system parameters governed by jump processes: a model for removal of air pollutants

System Parameters Governed by Jump Processes: A Model for Removal of Air PollutantsAuthor(s): Michael SteinSource: Advances in Applied Probability, Vol. 16, No. 3 (Sep., 1984), pp. 603-617Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1427289 .

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Adv. Appl. Prob. 16, 603-617 (1984) Printed in N. Ireland

?Applied Probability Trust 1984

SYSTEM PARAMETERS GOVERNED BY JUMP PROCESSES: A MODEL FOR REMOVAL OF AIR POLLUTANTS

MICHAEL STEIN,* Stanford University

Abstract

The asymptotic moments of the concentration of a pollutant subject to rainout and dry removal are given. Both the lengths of the wet and dry periods and the wet and dry removal rates are allowed to be random, although the removal rate is assumed to be fixed within any one wet or dry period. The results hold in a more general setting than removal processes, so its is hoped that they will be applicable to other problems.

CONCENTRATION OF ATMOSPHERIC POLLUTANTS; AIR POLLUTION

Introduction

A major problem in air-pollution research is the modeling of aerosol concentrations in terms of fluctuations in source, sink and transport processes. For some pollutants, dry deposition and rainout are the two major sinks. The removal rate due to rainout tends to be much higher than that due to dry deposition, and if this variation in removal rates is not accounted for, average concentrations will be underestimated (Grandell and Rodhe (1978)). Baker et al. (1979) have computed asymptotic moments of the concentration of a

pollutant when the rainy/dry process is Markovian and the wet and dry removal rates are constants. Grandell and Rodhe (1978) have found the

expected lifetime of a particle subject to removal when the rainout rate can

change from one rainfall to the next and the dry deposition rate is 0. Grandell

(1982) gives a summary of the major mathematical results to date dealing with this problem. Other important papers on this subject include Rodhe and Grandell (1972) and Rodhe and Grandell (1981). In this work, asymptotic moments of the concentration are found when both wet and dry removal rates are random.

If transport can be ignored (see Baker et al. (1979), p. 40, for a discussion on when this assumption is reasonable), then the concentration of a pollutant at

Received 8 August 1983; revision received 19 October 1983. *Postal address: Department of Statistics, Stanford University, Sequoia Hall, Stanford, CA

94305, USA.

603



604 MICHAEL STEIN

time t, denoted by c(t), will approximately obey

dc(t) = q(t, c(t)) - s(t, c(t)), dt

where q and s are source and sink, respectively. If the sink is due to removal, and the removal rate does not depend on c(t),

then

s(t, c(t)) = d(t)c(t),

where d(t) is the removal rate. If the source rate also does not depend on c(t), then

dc(t) (1)dc = q(t) - d(t)c(t). dt

In this paper, we shall assume that q(t) and d(t) can change each time rain

stops or starts, but otherwise remain fixed. The length of a rainy or dry period, and the values of the source and removal rates during a rainy or dry period are all allowed to be random variables.

The main technique used here is to relate the distribution of the concentration at an arbitrary time to the distributions of the concentrations at times when rain stops or starts. Then, because it is easy to compute the asymptotic moments of the concentrations when rain stops or starts, the asymptotic moments at an arbitrary time can be found. The main purpose of this paper is to present the results of this work; justifications of the results will be heuristic and incomplete. A rigorous development of the material will appear in future work.

Model

We shall consider a more general framework than the removal process, as it is no harder to deal with, and allows us to see what aspects of the problem are

important to this method of solution. Consider a state space with two alternating states, 1 and 0. Let

St = 0 = a time at which state 1 is entered,

(2) t, = (n - 1)th time after t, at which state 1 is entered, and

n, = first time after t, at which state 0 is entered.

So, O0= t t< ta z<t2 " 2 . Let

(3) Yy= l

- t,= length of nth visit to state 1,

Y(o=

t,+

- t-

= length of nth visit to state O.



