Specular Points and Wave Crests

Volume 2010, Article ID 621038,22pages doi:10.1155/2010/621038

Research Article

Level Sets of Random Fields and Applications:

Specular Points and Wave Crests

Esteban Flores

and Jos ´e R. Le ´on R

1Departamento de Matem´aticas, Facultad Experimental de Ciencias y Tecnolog´ıa, Universidad de Carabobo, Valencia 2001, Venezuela

2Escuela de Matem´atica, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47197 Los Chaguaramos, Caracas 1041-A, Venezuela

Correspondence should be addressed to Jos´e R. Le ´on R,jose.leon@ciens.ucv.ve Received 23 September 2009; Revised 22 February 2010; Accepted 22 February 2010 Academic Editor: Deli Li

Copyrightq2010 E. Flores and J. R. Le ´on R. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We apply Rice’s multidimensional formulas, in a mathematically rigorous way, to several problems which appear in random sea modeling. As a first example, the probability density function of the velocity of the specular points is obtained in one or two dimensions as well as the expectation of the number of specular points in two dimensions. We also consider, based on a multidimensional Rice formula, a curvilinear integral with respect to the level curve. It follows that its expected value allows defining the Palm distribution of the angle of the normal of the curve that defines the waves crest. Finally, we give a new proof of a general multidimensional Rice formula, valid for all levels, for a stationary and smooth enough random fieldsX:R^d → R^jd > j.

1. Introduction

In 1944, Rice1proposed the model

ζt

n

c_ncosσntε_n, 1.1

to describe the noise in an electrical current. In this relation,σ_n/2πdenotes the different frequencies,cnare Gaussian random variables, identically distributed and independent, and ε_nare random variables uniformly distributed in0,2π.

Later, in 1957, Longuet-Higgins2defined the following multidimensional general- ization of Rice’s model:

ζ t, x, y

n

cncos

unxvnyσntεn

. 1.2

Since then this model has been used to describe the movement of the sea.

The present work is aimed at studying functionals of random field level sets in order to understand certain phenomena occurring in random sea modeling such as the movement of the luminous points which appear over any water surface. These points are called specular points and originate when the light is reflected in agreement to Snell’s Law from different zones which act as small mirrors. They can be modeled as level sets of certain derivatives of the original random fieldζ. This type of phenomena leads us to study the sizecardinal, length, area, and volumeand other measurements of level sets for Gaussian random fields.

It is thus necessary to consider functionals over fields defined by 1.2 or their generalizations given inSection 3. Our study relates the expectation of such functionals with the moments of the spectral measure of this process. The latter is important for applications, as usually the spectral measure of the process as well as its moments may be estimated based on data measured by buoys or satellites. The main tools that we use are given by Rice’s multidimensional formulas.

Our main results include the probability density function of the velocity of the specular points studied by Longuet-Higgins in 3–5. First, we compute the probability density function of the Palm distribution of the speed of the specular points in an arbitrary, but fixed, direction. Then, using model 1.2we are able to compute the density of the Palm distribution of the speed of the specular points in a 2D spacesee6for applications of this type of densities. We are also interested in obtaining the expectation of the number of the specular points in two dimensions. We provide an expression for this expectation by using a multidimensional Rice formula recently proved in the books of Aza¨ıs and Wschebor7, page 163and Adler and Taylor’spage 267.

Also based on a multidimensional Rice formula we are able to study a curvilinear integral with respect to the level curve whose expected value allows defining the Palm distribution of the angle of the normal of the curve that defines the waves crest in a fixed direction, such type of objects was recently introduced in8.

All the expectations mentioned above can be rigorously computed by using the multidimensional Rice formula for Gaussian random fieldsX:R^d → R^d, recently proved in 7and by using another Rice formula for random fieldsX :R^d → R^jd > jestablished by Caba ˜na in 19859. For the sake of completeness, we also include a simplified proof of the latter, which allows a generalization of the original results. Namely, we show that the formula holds true over the complete level set, instead of over the intersection of the level set with the set of regular points, that is, those where the derivative of the random field has rank equal toj.

This work can thus be viewed as the implementation of several applications suggested in the book mentioned in7as well as a continuation of the second author’s articles10,11.

The paper is organized as follows. Section 2 studies the coarea formula and its application in the computation of the expectation of the Lebesgue measure of the level sets and some related surface integrals with respect to the measure over the level set, see 9, 12, 13. The formula holds true for all levels and this is a new result. Section 3 gives a stochastic integral representation of the Longuet-Higgins model and the relation between this model and the directional spectrum. Section 4 gives the probability density function for the speed of the specular points in a fixed but arbitrary direction. InSection 5, the multidimensional Rice formula cf. 7, 14 is used to obtain the expectation of the number of the specular points in two dimensions.Section 6provides the probability density function associated with the velocity of the specular points in all directions. These velocities are computed both for Gaussian and non-Gaussian random fields, thus formalizing and generalizing, the deep and inspired work of Longuet-Higgins. Finally,Section 7establishes

an application of Rice’s formula to study the asymptotic distribution of the normal angle to the crests.

