Model Fitting and Optimal Design of Atmospheric Tracer Experiments: Part I

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W O R K I N G P A P E R

MODEL FTITING AND

0-

DESIGN OF

ATMOSPHERIC

TRACER MPERIKDJTS:

PART I.

Y.)! Fedorov

SX.

R t o v r a n o v

July 1988 W-88-65

l n t e r n a t ~ o n a l l n s t ~ t u t e for Appl~ed Systems Analys~s

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MODEL FTlT1.G

AND

OF'TIMAL DESIGN OF ATMOSPHERIC TRACER

EXPEXMENTS:

PART I.

V V F e d o r o v S.E. P i t o v r a n o v

July 1988 WP-88-65

W o r k i n g P a p e r s are interim r e p o r t s o n work of t h e I n t e r n a t i o n a l Institute f o r Applied Systems Analysis and have r e c e i v e d only limited review. Views or opinions e x p r e s s e d h e r e i n d o not n e c e s s a r i l y r e p r e s e n t t h o s e of t h e I n s t i t u t e or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , Austria

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Foreword

Dr. M. Dickerson, Deputy Division Leader of Lawrence Livermore National La- b o r a t o r y (USA) visited IIASA briefly in t h e autumn of 1987. In his discussions with t h e a u t h o r s of t h i s p a p e r , Dr. Dickerson suggested t h a t some of t h e statistical methods developed

at

IIASA and in t h e USSR might b e useful in t h e analysis of

at-

mospheric

tracer

data. Subsequently, Dr. Dickerson provided t a p e s of d a t a col- lected during a s e r i e s of field studies

at

t h e Savannah River Laboratory (USA).

The r e s u l t s of a preliminary analysis of t h e s e d a t a were r e p o r t e d by S. Pito- vranov

at

a workshop on Optimal Design of Environmental Networks, organized by t h e E l e c t r i c Power Research Institute (Palo Alto, California, May 1988) and by V.

Fedorov and S. Pitovranov at a subsequent seminar at Lawrence Livermore Nation- a l Laboratory.

In my view. t h e r e s u l t s are quite remarkable and d e s e r v e r a p i d publication.

In p a r t i c u l a r , traditional a p p r o a c h e s t o locating sampling stations downwind of a point s o u r c e are shown t o b e inefficient, resulting in ill-conditioned problems of p a r a m e t e r estimation.

I a g r e e with D r . Dickerson who summarized as follows t h e benefits t o b e derived from t h i s study:

Better design of field tracer experiments used t o evaluate models;

Better objective estimates of accident parameters, e.g., height of r e l e a s e and s o u r c e s t r e n g t h ;

Better testing of t h e sensitivity of model p a r a m e t e r s t o determine those t h a t are most c r u c i a l t o providing dose estimates;

Better placement of samplers following a n accidental release of material.

R.E. Munn

Head. Environment Program

-

ⁱⁱⁱ

-

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The r e s u l t s p r e s e n t e d in t h i s p a p e r

are

t h a t p a r t of IIASA's activity r e l a t e d

to

t h e application of t h e s t a t i s t i c a l methods in t h e optimization of monitoring networks (see, f o r instance, Fedorov

et

al. 1987; Fedorov and Mueller, 1988; Mueller, 1980).

The main a p p r o a c h i s based on t h e optimal experimental design theory. Two things are essential f o r t h i s approach: an experimenter must have a model. o r

set

of competitive models, which d e s c r i b e t h e observed p r o c e s s a p p r o p r i a t e l y and h e must formulate quantitatively t h e objective of t h e experiments. In t h e forthcoming P a r t 11, t h e monitoring network design problem will b e considered f o r cases which include p r i o r uncertain inputs, i.e., weather conditions during a designed experiment. Some corresponding t h e o r e t i c a l r e s u l t s have been r e p o r t e d by Atkinson and Fedorov (1988).

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MODEL ITlTING AND

0-

DESIGN OF

ATMOSPHERIC

TRACER EXPERIMENTS:

PART I.