System parameters governed by jump processes 605

Assume {Yn}n=,

is an i.i.d. (independent and identically distributed) sequence of strictly positive random variables, and { Y?}=1 is an independent sequence of random variables which are also i.i.d. and strictly positive. Also assume that YI and Yo have finite variance and bounded densities. Denote

(4) T = EY1, (4)

To= EY?. For the rainy/dry process, t, would be a time at which rain starts, Y1 would

be the length of the rainy period, and t, would be when the rain stops. Let {W1,Y Y}= • be an i.i.d. sequence of random vectors, and let

{Wo, Yn,=l

be an independent i.i.d. sequence. W1 is the random vector of parameters of the process (i.e., source and removal rates) during the nth visit to state 1, and W? the parameters during the nth visit to state 0.

Define

1, if t, < t

=

t, for some n, i.e., if process in

(5) Ii(t)=

state 1 at time t,

0, if < t t~+l for some n,

Io(t) = 1 - MO(t). It is more usual to define the interval of time spent in a state as half-open on the right, but for technical reasons it is defined here as half-open on the left.

Now assume there is some process c(t) defined for t ?0 by

c(t) = c(0), for t= 0, c(0) some random variable

independent of { Y, Y%, W1, W.}.= I,

c(t) = c(t)Ii(t) + c(t)Io(t)

(6) = { c(t)al(Wl, t--

t.)+ bl(W1, t-

t,,)}I_<tm= n=1

?

o{c(i.)a?(W?, t-

f.)? b?(W?, t-

-n))}I{•,<t for t > 0,

where a1, b1, ao, and bo are continuous functions. Note that c(t) is well- defined for

t_ 0, since it is clearly well defined for t = 1f,

so c(F1)

is defined, then the process is also defined for t <t t , and so on.

In the special case of the removal process defined by (1),

WI= (DQ, 0l), where

(7a) D = removal rate during (ta, fi], the nth rainfall,

Ql source rate during (ta, t ].



606 MICHAEL STEIN

And,

WO = (D?, QO), where

(7b) Do removal rate during (f-, t+,,], the nth dry period,

QO = source rate during (,, t+j11.

This model allows the source rate to change when a rainfall starts and stops. Jan Grandell, in his comments on this paper, gives the following explanation as to how such a phenomenon could occur: 'Let us consider an air parcel rather

high up in the atmosphere. During a rainfall only parts of the concentration

generated by a source on the ground ever reach the air parcel, since the other

part is removed on its way up in the atmosphere.' For the removal process, if t > 7, then by solving Equation (1),

c(t) = c(-) exp -J d(s) ds

+exp {- jd(s) ds}j

q(s) exp fj d(u) du} ds.

Thus, for 7 = t,, and t. < t : 5i,

c(t)=c(tn) exp - D ds

+exp{- D ds}J Qexp D{ du ds

= c(tn)

exp (-Dl(t- t.)) + exp (-D(t- tn))QA

exp ((s - t )D') ds

Q1 c(tn) exp (-D(t - tn)) +- (1- exp (-Dl(t- tm))), D #0,

= D.

c(t,) + O1(t

- t.), DI=0.

Thus, for the removal process,

a w t- t) = exp (-Dl(t - tn))

Q1 (8)

(W ,nt- t

) (1 - exp (-Dl(t - tn))), D1 O,

b(W(t, t - t), = D0.

ao and bO are defined similarly, although for the general model given by (6), ao and bo are not required to be the same as aI and bl.




Another approach to this problem is to assume that the process extends

infinitely far into the past, which is the approach used by Grandell and Rodhe.

Then, c(t) will be a stationary process, and many of the difficulties in the next section can be avoided. However, if there are long-term trends in weather or source rates, assuming that the process extends indefinitely into the past may not be appropriate. By starting at an arbitrary initial state, we see explicitly that the steady state is obtained in the limit no matter what the initial state. More importantly, this approach is the appropriate one to use when we want to deal with the problems of non-stationarity. For example, it is easy to write down an exact expression for Ec(t,), and thus we can see how it differs from the limit of Ec(tk) as k -- oo.

Analysis

We first consider the general process defined by (6) at times when the state

changes. From (6),

c(tn+x) = ao(tnl - n, Wo)[aXl(F - tn, Wl)c(tn) + bl(t - tn, Wl)]

(9a) + bo(t+x- t-, W?) = Anc(tn) + B,,

where we define

A, = ao(tn+l- tn, W?))al(- - tn, Wl),

and

B, = ao(tn+ - i , Wl)bl(i, - tn, Wt) +

b(tn+x - - , WOn).

