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On numerical modelling of one-dimensional stochastic wave problems
97 6. VILENKIN S.YA.,Statistical processing of the results of a study of random functions rezul'tatov issledovanya slyuchainykh funktsii), Energya, (Statisticheskaya obrabotka Moscow, 1979. (Splainy v 7. STECHKIN S.B. and SUBBOTIN YU.N., Splines in computational mathematics vychislitel'noi matematike), Nauka, Moscow, 1976. for estimating second derivative 8. SAKALAUSKAS E.I., The method of B-spline-regularization of measurable functions, Dep. at LitNIINTI, No.875-82 DEP, 1982. Translated
U.S.S.R.
Printed
0041-5553/84 $lo.oo+o.oo 01986 Pergamon Press Ltd.
vo1.24,No.6,pp.97-99,1984
Comput.Maths.Math.Phys.,
in Great
by D.E.B.
Britain
ON NUMERICAL NODELLING OF ONE-DIMENSIONALSTOCHASTIC WAVE PROBLEMS* 1.0. YAHOSHCHDK A high-accuracy difference scheme is constructed and studied for the equations of invariant immersion, and on its basis the wave propagation in a one-dimensional randomly inhomogeneous medium is modelled numerically. The property of ergodicity is proved for the equations of immersion. The problem of wave propagation in a one-dimensional medium with random inhomogeneities Both here has attracted attention for a long time, and a bibliography may be found in /l/. and abroad, several attempts have been made at a numerical solution of stochastic boundary value problems, see e.g., /2, 3/, averaging over an ensemble of realizations being used to find the statistical characteristics of the wave field. This approach is not promising, It is quite useless however, since a large ensemble is needed to obtain a reliable statistic. for actual physical problems, e.g., on sound propagation in the ocean, where there is usually just one realization. The method of invariant immersion, recently developed for wave boundary value problems, see e.g., /4/, enables us to transfer from boundary value problems to differential equations in the immersion parameter with initial data, for which there are well-known standard methods of computer modelling (see bibliography in /5/). If moreover we have ergodicity with respect to this parameter, so that the average over an ensemble can be replaced by an average over the immersion parameter for one sample of the process, wide scope is opened up for numerical of the statistical characteristics of wave fields in actual media. computation Below we construct a difference scheme for the immersion equations, having high accuracy and allowing the hypothesis of ergodicity to be studied. 1. Formulation of the problem. Let a layer of randomly inhomogeneous medium occupy erp[-ix(r-H)] be incident on it from the right. the part of space Hd
continuity
conditions
z2[i+>(r)+ir]U(t)=O
for C(Z) and con
the layer boundaries:
U(H)+&yH)=*, U(Ha) -I x We shall
assume
that
;(I) is a Gaussian
(1)
U’(H,)
(2)
= 0.
K
delta-correlated
random
function:
(&))=O,
;e(l)e(l,))=20'16(r-r'). Using the ideas of the method of invariant immersion, and introducing the dependence of the wave field U(r) on H, it "as shown in /4, that problem (l), (2) is equivalent to a system of equations having the form in dimensionless variables f=zD, h=MD (D=x'ozI/2 1s the diffusion coefficient) (we drop the tilde on I):
-+h)=[ia+(:c(h)-p/z)urJv(z;h). t
V(z;
I)=
u,,
Uh=2~aClh-?la +(i.z(h)~/2)V,+',u,--i,
(3) (4)
where Uh=I+Rh, Rh is the complex reflection coefficient of the wave from the layer, the parais the dimensionless wave n.umber, fl=xy/D, and the "white noise" z(h) has tie a=x/D meter characteristics i -2 (E(h))=li
2. Each of the equations initial
of system
data, of the type
*Z!~.vgchisl.~~at.mat.Piz.,24,11,17~'18-1751,1?83 "es* 2.,.-e
3), (4) is a stochastic
differentia-
eguatlcr: >::z:
98 h (0) = ho. ~h(h)=l(h,h)+g(h,h)e(lr),
(6)
where ~(1,)is white noise with parameters (5). One posslbie approach to constructrng difference scheme for (6) is to use some numerrcal method for the equation
f
k(h)-
(7)
h (hh) = ?“A.
