Stochastic Simulation of Non-stationary Meteorological Time-series

Daily Precipitation Indicators, Maximum and Minimum Air Temperature

Simulation using Latent and Transformed Gaussian Processes

Nina Kargapolova

1,2

Institute of Computational Mathematics and Mathematical Geophysics, Pr. Lavrentieva 6, Novosibirsk, Russia

Novosibirsk State University, Novosibirsk, Russia

Keywords: Stochastic Simulation, Non-stationary Random Process, Air Temperature, Daily Precipitation, Extreme

Weather Event.

Abstract: In this paper a stochastic parametric simulation model that provides daily values for precipitation indicators,

maximum and minimum temperature at a single site on a yearlong time-interval is presented. The model is

constructed on the assumption that these weather elements are non-stationary random processes and their

one-dimensional distributions vary from day to day. A latent Gaussian process and its threshold

transformation are used for simulation of precipitation indicators. Parameters of the model (parameters of

one-dimensional distributions, auto- and cross-correlation functions) are chosen for each location on the

basis of real data from a weather station situated in this location. Several examples of model applications are

given. It is shown that simulated data may be used for estimation of probability of extreme weather events

occurrence (e.g. sharp temperature drops, extended periods of high temperature and precipitation absence).

1 INTRODUCTION

For solution of different applied problems in such

scientific areas as hydrology, agricultural

meteorology and population biology, it is quite often

necessary to take into account statistical properties

of different meteorological processes. For example,

it may be necessary to consider probability of

occurrence of meteorological elements combinations

contributing to forest fires spread, probability of

frost occurrence in spring and summer, average

number of dry days, etc. Since real data samples are

usually small, real data based statistical investigation

of rare and extreme weather events is in most cases

unreliable. Therefore, instead of small real data

samples it is necessary to use samples of simulated

data.

In this regard, in recent decades a lot of scientific

groups all over the world work at development of

so-called "stochastic weather generator". At its core,

"generators" are software packages that allow

numerically simulate long sequences of random

numbers having statistical properties, repeating the

basic properties of real meteorological series. Most

often series of surface air temperature, daily

minimum and maximum temperatures, precipitation

and solar radiation are simulated (Furrer, 2007;

Kargapolova, 2012; Richardson, 1981; Richardson,

1984; Semenov, 2002). Not only single-site time

series, but also spatial and spatio-temporal

meteorological random fields are simulated with the

use of "weather generators" (Kleiber, 2012;

Ogorodnikov, 2013; Kargapolova, 2016). It should

be noted that practically all “weather generators”

possess same drawback: a model that describes well

main properties of a weather process over some

region or at several locations may be totally

unsuitable over another region (with different

physiographic characteristics). At the same time,

models that reproduce well characteristics of a

weather process on a relatively short time-interval (a

week, a month) may not be applicable for longer

periods of time (season, year) and vice versa. It

means that for each specific applied problem

solution it is always a good idea to try several

“weather generators” and then to choose the one that

“works” better.

In this paper a stochastic parametric simulation

model that provides daily values for precipitation

indicators, maximum and minimum temperature at a

single site on a yearlong time-interval is presented.

The model is constructed on the assumption that

Kargapolova, N.

Stochastic Simulation of Non-stationary Meteorological Time-series - Daily Precipitation Indicators, Maximum and Minimum Air Temperature Simulation using Latent and Transformed

Gaussian Processes.

DOI: 10.5220/0006358801730179

In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017), pages 173-179

ISBN: 978-989-758-265-3

173

these weather elements are non-stationary random

processes and their one-dimensional distributions

vary from day to day. A latent Gaussian process and

its non-linear transformation (so called threshold

transformation) are used for simulation of

precipitation indicators. Parameters of the model are

chosen for each location on the basis of real data

from a weather station situated in this location.

Several examples of model applications are given. It

is shown that simulated data may be used for

estimation of probability of extreme weather events

occurrence.

2 MODEL DESPRIPTION

In this section a formal theoretical description of a

considered stochastic model is given. Assumptions

about properties of a real weather proses that were

used for model construction are specified.

A model is constructed for simulation of joint

time-series on twelve-month time interval. It is

supposed that one-dimensional distribution of daily

maximum and minimum temperature are Gaussian.

This assumption is in good agreement (in sense of



-criteria) with long-term observation data from

weather stations. Parameters of these Gaussian

distribution vary from day to day. Figure 1 illustrates

variation of daily maximum and minimum

temperature sample average on a yearlong interval.