And,

(9b) c(tn+1)

= Anc(~t)

+ B,, where

A, = al(tn+1 -

tn+l, WIn+)ao(tn+l - F, Wn),

and t- ,

1

Bn

= a1x(in+x

- tx, Wl+x)b?(t+x -

tn, W?) ? bX(in+x

- +x, Wn+x).

Note that A, A,, where 2 means 'in distribution'. Also, (An, B,) is independent of

c(t,), (A.,,B,) is independent of c(fi), and {A,,B,}n=1,

{A,, Bn,}=1 are both i.i.d. bivariate sequences. If E IA1J <1, we see that for large n, the distant past will not contribute

much to the value of c(t) and c(~f). Thus, if E IBjI and E JB~11 are finite, it can be shown that as n - >,

(10) c(tC) -

c(too), and c(i,) 4 c(7o),

where c(to) and c(fo) are proper random variables which we take to be



608 MICHAEL STEIN

independent. (Here t.. and Z., are notational devices and should not be

interpreted as the limits of t, and i,.) Clearly, considering (9a), (9b), and (10), we must have

(11) c(t) A1c(to)+B1,

and c(ft') -

A1c(fto)+B1, where c(t.),

c(Lo) are independent of A,, B1, A1, B1. Thus, assuming the

moments exist, we can conclude

Ec(too)k = E(Axc(t.)+

BI)k >

(12a) Ec(to) = (EAk)- k () EC(to)k = (1- EA)-1 YtkA

- i=0 I

Thus the kth moment of c(t.) can be found iteratively. Similarly, k-1

()\ (12b) Ec(tFo)k

= (1 kEAk)-l - )Ec(. k)E( )i

i=0 I

Let us now find the relationship between the asymptotic distribution of c(t) and c(to.), c(f.). Consider

c(t)Ix(t)= {c(tn)a1(W t-t)+b1(Wn, t - t.)}IItt?, n=l

for large t. Only those terms in the sum for which n is of order t will contribute

significantly to the distribution of c(t)Ii(t).

Then the crucial point is that for n of order t, c(t,) and tn will be approximately independent for t, in the range it is likely to occur; that is, for t, - Et, = O(n'). Note that t, depends on what

happens over the entire process, but up to an error of order n' it is determined

by the value of t,_c,12. But c(tj) depends mostly on the recent past, and thus is

mostly a function of what happens after t,_c,?.1

Since what happens up to

t[n1C21 and after t,_4C]

are independent, it follows that c(tj) and tn are, in some reasonable sense, asymptotically independent. Then, since c(t) 4 (to), it seems plausible that for n of order t and for tn - Et, = O(n1), the distribution of

c(tj) given the value of t, is approximately the distribution of c(t.). So, for

large t,

(13) c(t)Ii(t)

XZ { c(tQ)al(W1, t- t.)+ bl(W1, t_- t.)}JIt?,

since by construction, W1 and i, - t, = are also independent of c(t,) and t-. Next, we apply the renewal theorem to

SP {t-t,>s, WX=w,

t, <tt}, rt=l




where X=(X1, X2, Xk) Y= (Y1

Y2," , Yk)

means xi - yi for i = 1, 2, ? ?

-, k.

By a simple renewal argument,

P P{t-t,>s, Wl1-w, t,<t-,}= P{t>s, itt,

Wl?w} n=l

+ P{(t- y)- tn > s, W1-w,

t, < (t- y),}P{t2E-dy},

where t2 E dy means t2 E [y, y + dy). Now, P{t > s, ti t, W1 w} is monotonic for t > s, and it is majorized by

P{tl = t'}, which is integrable; hence, P{t > s, F1 > t, W1 = w} is directly integra-

ble. Therefore, we may apply the renewal theorem (Feller (1971), p. 363) and conclude

lim E P{t-tn>s, W1? w, t,<tsf,}=

P{t>s, tF> t, W1=w}

dt t-n=1 Et2

1 = E{(tF - s)+; Wl w },

To+ TI where

.X+ = x, X 0,

{0, x<0.