f(h, h) + g(h. ~)EI,
a
Ah rsthed~scretization step), where the white rn every interval [h,.hr.+Ah](here and below noise is replaced by a limiting model rn the form of random pulses, havrng a fixed value of (References may be found rn J/). er rn each mesh interval. [hh, h*~Ah], r.e., a Obviously, if Eq.(7) can be solved analytically or, each rnterval recurrence relation is obtained: LPL,=F(~A,
e,, Ah),
(8)
where F is a function, we entirely get rrd of the errors due to the numerical methods applied to (7). The errors in the statistical characteristics (of course, when using a reliable I(h) will solely be connected wrth replacement of the white noise statistic) of the process by a stepped model. Consider the evaluation of the stochastic rntegrais .xhen solving (7) with the aid of [IIL. recurrence relation (8). Noting that i.rrI1s the exact solution of (7) in the interval hi +YL] we expand it in a Taylor series:
dividing on ensemble-averaging, of Markov process h(k):
Hence,
see /5/, the scheme
by 1)~. and passing
corresponds
to the ?rmit, we obtain the drift factor
to integration
in a symmetrized
sense.
3. The above method is applied to the system of lmmersron equations (3), (4). We introduce the regular mesh h*=liAh. and replace the whrte noise by a stepped approximation (II*)= e*=pr/(Ah)" in each interval [h,,&+,I (pk are Gaussian random quantities with parameters [/I~. ha+,] a system of ordinary differ0, (pkpLI)=&). Denoting fIr=rer-~/2, we have rn each interval ential equations (for each per, obtained by a random number generator) ;
U(t;h)=(ra+B*U*)U(z;h). ~(r;Ji)~hr;h*=u(~;h*).
d dkUh=2iaUh-2ia+0rU~z, which
is easiiy
fxlIhohl=Uh*,
solved: (9a)
(9b)
where
;(/I) (and hence also E(h)) Notice that, in actual physical wave propagation process, as a white-noiseprocess is the result behaves as a smooth function, and its representation
v-7,2
I 0
Fig.1. Moments of wave field intensity. Aircrages: 1 for (I), and 2 for (IL)over ensemble; 3 overone realization L=300
’
1
7
Average wave f:el! lntenslty. Averdalno: I ensemble, L' 3‘CZ i::i.: realizdtroi. , .li”
_i
Fiq.2.
-‘:fr
99 of asymptotic expansion of the function with respect to a parameter linked with its correlation Hence, for the differential equations of immersion to be stable to a passage to the radius. limit, we have to understand them in the symmetrized sense, this being realized by recurrence relations (9) as shown above. 4. When modelling system (9), the parameters a 3.Ah have to be specified. The first CaBI. since, two depend on the physical case of interest; in particular, we usually consider in the analytic working, the method of averaging over fast oscillations is generally used /l/. The statistical meaning of the parameter p is studied in /6/. For instance, with JBl, With 8x1, the influence of the phenomenological theory of radiation transport works well. in a way a limit case, since, with 8~1, see fluctuations becomes important. The case p=I /6/', the statistical effect particularly appears for the moments, starting with the second, $=I and of the field intensity (statistical parametric resonance /l/j. Below we choose 0.08. It is also obvious that the modelling step must be much less than the characteristic dimensions of the wave function variation, i.e., must be less than the wavelength: eARal. Using (9) with a--25 and Ah=O.Oi, we studied the statistical characteristics of 1.:r.Iii and Hh in the problem of wave incidence on a half-space (ho---). The ergodicity property was All the statistical characteristics were obtained by proved for the immersion equations. averaging over one process realization and were compared with the results of ensemble averaging, SeE? /6/. An an illustration we show in Figs.l-3 the values (ILILI'I") with !3=1 (Fig.1) and J=OilS (Figs.2, 3), calculated form the relation ‘ (,CIU.,",=~S~r,~LT.(z:I+E),". n=i.2, 0 for sufficiently
large L.
