06 from browian motion to stochastic shoreline...

From Brownian motion to stochastic From Brownian motion to stochastic From Brownian motion to stochastic From Brownian motion to stochastic

shoreline evolutionshoreline evolutionshoreline evolutionshoreline evolution

Dr. Xingzheng WU

Prof. Ping Dong

project meeting for RF-PeBLE in Plymouth Dec 2007EPSRC Grant EP/C005392/1

EE--mail: mail: [email protected]@dundee.ac.uk

University of Dundee, UKUniversity of Dundee, UK

ContentsContentsContentsContents

�1 Brownian motion

�2 Ito Stochastic shoreline evolution model

�3 Euler solution

�4 Fokker-Planck equation numerical solution

�5 Fokker-Planck equation analytical solution

�6 Conclusions

�7 System reliability analysis

From Brownian motion to stochastic shoreline evolutionFrom Brownian motion to stochastic shoreline evolutionFrom Brownian motion to stochastic shoreline evolutionFrom Brownian motion to stochastic shoreline evolution

0 shoreline evolution process

1 Brownian motion

Brownian motion (Wiener process) in three-dimensional space (one sample path shown) is an example of an Itō diffusion. (from Wikipedia)

Which is the random movement of particles suspended in a fluid or the mathematical model used to describe such random movements, often called a [partical theory].

1 Brownian motion --- in stock

Alongs

hore

Jetty

Stock market

Nasdaq

FTSE 100

stock price follows Black-Scholes model

is the stock volatility , is the stock drift, and Bt is a Brownian motion. Wt is wiener process

0 100 200 300 400Time (day)

0

10

20

30

40

50

y m (

m)

ModelAnalyticalNumericalFPtttt WStSS ddd σµ +=

tS

σ µ

The mathematical model of Brownian motion has several real-world applications. hydraulic, environmental, and biological fields An often quoted example is stock marketfluctuations

1 Brownian motion

{ }h

ytytyhtyEytu

ht

=−+=

→

)()()(Lim),(

0

[ ]h

ytytyhtyE

yth

t

=−+

=→

)()()(2

1

Lim),(

2

0

2σ ))(( stT −σσx

q

dytu t ∂

∂−= l

c

1),(

Langevin equation is a stochastic differential equationin statistic physic describing Brownian motion

dt

dBytytu

dt

dy ttt

t ),(),( σ+=

dtdBW t

t = Wt is wiener process

1 Brownian motion --- Schemes

Alongs

hore

Jetty

1) Fokker-Planck equation, which provides a deterministic equation satisfied by the time dependent probability density.( path integration; Numerical; Analytical)

2) numerical solutions to capture the trajectories (Euler)3) Fully Monte Carlosimulation 4) Schrödinger equation(quantum mechanics)

Brownian

Motion

Shoreline

Evolution

2 One-line coastal recession model

12/12/3

w2/5

l0 16

1agHq γρ=

),(cos),(sin),(),( 0 txtxtxqtxq bbll αα=

10 )( aCEq gbbl =

gn

ka

ws )1)(1/(161

1 −−=

ρρ

x

txq

Dt

txy l

c ∂∂−=

∂∂ ),(1),( (1)

(2)

Only longshore transport component considered as a contributionThe general equation for the deterministic process

2wb 8

1gHE ρ= ghC

2

1gb =

Hh 78.0≈41.01 =k 65.2/ ws =ρρ

4.0=n 247.01 =a

qi-1

qi

∆x

∆y

dc

Original New

z

x

y

bα

Wave crests

2 Ito Stochastic shoreline evolution

model

)(mw tKH σξ+=

Random fluctuations, such as rapid environmental changes, affectthe system through external parameters. Erosion or accretion is a complex process consisting of many components such as wave storm, tidal, currents, etc. Trying to integrate all these processes into a physical model seems to be hopeless.