Now define random variables Z1 and W1 which are independent of c(to) and c (T .), and

1 (14) P{Z >

s, W w} =-E{(YI- s)+; W1

w}.

Also define a random variable HI which is independent of Z1, W1, c(tJ), and c (F.o), and

TI 1, with probability + T To + T,

(15) HI

= To

0, with probability

To+ T+

Then as t- oo

P pit - t, > s, Wl - w, t,< t

-}--

PI{Z > s, Wl

--w,

il = 1} (16) n=

1 - T E{(Yi-s); W1< w}.

To+ T1



610 MICHAEL STEIN

Since a1 and b1 are continuous, from (13) and (16)

(17a) c(t)Ii(t)

4 {c(tO)al(W1, Z1)-+ bl(W1, Zl)}111. Similarly,

(17b) c(t)Io(t) {c(-t)ao(Wo, Zo) + bo(Wo, Zo)}(1- HI),

where (Wo, ZO) is independent of c(t,), c(fO), W1, Z1, and HI, and

(18) P{Z > s, WO <w}=- E(Y - s); W=w}. To

Thus, there exists a random variable c(oo) such that

c(t) X c(oo) e= {c(t,)a1(W1, Z1)+ bl(W1, Z1)}II1

(19) + {c(too)ao(W, Zo) + bo(Wo, Zo)}(1- Hi).

If c(t)k is uniformly integrable, then as t-- oo,

Ec(t)k _ Ec.(oo)k = E{c(to)al(W1,

Z1)+ bl(W1, ZI)}k To+ T, (20)

+

E{c(.O)ao(Wo, ZO)+ bl(W1, Zl)}k

O To+ T1

The distributions of (W1, Z1) and (Wo, Zo) are known, and the moments of

c(t.) and c(t) can be found using (12a) and (12b), so we have, in theory, a way to calculate any moment of c(oo).

If { 1W}=

and { 1Y'}=

are independent, {W}=}1 and { Yn}r= are indepen-

dent, and the state space is governed by a Markov process, then (19) and (20)

simplify considerably. First, Z' and W1 will be independent, as will Zo and WO. And, from the Markovian assumption, YI and Yo will have exponential distributions with means T1 and To, respectively. So,

P{Z'> s, W1i s}= -E{(I1- s); W =?

w}

1

(21a) = TI

E{(Y - s)+}P{W1 <- w}

= exp (-sI/T)P{W -< w}

= P{ Y > s}P{ W1 w},

and

(21b) P{Z> s, Wo?-

w}= P{Yo> s}P{WO? w}.

Now, from (6), we have

c(tn) =

c(tn)aX(W1, Y1)+ bX(W1, Y1),




from which it follows

c(T) c (t.)al(W1, Yx)+ bl(W1, YD).

But, since (W1, Yi) (W1, Z1) by (21a), we have

c('i.oo) c(t.)al(W1, Zl)+ bl(W1, Zl).

Similarly,

c(Q )Z c(t.)ao(Wo, ZO)+ bO(WO, ZO).

Therefore, (19) and (20) simplify to

(22) c(oo) X c(t)I1

+ c(t.)(1 - - ),

and

(23) Ec(oo)k = Ec()k + o Ec(to)k To+ Ti To+ TI

Computation of expectations

While the distribution of (W1, Y1) determines the distribution of (W1, Z1) via (14), it would be helpful to have an expression for Ef(W1, Z1) directly in terms of the distribution of (W1, Y1). Proceeding formally from (14),

a1 P{Z' E dz, W1-< w}= E{(Y1- z)+; W1-

w} dz az T1

= T1 a (YI - z)+; Wl w dz

1

= E{-Ii> z; WI w} dz

(we may ignore what happens at YI = z since YI has a density)

1 = - P{ Y > z, W1h sw} dz.

Thus,

1 (24) P{Z1edz, W1edw}= P{YI>z, W1edw}dz.



612 MICHAEL STEIN

So,

Ef(W,

Z),=

f(w, z)P{ YI > z, W,1e dw} dz TEf(W' Z)=

Z =0

L=

wzf f(w,

z)P{Yll Edy,

WlE dw} dz 1 z=0 Y=z

= f(w, z)P{Y1 e dy, W1 e dw} dz 1 f=0 =0

= P{Y dy, W1edw}[z

f=O (w, z) dz]

= 1 E f(z, W1) dz.