Figure 1: Sample average of daily minimum (1) and

maximum (2) temperature. Years of observation: 1976 –

2009. Novosibirsk, Russia.

Daily precipitation indicator in a day number

j, j 1, N is defined as 1 if amount of precipitation

during this day in more of equal than

0.1 mm and as

0 otherwise. It is supposed that

365 (for

convenience data for February 29 is not taken into

consideration). This means that daily precipitation

indicator is a binary random process. Joint time-

series of mentioned above weather elements are

assumed to be non-stationary on twelve-month time

interval.

Each simulated model trajectory is a matrix





TTT

MI,A,E



, where column-vector



12 N

II,I,,I



 is a vector whose component

is daily minimum air temperature in a day

number

j ,



12 N

AA,A,,A



 is a vector of

daily maximum temperatures and column-vector



12 N

EE,E,,E



 is a vector of daily

precipitation indicators.

Elements of a joint time-series

M are calculated

with the help of transformations

II I

jj j

AA A

jj j

1, c ,

0, c ,

 

























(2.1)

where vectors

IAIA

11 11

IAIA

22 22

IAIA

NNNN

,,,

 











are mean and standard deviation vectors, j1,N .

Threshold values

are defined from equations



PE 1 exp dt p,j 1,N.









   







Values of

IAIA

jj jj

,,,,p

 

are estimated on a

basis of real data from a weather station. It is

obvious that such way of model parameters

definition make it possible to take into account

seasonal variations of real weather processes It

should be noted that for all

j1,N equality



is true if and only if

0.5

, inequalities

c0

and

0.5 are equivalent. Hereafter it is supposed

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

174

that

0, p 1, i 1, N. Variables





IAE

jj j





are components of a joint Gaussian process













TTT

IAE

 









 

with zero mean and

specific correlation matrix

II IA IE

AI AA AE

EI EA EE

GG G

GG G G .

GG G











Matrix G must be such that a process





TTT

I,A,E





after transformation (2.1) has a

correlation matrix

II IA IE

AI AA AE

EI EA EE

RR R

RR R R ,

RR R











that is equal to sample correlation matrix. Method of

matrix G calculation is described below. Dimension

of matrixes

G and

is 1095 1095 (3N 3N



Element



ri,j of a matrix block

R is a

correlation coefficient between

X and

(

  

X,Y I,A,E , i,j 1,2, ,N ). Element



gi,j is corresponding to



ri,j correlation

coefficient of a Gaussian process.

Let’s take a closer look at the matrix

G and find

equations that define this matrix when the matrix

is given. In (Ogorodnikov, 2009) a special case of

such equations was considered. Normalisation of

two correlated Gaussian random variables doesn’t

change a correlation coefficient between them,

which implies

II II AA AA

IA IA AI AI

GR,G R,

GR,GR.



(2.2)

Definition of a correlation coefficient leads to

equations



ij i j

EI E EI EE

ri,j

DI DE





(2.3)







gi,j

exp c 2 , i, j 1, N.

2p 1 p

  



These equations fully define matrix

G . Matrixes

EI AE EA

G,G ,G are defined in a similar way. Since













ij ij

iijj

ij i j

PE 1,E 1 pp

ri,j ,

p1p p1p

P E 1, E 1 P c , c , i, j 1, N









   

following equalities hold for i, j 1, N & i j,









ijEE ij

iijj

F c ,c ,g i,j p p

ri,j ,

p1p p1p







(2.4)

where











Fh,k,

exp x 2 xy y dxdy.







 









 











Obviosly,









EE EE

ri,igi,i1,i1,N.

It should be noted that equations (2.4) don’t have

any analytical solutions, but it is possible to solve

them numerically. So, equations (2.2) – (2.4) define

matrix

G and, finally, we may formulate the

simulation algorithm.

Algorithm:

Step 1. Estimate

IAIA

jj jj

,,,,p,j1,N

 

 on

a basis of real data.

Step 2. Solving equations (2.2) – (2.4) define

matrix

G .

Step 3. Simulate required number of trajectories

of a joint Gaussian process

 with zero mean and

correlation matrix

G .

Step 4. Using equalities (2.1) transform

trajectories of a Gaussian processes into trajectories

of a non-Gaussian process





TTT

M I ,A ,E





If verification of obtained trajectories gives

satisfying result, these trajectories may be used for

study of rare / extreme events.