wb

c1

2/12/3wl

)2(sin1

8

1H

xdagq

∂∂

= − αγρ

Based on one-line determinstic model, we have

2 Ito Stochastic shoreline evolution model

With a random variables wave height and boundary condition

Which is a standard random differential equation with a input term and random initial condition

is a Brownian motion in physic, called by Wiener process

dt

dWt

tW

denotes for a white noise

)(mw tWKH +=w

2/5 HH = mKCD v= 10 ≤≤ vC

The variance is D

),0(~)();,()]([d

),(dm DNtWtytWK

t

txy φ+−=

Substitute one-line model to random differential equation, then

a random process model of shoreline position can be given

Which can be reformulated as

)(),(),(d

),(dm tWtyGKty

t

txy += φ ),(),( tytyG φ=here (8)

(7)

Transfer function

Drift term (mean)& random diffusion (variations)

)/()2sin2(sin16

1),( cb

1b1

2/12/3w xdagty ii ∆−= − ααγρφ


)(d),(d),(),(d m tBtyGtKtytxy += φ

{ } 0)(d =tBE [ ]{ } tDtBE d2)(d 2 =

Above equation is equivalent (see Soong, 1973) to the stochastic Ito equation

which D is the entry of the covariance parameter of the white noise process

dt(9)


[ ]))()((),(),( m1 jjjj WWtyGtKtyyy ττφ −+∆+= −

3 Euler-Maruyama scheme

1)1,0(),(),(m

<∆

∆+∆=

y

NttyDttyKC

φφ

In order to obtain the statistical characteristics of shoreline position including their mean and variance based on a large number of Brownian paths in the numerical experiments for the longshore transport.

Euler-Maruyama scheme (Kloeden et al. 1994) of Ito equation (9) can be given by Monte Carlo sampling to obtain shoreline trajectory

4 SDE – Fokker Planck Equation

[ ] [ ]pGDGy

pKtyyt

typ T )(2

1),(

),(2

2

m ∂∂+

∂∂−=

∂∂ φ

∫ =max

min

1d),(y

yytyp

0),(

0),(

)(),(

min

max

00

==

=

typ

typ

yptyp

It is known that system described by equation (9) has a transition PDF (refer to Soong, 1973), satisfies the Fokker-Planck equation

(10)

Therefore the problem of determining the probability density istransformed into the problem of solving above partial differential equation.

normalization conditioninitial and boundary condition

4 results given by numerical model

The behaviour of the model is evaluated by applying it to two simple shoreline configurations

1) a single long jetty 2) a rectangular beach nourishment

4 results given by numerical model

0 500 1000 1500 2000 2500Longshore distance (m)

0

20

40

60

80

100

120

y m (

m)

ModelAnalyticalNumericalFPMonte Carlo

Comparison of shoreline changing ym by various model

-10000 -5000 0 5000 10000Longshore distance (m)

0

10

20

30

40

y m (

m)


Fig. 7b The coutours of PDF at 80th

Fig. 7a PDF surface varying with time at cell 80

4 Results PDF in 3d space at 80th cell

Seems like water flowing in a river with a gradient slope

The flowin the “river” is not stationary, initially no change due to no deposition at the beginning then drift constantly over time

5 Analytical solution on Fokker-Planck

−=

−

tH

xerfc

tH

xe

tHtxy tH

x

w

0

w

04w00

22

4tan),( w

20

εεπ

πεα ε

The numerical solutions of the stochastic models are now more likely to be obtained without any unnecessary simplifications. However, this type of model might require an intensive computational effort, as the model need to solve a parabolic partial differential equation.