5. The elementary wave problem has been used above to shown the scope and efficiency of modelling of wave propagation in stochastic media on the basis of the equations of invariant More complex boundary value wave problems can immersion, using the hypothesis of ergodicity. also be re-stated as initial value problems, see e.g. /7/. Computer modelling of such problems with subsequent averaging over one realization means that we need not be concerned about the statistics of the parameters, and natural data can be used; this undoubtedly opens up new scope for solving practical problems of radio physics and acoustics.
M.F .
The author thanks V.I. Klyatskin Ivanov for his interest.
for suggesting
the problem
and useful
comments,
and also
REFERENCES 1. KLYATSKIN V.I., Stochastic equations and waves in randomly inhomogeneous media (StokhastichNauka, Moscow, 1980. eskie uravneniya i volny v sluchaino-neodnorodnykh sredakh), 2. BELOV V.D. and RYBAK S.A., On the application of transport equations in the one-dimensional problem, Akust. zh. 21, No.2, 173-180, 1975. 3. KOHLER W. and PAPANICOLAOU G.C., Power statistics for wave propagation and comparison with radiative transport theory, J. Math. and Phys. 15, No.12, 2186-2197, 1974. 4. BABKIN G.I. and KLYATSKIN V.I., Wave intensity fluctuations in one-dimensional randomly inhomogeneous medium, III, Influence of absorption and transport equations, Izv. vuzov, Radiofiz, 23, No.10, 1185-1193, 1980. 5. TIKHONOV V.I. and MIRONOV M.A., Markovprocesses (Markovskie protsessy), Sov. radio, Moscow, 1977. 6. BABKIN G.I. et al., Wave intensity fluctuations in one-dimensional randomly inhomogeneous medium, V, Numerical integration of the radiation transport equations, Izv. WZ, Radiofizika, 24, No.8, 952-959, 1981. 7. BABKIN G.I., KLYATSKIN V.I. and LYUBAVIN L.YA., Theory of invariant immersion and waves in statisticallyinhomogeneous media, Dokl. Akad. Nauk SSSR, 250, No.5, 1980.
Translated
by D.E.B.
U.S.S.R.
Printed
0041-5553/84 $lo.oo+o.oo 01986 Pergamon Press Ltd.
vo1.24,No.6,pp.97-99,1984
Comput.Maths.Math.Phys.,
in Great
by D.E.B.
Britain
ON NUMERICAL NODELLING OF ONE-DIMENSIONALSTOCHASTIC WAVE PROBLEMS* 1.0. YAHOSHCHDK A high-accuracy difference scheme is constructed and studied for the equations of invariant immersion, and on its basis the wave propagation in a one-dimensional randomly inhomogeneous medium is modelled numerically. The property of ergodicity is proved for the equations of immersion. The problem of wave propagation in a one-dimensional medium with random inhomogeneities Both here has attracted attention for a long time, and a bibliography may be found in /l/. and abroad, several attempts have been made at a numerical solution of stochastic boundary value problems, see e.g., /2, 3/, averaging over an ensemble of realizations being used to find the statistical characteristics of the wave field. This approach is not promising, It is quite useless however, since a large ensemble is needed to obtain a reliable statistic. for actual physical problems, e.g., on sound propagation in the ocean, where there is usually just one realization. The method of invariant immersion, recently developed for wave boundary value problems, see e.g., /4/, enables us to transfer from boundary value problems to differential equations in the immersion parameter with initial data, for which there are well-known standard methods of computer modelling (see bibliography in /5/). If moreover we have ergodicity with respect to this parameter, so that the average over an ensemble can be replaced by an average over the immersion parameter for one sample of the process, wide scope is opened up for numerical of the statistical characteristics of wave fields in actual media. computation Below we construct a difference scheme for the immersion equations, having high accuracy and allowing the hypothesis of ergodicity to be studied. 1. Formulation of the problem. Let a layer of randomly inhomogeneous medium occupy erp[-ix(r-H)] be incident on it from the right. the part of space Hd
continuity
conditions
z2[i+>(r)+ir]U(t)=O
for C(Z) and con
the layer boundaries:
U(H)+&yH)=*, U(Ha) -I x We shall
assume
that
;(I) is a Gaussian
(1)
U’(H,)
(2)
= 0.