In what followsλ_d and σ_d−m will denote, respectively, the Lebesgue measure in the spaceR^dand the Hausdorffmeasure defined in the subspaces of dimensiond−m, trivially by definitionλdσd.

2. The Coarea Formula and Its Application to Rice’s Formula

Before proving our main result let us give an overview of the area formula and its probabilistic consequence, the Rice formula. Letg :R^d → R^dbe a continuously differentiable function. If we define

N^g_Ay #

s∈A⊂R^d:gs y

, 2.1

then ifd1, one has

A

f

gsgsds

R^dfyN_A^gydy, 2.2

where f : R → R is a continuous and bounded function. This formula was obtained by Banach in 192515.

If∇gxis the Jacobian ofg inx andf :R^d → Ris a continuous bounded function, then the version of Banach’s formula ford≥1 is

A

f

gsdet∇gsds

R^dfyN_A^gydy. 2.3

This expression is usually called the area formulacf.12.

Now, letX : R^d → R^d be a Gaussian random field with continuously differentiable trajectories. The random number of times thatXtakes the valuey in the setAis defined as

N^X_Ay #

s∈A⊂R^d:Xs y

. 2.4

Letp_Xsdenote the marginal density ofXs. By using formula2.3, Fubini’s Theorem and duality we get for a.s.y∈R^d

E N_A^Xy

A

E|det∇Xs|Xs yp_Xsyds. 2.5

The fact that this formula is true for ally∈R^dis not trivial. The book by Aza¨ıs and Wschebor 7, page 163contains a definitive proof. The motivated reader can also read the interesting discussion given in Sections 11.2 and 11.4 of Adler and Taylor’s recent book 14and the references therein.

We will study below the more difficult case when function g has a domain whose dimension is greater than the dimension of the rank, namely, g : R^d → R^j d > j is a continuously differentiable function, with Jacobian∇g·defining the level set

CQy

x∈Q:gx y

g⁻¹y∩Q, 2.6 where Q is a compact set ofR^d. The following two results are well known as the Coarea formula cf. Federer 12, pages 247–249 and Caba ˇna 9. The reader may consult the excellent set of lectures by Weizs¨aker and Geibler of the University of Kaiserslautern 13 for an up to data exposition.

Theorem 2.1. Letf :R^j → Rbe a continuous and bounded function. Restricted to the set x∈R^d:∇gxhas rankj

, 2.7

the following formula holds:

Q

f

gx

det

∇gx∇gx^T_1/2 dx

R^j

fyσd−j C_Qy

dy. 2.8

Corollary 2.2. LetY :R^d → Rbe a continuous and bounded function under restriction2.7, then

R^j

fy

CQyYxdσd−jx

dy

R^df

gx

Yxdet

∇gx∇gx^T_1/2

dx. 2.9

Remark 2.3. Formula2.8and2.9hold true without restriction2.7. In fact it can be proved that for

A⊂

x∈R^d :∇gxhas rank< j

, 2.10

σ_d−jg⁻¹y∩A 0 holds for almost ally. This also implies that

A∩CQyY dσ_d−j

≤ Y_∞σ_d−j

g⁻¹y∩A

0. 2.11

Let us define for a compactQthe functions Gy σ_d−j

g⁻¹y∩Q

, Fy

CQyYxdσd−jx. 2.12

The following lemma holds true.

Lemma 2.4. Under hypothesis ofTheorem 2.1, functionsGandFare continuous.

The following results are simple consequences ofTheorem 2.1andCorollary 2.2.

LetX:Ω×R^d → R^jd > jbe a stationary random field belonging toC¹R^d,R^jand suppose that for allx∈R^d, the density ofXx, pXx·existsin the Gaussian case this holds whenever VarXx>0. We have

iFor almost ally∈R^j,

E σ_d−j

CQy

Q

p_XxyE

det

∇Xx∇Xx^T_1/2

|Xx y

dx. 2.13

iiFor almost all y∈R^j,

E

CQyYxdσd−jx

Q

p_XxyE

Yxdet

∇Xx∇Xx^T_1/2

|Xx y

dx.

2.14

Let us remark that formula 2.13 and 2.14 hold for almost all y ∈ R^j. However in applications, as we will see in the next sections, they are needed for a fixedy. We will prove in what follows that the formulas hold for ally.

Define the setD^r {x : ∇Xxhas rank j}.We will establish the continuity of the left-hand side term in formulas2.13and2.14restricting ourselves first to this set. Thus let us defineC^D_Q^ry CQy∩D^r.The following theorem was proved in 1985 by Caba ˇna9. The article was written in Spanish and had a very limited diffusion. We give a new and slightly more general proof. We point out that Theorems 6.8 and 6.9 of7yield the same result as our Theorem 2.8. However, in this book the proofs of these results are only sketched.