2

2

w x

yH

t

y

∂∂=

∂∂ ε

c

w2

d

C=ε 12/12/3

ww 8

1agC −= γρ

Under the assumption of small angles

For special case, a straight impermeable jetty (Larson et. al., 1987)


∫−

−+−= tH

xtH

x

exxetH

txy w

02

w

20

200000

4w00 dtantan

4tan),( ε γε γαα

πεα

tH

x

et

Hty w

20

4w0tand/d ε

πεα

−=

0xx = ∫−=

z t tezerf0

d2

)(2

π

we should like to explain its derivative with respect to time

For a special transect

This transformation of the shoreline change rate at a specified position, seems to escalate the complexity of the problem dramatically, actually this expression is expected to exhibit more interesting properties in the follow stochastic model


[ ] [ ] ),0(~)(;)(

tand

),(d )(4w0

w

20

DNtWet

tWH

t

txy ttWH

x

+−+= ε

πεα

tk

x

et

ktu m

20

4m0tan)( ε

πεα

−= Dt

x

et

Dtv ε

πεα 4

0

20

tan)(−

=

)(mw tWKH += )(mw tKH σξ+=or )1,0(~)( Ntξwith

The best candidate to describe this time varying behavior is assumed to follow some variations of Brownian motion with drift process

In the variation of Brownian motion the average over the ensemble of the fluctuating force is calculated by deterministic one-line model. The deviation represents the average shoreline position uncertainty of the motion.

Drift term (mean)& random diffusion (variations)


)(d)(d)(),(d tBtvttutxy +=

[ ] [ ]ptvy

ptuyt

typ)()(

),( 22

2

∂∂+

∂∂−=

∂∂

Ito stochastic differential equation (refer to Soong, 1973; Gardiner, 2004 )

Fokker-Planck equation

one-dimension second-order equation with linear but time-dependent drift and time-dependent diffusion coefficient PDE

)(tu )(tv )(tysince the transfer function and are independent of

At a specified position, a more promising and alternative exact solution can be solved by Lie-Algebra approach (Desai and Zwanzig, 1978; Lo, 2005)


)0,()(),( yptUtyp =

( )

( ) ( )( )z

t

tyzzpt

zt

yzzpt

yt

ypy

ty

ttyp

y

y

y

d)(4

)(exp)0,()(4

d)(4

)(exp)0,()(4)(exp

)0,()(exp)(exp),(

max

min

max

min

y

22/1

22/1

2

2

∫

∫

+−−×=

−−×

∂∂=

∂∂×

∂∂=

−

−

γαπγ

γπγα

γα

We may define the evolution operator

1)0(),()(d

)(d == UtUtAt

tU2

22 )()()(

yt

yttA

∂∂+

∂∂−= νµ

∂∂

∂∂=

2

2

)(exp)(exp)(y

ty

ttU γα

∫−=t

sst0

d)()( µα ∫=t s sest0

)(2 d)()( ανγ

)(tα )(tγSolution is Gaussion type with time-varying mean variance


0 50 100 150 200 250 300Offshore distance (m)

0.00

0.05

0.10

0.15

0.20

p(y,

t)

Time (Year)12510Num

100 150 200 250 300 350Offshore distance (m)

0.00

0.02

0.04

0.06

0.08

p(y,

t)

Km0.1560.250.3750.53

Jetty case

Typical instantaneous PDFs at various times PDFs with mK


31 32 32 33 33 34 34 35 35Offshore distance (m)

0.00

0.03

0.06

0.09

0.12

p(y,

t)

Cv0.150.250.350.45

vCPDFs with


+

+

−

=a

x

t

a

a

x

t

aYy F 1

2erf1

2erf

2 εε

2

30

142

30

14 )(

exp8

)(exp

8d

d2

02

0−

+−−

−− +−−−= txaY

txaY

t

y a

x

t

a

Fa

x

t

a

F

εεεε

Rectangular beach fill (Dean, 1984 )

With the possibility of a highly variable wave climate, the difficulties of developing accurate predictions of maintenance nourishment requirement are not surprising. Stochastic differential equation may provide more qualitative information on beach profile relative to maintenance requirements.