K K

Fedorov and S.E. Pitovranov

Introduction

The t r a c e r experiments performed by Savannah River L a b o r a t o r y (SRL) in 1983, as p a r t of i t s Mesoscale Atmospheric T r a n s p o r t Studies (MATS), were used f o r examining t h e skill of t h e MATHEW/ADPIC coupled model in t h e prediction of t h e pollutant s p a t i a l distribution downwind from a point s o u r c e (Rodriguez and Rosen. 1984).

A comparison of p r e d i c t e d concentrations and t h e o b s e r v e d d a t a made by Ro- driguez and Rosen (1984) showed considerable d i s c r e p a n c i e s between model pred- ictions and t h e o b s e r v e d pollutant spatial concentration distributions in some of t h e tracer experiments. I t w a s recognized t h a t improvement of t h e simulation p e r - formance needs a b e t t e r experimental design f o r model evaluation including location of t h e r e c e p t o r s i t e s and t h e choice of meteorological conditions surrounding t h e r e l e a s e s .

In t h i s work w e c o n c e n t r a t e o u r attention o n t h e f o r m e r a s p e c t , namely t h e determination of t h e number a n d location of sampling sites. For t h e s a k e of c l a r i - ty. a simple Gaussian model h a s been chosen as a

test

f o r t h e proposed methodolo- gy of t h e experimental design.

MATHEIY s u p p l i e s a three-dimensional a r r a y o f winds t o ADPIC which sums t h e a d v e c t i v e and dif- f u s i v e components o f t h e v e l o c i t i e s t o d e s c r i b e t h e movement o f Lagrangian p a r t i c l e s i n a Eulerian framework.

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1. A short description of the SRL tracer experiment

The SRL r e c o r d s contain t h e r e s u l t s of t h e 1 4

tracer

experiments in which SF, was released

at

t h e Savannah River Plant

at

a controlled

rate

during 15-minute periods, from a height of 62 m.

P r i o r t o any r e l e a s e , a meteorologist predicted t h e most probable p a t h of t h e effluent cloud. This information

was

used

to

^guidea field operation team in t h e de- ployment of samplers. The sampling interval lasted 20-minutes and t h e s o u r c e re- c e p t o r distance w a s 30 km. The s e p a r a t i o n distance between samplers w a s approximately 1 t o 1.5 km. Figure 1 shows a typical sampler experimental layout.

From Figure 1 it c a n b e s e e n t h a t t h e samplers

are

located along t h e a r c (which in f a c t coincides with a r o a d ) in relation to t h e s o u r c e , marked by t h e l e t t e r S ( f o r details s e e Rodriguez and Rosen, 1984).

The d a t a base a l s o includes meteorological parameters. Wind speed and direction observations were analyzed t o obtain 15 minute a v e r a g e s as well as t h e standard deviations of t h e wind direction fluctuations. The locations f o r a subset of t h e stations a r e shown in single l e t t e r s in Figure 1. A t about 30 km north of t h e s o u r c e , a 304 m. television tower w a s instrumented

at

seven levels t o obtain t u r - bulent, wind and t e m p e r a t u r e d a t a which are a l s o a v e r a g e d e v e r y 15 minutes. All 14 experiments were conducted during daytime, between t h e h o u r s of 14:OO and 16:30. The wind speed in all experiments w a s in t h e r a n g e from 2 t o 5 m/sec.

2. Yodel description

The simplest and most extensively used model f o r local s c a l e dispersion i s t h e Gaussian model. The concentration distribution from a single r e l e a s e i s

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Figure 1: Location of the source ( S ) , 62m meteorological t o w e r s (single letters) and the samplers (small squares) during the experiment a t SRL

(MATS

8 , 22 July, 1983).

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where ^qmII i s a pollutant concentration; z l ,

zps z 3

are t h e coordinates of a sampler; t is t h e t r a v e l time; 6, and 19, are t h e mean wind speed along horizontal components

z

and

z 2

correspondingly; b1 is t h e total amount of material released a t time t =0; -02 i s t h e effective height of release; 1 9 ~ and $6 are t h e coordinates of source.

Variances o,,

uy

and a, are functions of t r a v e l time (see e.g., Doury, 1976).