K
delta-correlated
random
function:
(&))=O,
;e(l)e(l,))=20'16(r-r'). Using the ideas of the method of invariant immersion, and introducing the dependence of the wave field U(r) on H, it "as shown in /4, that problem (l), (2) is equivalent to a system of equations having the form in dimensionless variables f=zD, h=MD (D=x'ozI/2 1s the diffusion coefficient) (we drop the tilde on I):
-+h)=[ia+(:c(h)-p/z)urJv(z;h). t
V(z;
I)=
u,,
Uh=2~aClh-?la +(i.z(h)~/2)V,+',u,--i,
(3) (4)
where Uh=I+Rh, Rh is the complex reflection coefficient of the wave from the layer, the parais the dimensionless wave n.umber, fl=xy/D, and the "white noise" z(h) has tie a=x/D meter characteristics i -2 (E(h))=li
2. Each of the equations initial
of system
data, of the type
*Z!~.vgchisl.~~at.mat.Piz.,24,11,17~'18-1751,1?83 "es* 2.,.-e
3), (4) is a stochastic
differentia-
eguatlcr: >::z:
98 h (0) = ho. ~h(h)=l(h,h)+g(h,h)e(lr),
(6)
where ~(1,)is white noise with parameters (5). One posslbie approach to constructrng difference scheme for (6) is to use some numerrcal method for the equation
f
k(h)-
(7)
h (hh) = ?“A.
f(h, h) + g(h. ~)EI,
a
Ah rsthed~scretization step), where the white rn every interval [h,.hr.+Ah](here and below noise is replaced by a limiting model rn the form of random pulses, havrng a fixed value of (References may be found rn J/). er rn each mesh interval. [hh, h*~Ah], r.e., a Obviously, if Eq.(7) can be solved analytically or, each rnterval recurrence relation is obtained: LPL,=F(~A,
e,, Ah),
(8)
where F is a function, we entirely get rrd of the errors due to the numerical methods applied to (7). The errors in the statistical characteristics (of course, when using a reliable I(h) will solely be connected wrth replacement of the white noise statistic) of the process by a stepped model. Consider the evaluation of the stochastic rntegrais .xhen solving (7) with the aid of [IIL. recurrence relation (8). Noting that i.rrI1s the exact solution of (7) in the interval hi +YL] we expand it in a Taylor series:
dividing on ensemble-averaging, of Markov process h(k):
Hence,
see /5/, the scheme
by 1)~. and passing
corresponds
to the ?rmit, we obtain the drift factor
to integration
in a symmetrized
sense.
3. The above method is applied to the system of lmmersron equations (3), (4). We introduce the regular mesh h*=liAh. and replace the whrte noise by a stepped approximation (II*)= e*=pr/(Ah)" in each interval [h,,&+,I (pk are Gaussian random quantities with parameters [/I~. ha+,] a system of ordinary differ0, (pkpLI)=&). Denoting fIr=rer-~/2, we have rn each interval ential equations (for each per, obtained by a random number generator) ;
U(t;h)=(ra+B*U*)U(z;h). ~(r;Ji)~hr;h*=u(~;h*).
d dkUh=2iaUh-2ia+0rU~z, which
is easiiy
fxlIhohl=Uh*,
solved: (9a)
(9b)
where
;(/I) (and hence also E(h)) Notice that, in actual physical wave propagation process, as a white-noiseprocess is the result behaves as a smooth function, and its representation
v-7,2
I 0
Fig.1. Moments of wave field intensity. Aircrages: 1 for (I), and 2 for (IL)over ensemble; 3 overone realization L=300
’
1
7
Average wave f:el! lntenslty. Averdalno: I ensemble, L' 3‘CZ i::i.: realizdtroi. , .li”
_i
Fiq.2.