Remark 1. Due to a physical sense of daily

minimum and maximum temperatures, an inequality



must be true for all

j1,N

. But

transformation (2.1) doesn’t guarantee it. This

means that one must eliminate from consideration all

trajectories in which this inequality violates. In

Stochastic Simulation of Non-stationary Meteorological Time-series - Daily Precipitation Indicators, Maximum and Minimum Air

Temperature Simulation using Latent and Transformed Gaussian Processes

175

practice, it is typical that



 and



are

relatively small, so usually there are few trajectories

with

IA

Remark 2. Equations (2.4) are solved

numerically, so some computational errors may

appear. These errors influence on the matrix

G and

it may happen that obtained matrix

G is not

positively-defined. In this case before a Gaussian

process simulation a normalisation of the matrix

must be done (see, Ogorodnikov, 1996). There are a

lot of algorithms for simulation of a Gaussian

process with given correlation matrix. The most

common are algorithms based on

LU  decomposition of the correlation matrix and

on its spectral representation.

Remark 3. Numerical solution of equations (2.4)

is a time-consuming problem. There is a way to

reduce computational time. So-called Owen’s

formulas (Owen, 1956) give a representation of

function



Fh,k,



via one-dimensional integrals:



ijEE i j

i1 j2

F c ,c ,g i,j c c

Tc,a Tc,a ,

  

 

c0,c0 or

c0,c0, and



ijEE i j

i1 j2

F c ,c ,g i,j c c

Tc,a Tc,a ,

  



c0,c0

c0,c0,

where







c exp t 2 dt,

 













j i EE i j EE

EE EE

c1t

1dt

Tc,a exp ,

c cg i,j c cg i,j

a,a .

c1gi,j c1gi,j



























This representation together with the fact that



c,i1,N

 

let to replace equations (2.4) with equations



ijij

iijj

i1 j2

iijj

pppp

ri,j

p1p p1p

Tc,a Tc,a

p1p p1p















(2.5a)

cc 0



ij ij

iijj

i1 j2

iijj

111

ppp

222

ri,j

1p p1p

T c ,a T c ,a

p1p p1p













(2.5b)

cc 0





gi,j

2T c ,

1g i,j

ri,j ,

p1p

















(2.5c)

c0,c0,







gi,j

2T c ,

1g i,j

ri,j

p1p

















(2.5d)

c0,c0



 and



2arcsing i,j

ri,j



(2.5e)

cc0



Numerical experiments show that computational

time required for solution of equations (2.5a) –

(2.5e) is approximately 4 times less than

computational time required for solution of

equations (2.4). This is due to the fact that

computation of a one-dimensional integral is much

simpler than computational of a bivariate integral.

Remark 5. In some numerical experiments,

obtained matrix G was ill-conditioned and didn’t let

accurate simulation of a Gaussian process. It calls

for further investigations to find out conditions when

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

176

matrix

G is ill-conditioned. Ways of matrix

correction are also have to be found.

Remark 6. Since correlation coefficients



gi,j

may be found from equations (2.4) (or

(2.5a) – (2.5e)) independently from each other and

trajectories of the Gaussian process are also

simulated independently, parallel computing

technologies may be easily applied for simulation of

the process





TTT

MI,A,E





3 NUMERICAL EXPERIMENTS

Described above stochastic model was used for

simulation of joint meteorological non-stationary

time-series on more than 50 weather stations situated

in different climatic zones in Russia. Verification of

the model shows that the model gives satisfactory

results for most of the stations. Here is an example

of a process characteristic that was used for the

model verification. Average numbers of days in a

month, when minimum temperature is below

and maximum temperature is above

(

I0,A0), estimated on basis of real and

simulated data, were compared. This characteristic is

not the model input parameter, so it can be used for

verification. Table 1 presents values of this

characteristic. It can be seen from Table 1, that the

model reproduces this characteristic accurately (up

to a statistical mistake).

Table 1: Average number of days with

I0,A0

. St.

Petersburg, Russia.

Month

Average number of days

Real data Simulated data

Octobe

4.7 4.9

Novembe

8.9 9.1

Decembe

5.7 5.3

January 9.3 9.4

February 7.7 7.4

March 16.3 16.8

Since the model is adequate to real weather

processes, it may be used for study of rare / extreme

events. Here are several examples. Hereafter all

estimations on basis of real data were done for years

of observation from 1976 to 2009 and estimations

based on simulated data were done over

trajectories.