A simple hypothesis is that:

More sophisticated functions can b e used in (2) but i t is not v e r y crucial f o r o u r considerations. The instantaneous s u r f a c e concentration is defined by t h e in- t e g r a l

where is t h e duration of t h e release.

Each sampling interval lasts t j

-

^{t j}

⁼

²⁰^min.. ^j^{= l ,}

...,

^k^,and t h e measured value is

It w a s assumed t h a t t h e observations contained a n additive "error":

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The

t e r m

cij comprises observational e r r o r s , random t u r b u l e n c e of atmos- p h e r i c flow, deviation of t h e m o d e l from t h e "true" behavior, i r r e g u l a r i t y of

ter-

r a i n , etc. In what follows i t i s assumed t h a t

zij

a r e random values with z e r o mean E(ri,)

=

0, independently identically distributed with f i n i t e v a r i a n c e

d.

More mm- plicated assumptions o n t h e v a r i a n c e of zij ( f o r instance 06-q2(zi , t j , d ) ) d e s e r v e t o b e considered, a n d t h i s will b e done in subsequent publications where more real- i s t i c q ( x , t ,$) will b e analyzed.

3. Model fitting.

Let u s consider 19

= . . .

^,

%)'

as unknown p a r a m e t e r s which should b e identified o n t h e basis of t h e o b s e r v a t i o n s

vij.

The d a t a of a

tracer

experiment MATS-8 (22 July, 1983) h a s been chosen

as

a p a t t e r n f o r t h e model fitting, see Ro- driguez and Rosen (1984). Under t h e abovementioned assumption o n cij i t i s reasonable from t h e s t a t i s t i c a l point of view

to

u s e t h e l e a s t s q u a r e estimator (1.s.e.) f o r identification of unknown p a r a m e t e r s .

The i t e r a t i v e second-order algorithm without calculation of d e r i v a t i v e s h a s been applied ( s e e Fedorov and Vereskov, 1985). This algorithm i s based o n ideas developed by Peckham (1970) a n d Ralston and J e n n r i c h (1978). Though t h e algorithm demands r a t h e r extensive intrinsic calculation, i t a p p l i e s only o n c e

at

e v e r y i t e r a t i o n

to

t h e subroutine where t h e s q u a r e residuals sum

v2(*)

= Cbij -

q(xi.tj*4)12 ( 6 )

# j

is calculated. The majority of o t h e r methods e i t h e r u s e t h i s subroutine

at

l e a s t

m

+1 times (

m

i s t h e number of unknown p a r a m e t e r s ) or demand t h e calculation of m derivatives av2($)/ 39

at

e v e r y iteration.

The computation shows t h a t t h e problem of simultaneous estimation of a l l unknown p a r a m e t e r s i s ill-posed. The variance-covariance matrix ( o r more a c c u r a t e l y

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its f i r s t o r d e r approximation) of estimated p a r a m e t e r s i s essentially ill- conditioned. This f a c t indicates t h a t t h e organization of t h e experiment (the design of t h e experiment) i s not a p p r o p r i a t e f o r t h e s t a t e d problem. Therefore, s e v e r a l reduced estimation problems were considered, each including only part of t h e nine abovementioned parameters.

R a t h e r reasonable r e s u l t s were found when p a m m e t e r s QJ, g4, 9,, Q8 were estimated (see Table 1 (a.b)).

Table 1: The estimates of dispersion p a r a m e t e r s and wind speed components 9,,9,.9,, 9,

(b) Variance-covariance and correlation matrices

A comparison of observed and computed concentrations f o r different sampling times can b e s e e n in Figure 2. The comparison shows t h a t t h e computed

results are

in agreement with t h e observed data.

Frequently, t h e estimation of power and time (Q9) of pollutant release i s a n important problem f o r practitioners, f o r instance, when t h e c h a r a c t e r i s t i c s of an accident are evaluated.

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In t h e t r a c e r experiments u n d e r consideration, t h e s e p a r a m e t e r s were d i r e c t l y controlled and t h e r e f o r e known r a t h e r a c c u r a t e l y .