-‘:fr
99 of asymptotic expansion of the function with respect to a parameter linked with its correlation Hence, for the differential equations of immersion to be stable to a passage to the radius. limit, we have to understand them in the symmetrized sense, this being realized by recurrence relations (9) as shown above. 4. When modelling system (9), the parameters a 3.Ah have to be specified. The first CaBI. since, two depend on the physical case of interest; in particular, we usually consider in the analytic working, the method of averaging over fast oscillations is generally used /l/. The statistical meaning of the parameter p is studied in /6/. For instance, with JBl, With 8x1, the influence of the phenomenological theory of radiation transport works well. in a way a limit case, since, with 8~1, see fluctuations becomes important. The case p=I /6/', the statistical effect particularly appears for the moments, starting with the second, $=I and of the field intensity (statistical parametric resonance /l/j. Below we choose 0.08. It is also obvious that the modelling step must be much less than the characteristic dimensions of the wave function variation, i.e., must be less than the wavelength: eARal. Using (9) with a--25 and Ah=O.Oi, we studied the statistical characteristics of 1.:r.Iii and Hh in the problem of wave incidence on a half-space (ho---). The ergodicity property was All the statistical characteristics were obtained by proved for the immersion equations. averaging over one process realization and were compared with the results of ensemble averaging, SeE? /6/. An an illustration we show in Figs.l-3 the values (ILILI'I") with !3=1 (Fig.1) and J=OilS (Figs.2, 3), calculated form the relation ‘ (,CIU.,",=~S~r,~LT.(z:I+E),". n=i.2, 0 for sufficiently
large L.
5. The elementary wave problem has been used above to shown the scope and efficiency of modelling of wave propagation in stochastic media on the basis of the equations of invariant More complex boundary value wave problems can immersion, using the hypothesis of ergodicity. also be re-stated as initial value problems, see e.g. /7/. Computer modelling of such problems with subsequent averaging over one realization means that we need not be concerned about the statistics of the parameters, and natural data can be used; this undoubtedly opens up new scope for solving practical problems of radio physics and acoustics.
M.F .
The author thanks V.I. Klyatskin Ivanov for his interest.
for suggesting
the problem
and useful
comments,
and also
REFERENCES 1. KLYATSKIN V.I., Stochastic equations and waves in randomly inhomogeneous media (StokhastichNauka, Moscow, 1980. eskie uravneniya i volny v sluchaino-neodnorodnykh sredakh), 2. BELOV V.D. and RYBAK S.A., On the application of transport equations in the one-dimensional problem, Akust. zh. 21, No.2, 173-180, 1975. 3. KOHLER W. and PAPANICOLAOU G.C., Power statistics for wave propagation and comparison with radiative transport theory, J. Math. and Phys. 15, No.12, 2186-2197, 1974. 4. BABKIN G.I. and KLYATSKIN V.I., Wave intensity fluctuations in one-dimensional randomly inhomogeneous medium, III, Influence of absorption and transport equations, Izv. vuzov, Radiofiz, 23, No.10, 1185-1193, 1980. 5. TIKHONOV V.I. and MIRONOV M.A., Markovprocesses (Markovskie protsessy), Sov. radio, Moscow, 1977. 6. BABKIN G.I. et al., Wave intensity fluctuations in one-dimensional randomly inhomogeneous medium, V, Numerical integration of the radiation transport equations, Izv. WZ, Radiofizika, 24, No.8, 952-959, 1981. 7. BABKIN G.I., KLYATSKIN V.I. and LYUBAVIN L.YA., Theory of invariant immersion and waves in statisticallyinhomogeneous media, Dokl. Akad. Nauk SSSR, 250, No.5, 1980.
Translated
by D.E.B.