31 32 33 34 35 36 37 38Offshore distance (m)

0.00

0.05

0.10

0.15

0.20

0.25

p(y,

t)

Km0.1560.250.3750.53

-10000 -5000 0 5000 10000Longshore distance (m)

0

10

20

30

40

y m (

m)


Nourishment beach fill

axa +<<−FY

a=2500m

=50m

10 years later

5 Summary on Analytical scheme

tk

x

et

ktu m

20

4m0tan)( ε

πεα

−=

Dt

x

et

Dtv ε

πεα 4

0

20

tan)(−

=

)(d)(d)(),(d tBtvttutxy +=

Drift

Diffusion

SDE Ito

FPE

)()0(),()()(),()(),(d

),(dbmm tWtyKtWtytyK

t

txy ψϕαψϕαψϕ ++−=Angle

[ ] [ ]ptvy

ptuyt

typ)()(

),( 22

2

∂∂+

∂∂−=

∂∂

( ) ( )( )z

t

tyzzpttyp

yd

)(4

)(exp)0,()(4),(

max

miny

22/1

∫

+−−×= −

γαπγSolution

5 Liouville shoreline evolution model

Substitute one-line model to random differential equation, then a random process model of shoreline position can be given

In order to employ Liouville model Which can be reformulated as

(8)

(7)

Transfer function

),,(d

),(dw txyH

t

txy ζ−=

)/()2sin2(sin16

1),( b

1b1

2/3w xagty ii ∆−= − ααρζ

=

=

0

w

)0,(

),,(d

),(d

ZxZ

HxyLt

txZ

=

w

),(),(

H

txytxZ

=

0

ηL

=

w

00 H

yZ

),(),( w tyHty ζη =

5 Liouville shoreline evolution model

Mathematical methods for determining the solution process )(ty

governed by equation (18) are available following Liouvilleequation (Syski, 1967; Soong, 1973)

)(ty wH

),),(( wwthtypYH

A differential equation for the joint PDF of and

0)],,(),([),,( ww ww =

∂∂

+∂

∂y

thypty

t

thyp YHYH η(9)

5 Liouville Shoreline Evolution model

Eq. (9) can be simplified by a one-dimensional advection

The probability density flow does not have sources, means

0/),( =∂∂ ytyη (Shvidler and Karasaki, 2003)

0),,(

),(),,( w

ww ww =

∂∂

+∂

∂y

thypthY

t

thyp YHYH &

ytythY ∂∂= /),(),( w η& is the ‘velocity’ of the response for a prescribed

wh called advecting velocity coefficient

which means shoreline travels by this velocity

),,( wwthypYH is numerically solvable

(10)

6 Conclusions

• Shoreline random evolution can in principle be treatable using the formalism and techniques that are available for solving stochastic partial differential equations (SPDE) , probabilistic solution of shoreline was transferred to be a deterministic differential equation

• Comparing the results with some other solutions by deterministic or stochastic approach, very similar trend and reasonable distributions have been predicted. It is shown that the proposed random models can reflect the effects of uncertainties of parameters and boundary condition on the shoreline position and with further extension can be used for predictive purpose

6 Conclusions

If the exact solution in closed form for the other different beach situations and the differential term of shoreline change are existed, density function can be derived following above solving procedure. Additionally, if wave breaking angle is not small, no explicit expression on shoreline change rate for this special case, which is hardly solved by this analytical approach.

However, the complexity of beach change implies that results obtained from an analytic model should be interpreted with care and with awareness of the underlying assumption. Analytic solutions of mathematical models can not be expected to provide quantitivelyaccurate results in situations involving complex boundary conditions and complex wave inputs, deriving a non-linear stochastic differential equation, in that case, a numerical model of stochastic shoreline evolution would usually be more appropriate.

Acknowledgements

�We would like to acknowledge the financial support of UK EPSRC Grant No GR\L53953 and research group from Univ. of Plymouth

�Thanks to Prof. Ping Dong and other colleagues in Dundee geotechnical group

Further study

System reliability

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06 from browian motion to stochastic shoreline...

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