For both p a r a m e t e r s t h e p r i o r values (which are initial f o r i t e r a t i v e least s q u a r e s p r o c e d u r e ) were chosen with 100% e r r o r s , i.e., 1 9 ~

=

1 2 0 g / s e c instead of t h e t r u e value 66.7 g/sec and 1 9 ~

=

1800 sec instead of 900 sec. Only g1 and 'rPg were estimated a n d all o t h e r p a r a m e t e r s were fixed ( 1 9 ~

=

7 2 m . 1 9 ~

=

0.26m / smc, gq = 7 . 4 m / G , 1 9 ~ = o . o ~ ,

g6

= o . o ~ , i~, = 3 . 7 m / s e c ,

=

+ . 5 m / s e c ).

Table 2: Estimates of t h e power and duration of release.

a

I

^a

dl g / s e c 1 9 ~ s e c

(b) Variance-covariance and c o r r e l a t i o n matrices

Similar numerical experiments were made with t h e estimation of t h e location of t h e s o u r c e . The computations showed t h a t t h e least s q u a r e s p r o c e d u r e a l s o al- l o w s o n e to identify t h e c o o r d i n a t e s of release with sufficient a c c u r a c y . (The actu- a l location was

at

t h e origin of t h e coordinate system.)

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- 9 -

Table 3: Estimates of t h e coordinates of r e l e a s e . (a) u2($)

=

0 . 3 8 . 1 0 ~

(b) Variance-covariance and c o r r e l a t i o n matrices

I t w a s found impossible t o identify t h e effective height of release from t h e S R L t r a c e r experiment sampler locations. Approximately t h e same residuals were f o r 19~ equal t o 210

meters

u2($) (0.37

-

^lo5)^asf o r 1 0 meters u2($) (0.39.

lo5

^).

The standard deviation of assessment of t h e effective height i s equal t o 1 2 7

m ,

4. Optimal design of experiment.

4.1. Uncontrolled s a m p l i n g i n t e r v a l s .

The set .$

=

Ipi ,xi i s usually called (see, f o r instance, Fedorov 1972) a design, where "weights" pi could b e t h e duration, frequency o r t h e precision of t h e observation which h a s t o b e made

at

a point zt (this i s called t h e "supporting point"). Searching f o r optimal design

#

providing in t h e sense of some objective function, t h e best estimations of unknown p a r a m e t e r s of

a

r e g r e s s i o n model (in o u r c a s e , model (5)) i s a traditional problem in optimal design t h e o r y ( s e e Fedorov, 1972; Silvey, 1980). The non-standard element in r e g r e s s i o n model (5) i s t h e dependence of t h e model upon t j which are known, b u t cannot be controlled.

The asymptotic information matrix in this case h a s t h e following s t r u c t u r e (for details see Fedorov and Atkinson, 1988):

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where f ( z , t , 9 )

=

a77/ 89 and 9, are t h e p r i o r values of t h e estimated p a r a m e t e r s . The optimal e x a c t design i s a solution of t h e following minimization problem:

ti ⁼

^Argmin O[n -I n m (zi )]

.

cn i =I

I t h a s to b e pointed out t h a t (8) admits r e p e a t e d observations

at

some points xi. In t h e sampler location problem, t h i s could c a u s e some difficulties: o n e cannot locate t w o or more samplers

at

t h e same site. Of c o u r s e , x; c a n b e considered as t h e c e n t r a l point of some relatively s m a l l area, where all t h e s e samplers c a n b e neigh- b o r s . For more information, see Section 5.

I t is c r u c i a l t h a t (7) h a s a n additive s t r u c t u r e a n d t h e r e f o r e t h e traditional r e s u l t s and algorithms ( s e e f o r example Fedorov, 1972; Fedorov

et

a l , 1987) c a n b e applied. The D-criterion (i.e.. *(D)

=

In

I

M ( , where D

= M

-'(t,) or some p a r t of it) w a s used as t h e optimality c r i t e r i o n in t h i s study.

