= ..... o & 14-

= calculated by model ( 32 days I month)

10

6-

/•/i

DECEMBER

42~

-840-760-660-600-520-440-360-280-200-120-40

, \',~'-,, 0 40 120 200 280 360 440 520 600 heating-coollng load [W]

Fig. 14. Distribution functions of heating and cooling load.

CONCLUSIONS A synthetical reference year can be derived in the form of a mathematical model, which generates hourly data with the same statistical properties as the real climate. For a generating period of the model of 32 days per month the agreement with real meteorological data over a period of 10 years is of a good order. The yearly heating and cooling, calculated by means of both types of weather data, show a deviation of only 2% for heating and 3% for cooling. The amount of weather data can be reduced to 17.5% of that of a natural reference year by using 16 days per quarter, when deviations of 3.3% and 8%, respectively, can be accepted. The synthetical reference year can also be used for design purposes.

ACKNOWLEDGEMENTS The authors acknowledge with thanks the assistance of Prof. Dr. ir. H. G. Stassen, it.

160

O. H. Bosgra and P. Valk, in the field of the spectral analysis and the assistance of Prof. ir. A. W. Boeke in the field of air conditioning. The authors also thank ir. W. A. C. Bodewes for his contributions as a student at the initial stage of this project.

REFERENCES 1 G. M. Jenkins and D. C. Watts, Spectral Analysis and its Applications, Holden day, San Francisco, 1968. 2 Z. O. Cumali, Spectral analysis of coincident weather data for application in building heating, cooling load and energy consumption calculations, A S H R A E Trans., 76 (2159) (1970) part II. 3 R. J. Polge, E. M. Hollidag and B. K. Bhagavan, Generation of a pseudo-random set with desired correlation and probability distribution, Simulation, May 1973. 4 A. H. C. van Paassen and A. G. de Jong, The synthetical reference outdoor climate, Report WTHD 106, Dept. of Mechanical Engineering, Delft University of Technology, The Netherlands, May 1978.

APPENDIX

1

Generating independent signals In order to generate independent signals, satisfying a given probability function f(x) and autocorrelation function Kxx(r), the following procedure can be used. A "pseudo random n u m b e r generator" is applied, which delivers random values between 0 and 1 by means of the following equation:

the random series of numbers r into a new series of numbers y according to the delaying equation: (rn + TYn-1)/(~ + 1)

The time constant r is of the order of several hours, or days depending on the kind of signal. Next the order of the values in this time series y is used to rearrange the sequence of the original time series x in such a way that the arrangement according to the size of the values becomes the same as that of the time series y. By this procedure the time series x assumes approximately the desired autocorrelation. The time series x starts each time the generator starts, at the same value. Consequently various c o m p u t e r runs deliver the same time series, so that calculation results are reproducible.

APPENDIX 2

The crosscorrelation function between the stochastic variables ~ and y is defined as:

C~(r) K~

(r) -

-

o~o~

with 1

o~z = lin

--

T --*°o T

T/2

f

(x(t)-- ~ ) 2 dt

-T/2

= standard deviation

rn+l = {am + c ( m o d m)}/m The desired probability distribution function f(x) can be realised by using these random values r as values of the cumulative distribution function F(x) corresponding to f(x). The values x, which can be f o u n d by solving the equation F(x) = r, deliver values of a series satisfying the desired probability function f(x). To obtain the desired autocorrelation function a correlation procedure, as described by Polge et al. [ 3 ] , is used. These authors describe a simple m e t h o d , based on the assumption that the coherence is more defined by the way the values succeed each other than by the values itself. The m e t h o d used for the weather model is as follows. First a new series o f numbers y with the desired autocorrelation is generated. Here this is done by converting

-

1 =

lin

T/2,.

--

|

J T --*°° T -T/2

x(t) dt

= m e a n value

C~}(r)=

lin

17

--

T --*oo T

{x(t)--rT~}X

-TI2

{y(t + r ) - - ~ } dt = covariance function r = variable time interval If there is no correlation K ~ (r) = 0 and in case of a strong correlation K ~ (r) = 1. These functions can be Fourrier transformed. Information about the auto- and crosscorrelation for each frequency can then

161

be obtained. For example the Fourrier transformation 9f the cross-covariance function C ~ (r) is called the cross spectral density function Sxy(f). The coherence for each frequency between variables ~ and y is expressed by the coherence functions, defined as:

l IS~ (f)12 f ~'2

r~(f) = s~- ~ - ) ~ if)

The cross spectral density function can be described by the amplitude spectrum and the phase spectrum, which can be derived from respectively the real and immaginary parts of x(f) and y(f). In case x = y the functions mentioned above indicate the way values of one variable succeed another.