First considered characteristic is a probability of

low temperatures and light frosts in spring and

summer. These weather events may negatively

influence on open-ground planted crop species.

Since different species have different resistance to

frost, it is necessary to take probability of low

temperatures and light frosts into account when

choosing a varieties or species of plants. Formally,

considered characteristic may be written as







 when j varies from 121 (May 1) to 243

(August 31). Here



C (deg. Celsius) is a given

temperature level. Table 2 presents estimations of







 obtained on basis of real and simulated

data. For real and simulated data estimations two

and three, respectively, fraction digits are

significant. During years of observation, there were

no days in considered period with temperature below



C (this is subminimum temperature for most of

plants species), but it doesn’t mean that such

temperature drop is impossible. Simulated data

provides an estimation of probability of this rare and

severe weather condition.

Table 2: Estimations of





PI

obtained on basis of

real and simulated data. Novosibirsk, Russia.







P I , j 121,234 

Real data Simulated data

2 0.081 0.085

0 0.030 0.031

-2 0.014 0.013

-6 0.000 0.001

Table 3: Average number of summer time-intervals lasting

k days, with absence of precipitation and daily minimum

temperature above 20

C. Astrakhan, Russia.

Period

length, days

Average number of time-intervals

Real data Simulated data

k=1 5.3 5.72

k=2 2.0 2.20

k=4 0.7 0.76

k=5 0.6 0.59

k=6 0.0 0.53

k=8 0.0 0.37

k=9 0.3 0.32

k=10 0.1 0.09

Another weather event that may be dangerous

both to individuals and to agricultural industry is

long-term combination of high air temperature and

absence of precipitation. Such combination may

negatively influence on individuals’ health and may

cause soil drying up. Table 3 presents average

number of time-intervals lasting

k days, when daily

minimum temperature was above 20

C and there

were no precipitations (only significant digits are

Stochastic Simulation of Non-stationary Meteorological Time-series - Daily Precipitation Indicators, Maximum and Minimum Air

Temperature Simulation using Latent and Transformed Gaussian Processes

177

given). Averaging was done over summer months.

Once again, described in the paper model reproduces

this characteristic for short time-intervals

satisfactory, so model results for longer time-

intervals may be considered as reasonable.

Finally, let’s consider such unpleasant weather

event as sharp temperature drop or rise during one

day (formally,

AI, where 

C is given

level.). Numerical analysis shows that this

characteristic is reproduced well for





5,14 . For

14

C real data estimations are unreliable. This

means that for applied problems solutions it is better

to use simulated data estimations. Table 4 presents

seasonal probabilities of such temperature variation

with

20

Table 4: Seasonal probabilities of

AI 20

C. Ulan-

Ude, Russia.

Season

Seasonal average number of days

Real data Simulated data

Winte

0.011 0.009

Spring 0.120 0.124

Summe

0.027 0.025

Autumn 0.017 0.021

4 CONCLUSIONS

In this paper a model for simulation of

meteorological time-series was considered. It was

also shown that simulated trajectories may be used

for study of rare / extreme events.

There are several ways of the model

improvement. For example, instead of Gaussian one-

dimensional distribution of daily minimum and

maximum temperatures a mixture of 2 Gaussian

distributions may be used. This will make

computation of the matrix

G much more complex,

because it will require usage of the inverse

distribution function method, but it will give a

chance to reproduce temperature behavior more

precisely. Simulation of precipitation indicators



may be replaced also by simulation of daily

precipitation amount



12 N

DD,D,,D





in a

form of a multiplicative process

jjj

DEC,j1,N,

where



12 N

CC,C,,C





is a conditioned

non-Gaussian random process describing amount of

daily precipitation on the assumption of their

presence. Such process representation is used in a

well-known “weather generator” WGEN

(Richardson, 1984), but indicator process in WGEN-

model differs fundamentally from process



considered in this paper. These two model’s

modifications are subject of further research.

ACKNOWLEDGEMENTS

This work was supported by the Russian Foundation

for Basis Research (grants No 15-01-01458-a, 16-

31-00123-mol-a, 16-31-00038-mol-a) and the

President of the Russian Federation (grant No MK-

659.2017.1).

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Stochastic Simulation of Non-stationary Meteorological Time-series - Daily Precipitation Indicators, Maximum and Minimum Air

Temperature Simulation using Latent and Transformed Gaussian Processes

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