4.2. E z p e r i m e n t s a d m i t t i n g d m e r e n t "weights".

The f i r s t s e r i e s of experiments were done to s e e k a n optimal design f o r t h e estimation of p a r a m e t e r s 1J3 a n d 1 9 ~ with various t r a v e l times. To avoid t h e calculation (which c a n b e v e r y extensive f o r a more sophisticated model) of t h e p a r t i a l d e r i v a t i v e s f ( z

, t

. 9 )

at

e v e r y i t e r a t i o n , t h e y were computed and s t o r e d on a regu- l a r g r i d with a mesh s p a t i a l s c a l e 1.0 X 1.0 km, with t h e help of t h e auxiliary program b e f o r e s t a r t i n g t h e i t e r a t i v e p r o c e d u r e . The same idea was used by Gribik et d . (1976) in o n e of t h e f i r s t a t t e m p t s

to

optimize regional air pollution monitor networks. The computed designs f o r s e v e r a l t r a v e l times c a n b e s e e n in Figure 3.

The optimal number of supporting points is e i t h e r 3 or 5. One station i s on t h e plume c e n t r e l i n e (in all exeriments i t i s assumed t h a t t h e wind s p e e d in

z z

d i r e c -

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Figure 3: Optimal sampler location for estimation of dispersion parameters for various travel times.

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tion i s equal

to

zero), and o t h e r s are allocated symmetrically. The distance between t h e c e n t e r l i n e station and o t h e r stations i n c r e a s e s from 2 t o 3 km when w e i n c r e a s e t h e t r a v e l time. The value of t h e determinant of t h e information matrix is t h e c h a r a c t e r i s t i c of informativeness of t h e observing network. From Table 4 one c a n see t h a t t h e value of t h e determinant changes considerably

as

a function of t h e t r a v e l time.

Table 4: Determinants of information matrix f o r various t r a v e l times.

Travel time, min 30 4 5 60 75 90 105 120

Assessment of a n accidental release needs knowledge of t h r e e main parame- t e r s : t h e s o u r c e s t r e n g t h (19~). time of release (19~), and effective height of release (19~). T h e r e f o r e , t h e corresponding optimal designs were computed f o r t h e model under consideration. The median wind direction during a n accidental non-elevated r e l e a s e i s assumed t o b e known. The problem i s t o allocate samplers t o estimate p a r a m e t e r s g l , 1 9 ~ and 1 9 ~ as precisely as possible. Optimal design f o r t h i s c a s e contains t h r e e supporting points allocated along t h e c e n t e r l i n e direction (Figure 4). The dependence of t h e determinant of variance-covariance matrix of estimated p a r a m e t e r s 1 9 ~ and 1 9 ~ f o r various t r a v e l times i s given in Table 5.

Table 5: Determinants of information matrix of estimates of release power, r e l e a s e time, and release height f o r various t r a v e l times.

Travel time, min 4 5 9 0 120

I t c a n b e seen t h a t t h e determinant d e c r e a s e s s h a r p l y when t r a v e l time i n c r e a s e s from 45 min t o 90 min. The s t a n d a r d deviation of t h e effective height of t h e release

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Figure 4: Optimal sampler locations f o r estimation of source strength, e f f e c t i v e height, and r e l e a s e duration f o r various travel times.

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estimate i n c r e a s e s from 6 t o 127

m e t e r s

correspondingly. This e f f e c t i s r e l a t e d t o t h e f a c t t h a t a t some distance from t h e s o u r c e , t h e r e l e a s e d material becomes well-mixed due t o v e r t i c a l dispersion. Therefore, a t a distance of 20 km from t h e s o u r c e i t i s impossible t o identify t h e height of t h e release.

5. Experiments with the prescribed number of samplers.

Assume t h a t a number of samplers i s available. The problem i s t o allocate t h e s e samplers in a n optimal way, i.e.,

to

find t h e solution of problem (8). under t h e constraint t h a t supporting points zi*, i

==,

cannot coincide and t h a t all weights pi are equal n -I. To find t h e corresponding solution one c a n apply t h e exchange t y p e algorithm developed by Fedorov, 1986.