So,

(25) Ef(Zi, W1) = 1

E f(z', W) dz

gives us a way to compute expectations of functions of (Z1, W1) directly in terms of (YI, WI). A similar formula holds for (Zo, W?) and (Y, W~i).

Application to removal process

In this section, we find the asymptotic expected concentration of an aerosol under the removal model described earlier. Now, assuming DI 0 and Do 0, and using (8), (9a), and (9b),

A1 = exp (-DI Yl - Do Y), and

B1 e- D Y?)

1 - exp (-D11Y) 1-exp (-D Y?) B =exp (-DoYO)Q+ D011 D1 Do

Define

at = E exp (-DI Y1), ao = E exp (-Do Yo),

D1

and

= E(Qo 1- exp (-D?Y?))

Then, EA1= aoal, and EB1 = ao301+3o.




Similarly, EAi1=aa 1, and EB,= a1300+31.

Now, from (20) with k = 1,

Ec(oo) = E[c(to)

exp (-DIZ1)+ 1 1-exp

(-D1Z1)] T1 D' J To+ T1

+E[c()exp (-DOZ 1- exp (-DOZo)] To D) exTo + TI

where the distributions of (Z1, Q1, D1) and (Zo, Q0, Do) are given by (14) and

(18), respectively. From (12a) and (12b) with k = 1,

EB 1 _

ao31+ 3o EB, a o1 + 1

Ec(to)= E - , and Ec(tc)= 1 - EA 1

-oa• 1 - 1 - ao a

And, by (25)

1

Io1

1-exp (-DI Yi) E exp (-D1Z1) =

• EJ exp (-D{Z) dz =

-T E

D =

where

(1 - exp (-D Y1))

Similarly E exp (-DOZO)= 80/To,

where

8= E(1- exp (-D?

Y?)) Do Also by (25),

E( ll-exp (-DZ1) 1 Y 1- exp (-D{Z)

EJ- Q E 1 dz

D' T, fo D

1 E(exp (-D'Y')- 1 + D Y

T, I (D1)2

= y/T1,, where we define

=E(exp (-DOYY)- 1+ D1Y1)

1(D)2

Similarly,

E(Qo 1- exp (-DoZo)



614 MICHAEL STEIN

where

y exp (-D Y) - 1 + D Y?) To= E(2(D0)2

Thus,

[ a +3aoj8+0. 1 y1 T1 [a 1 30+g3 o1 80 Yo To - a0a1 T

TI To + T i 1 -

ao To ToJ To+T1 (26)

=61 a?615

+ ?6o+

+ 36?0

+ a36? ]

To+ TI 1 - aoal0Y1

We consider two special cases. If OQ = Q0 = Q with probability 1, D1 and YX are independent, Do and YO are independent, and the rainy/dry process is Markovian, then (26) simplifies to give

O (27) Ec(oo)= (T2ao + T2ax +2ToTxox). (To + T1)(1 -

a0ao1)

And, in this special case,

ai = E exp (-DIY) )= E{E{exp (-DI Yl) I DI}= + TD '

and

ao= E(l +D).

This result is derived by somewhat different means in Stein (1982). If

Do= 1/Td and DI = 1/TP

with probability 1, where Td and Tp are constants, then Equation (8) in Baker et al. (1979) is recovered.

By Jensen's inequality,

1 1 1 al1= E

DI-- T ,

and ao >-- 0 1+ TDi 1++ T+ED1 1+ ToED,

Applying these inequalities to (27), we see that Ec(oo) is underestimated if

D1 and Do are replaced by their expected values, ED' and EDO. For the second special case, assume = 0o= 1, and Do =0, all with

probability 1. Note that the results of this section hold even if DO =0 (or DI1= 0) as long as we define the various functions for D? = 0 (or DI = 0) by continuity. For example, define

1- exp (-DO Yo) = l 1- exp (-DO Yo), DO

O=0 D?~o DO




Then (26) yields

Ec(1)=T2 [f32+2To3i+a1To+Y+ (Y))] Ecl(0)

+=1.•2 o

+1E (( YO)2)

To +

Tr

1

-- 2t1 (28) (28)1= + Ty +1(Var yO_ T2)

To+ T, 1-a-xl+

1

which is the same as the average lifetime of a single particle given by Grandell and Rodhe (1978). Considering the relationship between residence times and concentrations given in Grandell (1982) in Equation (3), we see that (26) with

OQ = OO = 1 with probability 1 gives the expected lifetime of a single particle subject to both wet and dry removal.