The comparison of optimal allocation of 20 stations f o r t h e estimation of dispersion p a r a m e t e r s and

SRL

samplers allocation design c a n b e s e e n in Figure 5.

The samplers should b e allocated r a t h e r close t o t h e s o u r c e and t h e i r distribution o v e r t h e region should r e f l e c t t h e s h a p e of t h e pollutant cloud. The d e t e r - minant of information matrix of estimated p a r a m e t e r s f o r such a n allocation i s D

=

0.1. while t h e determinant f o r t h e

SRL

t r a c e r experiment i s equal t o 0.002.

6. A remark on the empirical design of sampler locations.

I t i s evident t h a t t o some e x t e n t any s e r i o u s physical experiment i s designed t o make i t sensitive

to

t h e p a r a m e t e r s of i n t e r e s t . Possibly t h i s w a s done when t h e original sampler allocation ( s e e Figure 1 ) w a s chosen.

For t h e model considered in t h i s p a p e r even t h e empirical approach leads nevertheless

to

d i f f e r e n t sampler allocations. This f a c t emphasizes t h e evident statement t h a t t h e optimal allocation essentially depends upon t h e model.

Suppose one wishes

to

evaluate t h e dispersion coefficients in m o d e l (1). Very roughly, t h e empirical p r o c e d u r e f o r t h e construction of t h e optimal sampler allo-

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Figure 5: The optimal allocation of 20 stations f o r t h e determination of dispersion parameters f o r various t r a v e l times.

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cation c a n b e d e s c r i b e d in o u r case as follows:

-

If t h e t e r r a i n i s uniform. t h e r e h a s

to

b e a symmetry in t h e sampler locations because of t h e symmetry of t h e considered m o d e l .

-

_A number of samplers h a v e

to

b e located along t h e c e n t e r l i n e d i r e c t i o n

to

measure t h e peak concentration and t h e

rest

of t h e available samplers h a v e

to

b e symmetrically

remote

from them. The distance between t h e s o u r c e of emission and t h e samplers

at

t h e c e n t e r l i n e direction i s mainly defined by t h e wind speed. Samplers have

to

b e located where t h e ground peak concentration i s sufficiently high (maybe t h e highest)

to

b e reliably measured.

- The o t h e r samplers have

to

b e located

at

points where i t i s possible to o b s e r v e t h e g r a d i e n t of t r a c e r concentration but, n e v e r t h e l e s s , t h e signal/noise r a t i o h a s

to

b e sufficiently high

to

obtain r e l i a b l e observations.

Manipulation with t h e known wind speed (- 5m/sec) and t h e most p r o b a b l e values of t h e dispersion coefficients f o r t h e given t y p e of weather conditions leads t o samplers allocation similar

to

Figures 3 and 5.

It is c l e a r t h a t in s p i t e of t h e use of some mathematics, o u r answer h a s a p a r t l y qualitative c h a r a c t e r . A t t h e same time t h e methods considered in t h e p r e - vious sections allow o n e to put t h e solution of t h e design problem on a well- formalized basis, converting t h e optimal design of sampler allocation into a r o u t i n e computing operation.

Probably f o r more sophisticated models t h a n (1) and more complicated e x p e r - imental situations, o n e h a s t o combine both a p p r o a c h e s .

1 . The allocation of a l l available samplers along one arc (see Figure 1 ) r e s u l t s in ill-conditioned problems of p a r a m e t e r estimation.

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2. The t r a c e r experiment should be designed with t h e linkage of t h e p a r a m e t e r estimation problem f o r c e r t a i n air-pollutant t r a n s p o r t model ( o r models, when one h a s t o choose between them). The optimal location of samplers is sensitive t o t h e s t ~ c t u r e of t h i s model ( o r models).

3. The optimal design depends upon t h e objective function, which h a s

to

r e f l e c t t h e experimenter's needs e x p r e s s e d in qualitative form.

References

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Acknowledgements

The a u t h o r s are grateful

to

M. Dickerson, R. Harvey and P. Gudicksen from Livermore Laboratory, and t o M. Antonovsky and R. Munn, f o r t h e i r constructive comments.