Multivariate systems. The previous results can be generalized to allow c(t) to be a k x 1 vector. Then b0 and b 1 would be k x 1 vectors of functions, and ao and a' would be k x k matrices of functions. Then under appropriate conditions, (19) will still hold. The crucial condition is

max Ahi < 1,

where Ak, 2., k are the eigenvalues of E(lail), where A1 = (ai).

Other possible generalizations. In Equation (6), c(t) is a linear function of

c(t,) and c(t,). This model, while it may describe many systems, is still

unnecessarily restrictive. Consider the process

c(t)= E {f(c(tn), Wl, t

t,)}I <tt,, n=l

Ifo(c(tn), wo, t-tn

The results can also be extended to situations in which there are more than two states. However, relations between the distribution of c(t) at an arbitrary time and at times when the process enters a new state are not of practical value if nothing can be said about the distribution at times when states change. In the case of a process with arbitrary transition probabilities, it is difficult to say anything useful about the distribution of c(t) at times when states change. However, if the matrix of transition probabilities has certain simple structures, it is still possible to find the asymptotic moments at times when a specific state is entered. These cases will be discussed in future work.



616 MICHAEL STEIN

Conclusions

In theory, the results presented here give a method to find any asymptotic moment of a wet/dry removal model with random wet and dry removal rates. In practice, the formulas are hard to apply. The formula for the first moment, (26), is rather complicated; expressions for second and higher moments would be much more complex and difficult to interpret. Also, throughout this work, the distributions of the rainy/dry period lengths and removal rates have been assumed to be known. In practice, properties of these distributions must be estimated. Grandell and Rodhe (1978) and Grandell (1983) have done some work on the estimation of parameters in this problem, but more work needs to be done. Probably the most interesting practical result in this paper is the

expression for the asymptotic expected concentration when the source rate is constant, the rainy/dry process is Markovian, and the removal rate is independent of the length of the rainy/dry period, given by (27). This result is of

practical value because it is relatively simple and it shows that replacing the random wet and dry removal rates by two constant rates, as in Baker et al. (1979), causes the asymptotic expected concentration to be underestimated.

Since the results in this paper do not depend on the specific structure of the removal process, we hope that they can be applied to other problems. Any process in which parameters remain fixed for a random length of time and then all change simultaneously when the 'state' of the system changes could conceiv-

ably be approximated by the models described in this paper.

Acknowledgements

This work was supported by the SIAM Institute for Mathematical Studies (SIMS). I would like to thank Paul Switzer for suggesting the problem, David

Siegmund for advising me during this work, and Jan Grandell for his many useful comments on the paper.

References

BAKER, M. B., HARRISON, H., VINELLI, J. AND ERICKSON, K. (1979) Simple stochastic models for the sources and sinks of two aerosol types. Tellus 31, 39-51.

FELLER, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.

GRANDELL, J. (1982) Mathematical models for the variation of air-pollutant concentrations. Adv. Appl. Prob. 14, 240-256.

GRANDELL, J. (1983) Estimation of precipitation characteristics from time-integrated data. Trans. 9th Prague Conf., Reidel, Dordrecht, 263-268.

GRANDELL, J. AND ROHDE, H. (1978) A mathematical model for the residence time of aerosol particles removed by precipitation scavenging. Trans. 8th Prague Conf. A, Riedel, Dordrecht, 247-261.




ROHDE, H. AND GRANDELL, J. (1972) On the removal time of aerosol particles from the

atmosphere by precipitation scavenging. Tellus 24, 442-454. ROHDE, H. AND GRANDELL, J. (1981) Estimates of characteristic times for precipitation

scavenging. J. Atmospheric Sci. 38, 370-386. STEI, M. (1982) Computing expected aerosol concentrations when source and deposition rates

are random.



system parameters governed by jump processes: a model for removal of air pollutants

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