Before stating the proof we include two useful conditions.

iA₁: for allx ∈R^d,p_Xx·exists and is continuous. For a continuous functionH : R^d×R^j → Rthe following expression:

Q

p_XxyEH∇Xx|Xx ydx 2.15

is a continuous function in they variable.

iiA₂: the expression

Q

p_XxyEYxH∇Xx|Xx ydx 2.16

is a continuous function in they variable. Let us note that ifYx Y∇Xx,then A₁is sufficient forA₂to hold.

Theorem 2.5. ConsiderX:Ω×R^d → R^jd > ja random field belonging toC¹R^d,R^j. iThen underA₁for ally∈R^j,

E σ_d−j

C^D_Q^ry

Q

p_XxyE

det

∇Xx∇Xx^T_1/2

|Xx y

dx. 2.17

iiIfY is an almost sure continuous function underA₂, for ally∈R^jone has

E

C^D_Q^ryYxdσd−jx

Q

p_XxyE

Yxdet

∇Xx∇Xx^T_1/2

|Xx y

dx.

2.18 Proof. We begin proving formula2.17. Let the differentiable functionψ : R → 0,1be such thatψt 1 If 0≤t≤1 andψt 0 ift≥2. Let us define the function

Y_m∇Xx ψ 1

mdet

∇Xx∇Xx^T ψ

⎛

⎜⎝ 1 mdet

∇Xx∇Xx^T

⎞

⎟⎠, 2.19

Y_m∇Xx 0 ifx∈ {y : det∇Xx∇Xx^T >2m} ∪ {y : 1/det∇Xx∇Xx^T>2m}.

Forx belonging to the complement of this set and definingλ₁x≤λ₂x≤ · · · ≤λ_jx the eigenvalues of∇Xx∇Xx^T, it holds

det∇Xx∇Xx^T ≤2m, 1 det

∇Xx∇Xx^T ≤2m, 2.20

and moreover, definingV ker∇XxandV^⊥its orthogonal subspace, we have ∇Xx|_V⊥⁻¹ 1

λ1x

⎛

⎜⎝

!j i2λ_ix det

∇Xx∇Xx^T

⎞

⎟⎠

1/2

≤√

2m× ∇Xx|_V⊥^j/2. 2.21 Observe that the hypothesis of continuity of∇Xx|V⊥ and the compactness of Q imply a uniform bound for the inverse.Lemma 2.4implies that the following function:

F^n,m_X y:ψ 1

nσ_d−j C^D_Q^ry

C^Dr_Qy

Y_m∇Xxdσ_d−jx 2.22

is a.s.continuous, also the sequenceF_X^n,m is nondecreasing in both indexes. Moreover, the inequalityF_X^n,my ≤ nyields thatEF_X^n,myis a continuous function. Using formula2.9

applied to the fieldXand the functionF_X^n,m, we have

R^jψ 1

nσ_d−j C^D_Q^ry

C^Dr_Qy

Ym∇Xxdσd−jxdy

Q

ψ 1

nσ_d−jC^D_Q^rXx

Y_m∇Xxdet

∇Xx∇Xx^T_1/2 dx.

2.23

From this we have that for almost ally∈R^j, E F_X^n,my

Q

p_XxyE

ψ 1

nσ_d−jC^D_Q^rXx

Y_m∇Xxdet

∇Xx∇Xx^T_1/2

|Xx y

dx.

2.24

Thus,

E F_X^n,my

≤

Q

E

det

∇Xx∇Xx^T_1/2

|Xx y

p_Xxydx. 2.25

Condition A1 implies that the function in the right-hand side is continuous, hence the inequality holds for all y. Taking limits asn → ∞ and m → ∞ and using Beppo-Levi’s Theorem, we have that

E σ_d−j

C^D_Q^ry

≤

Q

E

det

∇Xx∇Xx^T_1/2

|Xx y

p_Xxydx. 2.26

To prove the other inequality, letyN → y be such that for allNequality2.24holds. This is possible because the equality is satisfied for almost ally. Thus by applying Fatou’s Lemma, we obtain

E F_X^n,my lim

N→ ∞E F_X^n,myN lim

N→ ∞

Q

E

ψ 1

nσ_d−j

C^D_Q^ryN

×E

Y_m∇Xxdet

∇Xx∇Xx^T_1/2

|Xx yN

p_XxyNdx

≥

Q

E

ψ 1

nσ_d−j

C^D_Q^ry

×E

Y_m∇Xxdet

∇Xx∇Xx^T_1/2

|Xx y

p_Xxydx.

2.27

By using Fatou’s Lemma again and thatψ·/nis a nondecreasing sequence, we obtain

mlim→ ∞ lim

n→ ∞E F_X^n,my

Q

E

det

∇Xx∇Xx^T1/2

|Xx y

p_Xxydx. 2.28

Finally

F_X^n,my≤

C^Dr_Qy

Ym∇Xxdσ_d−jx≤σ_d−j

C^D_Q^ry

<∞ a.s. 2.29

MoreoverF_X^n,my↑

C^Dr_Q yY_m∇Xxdσ_d−jxwhenn → ∞. Clearly applying Beppo-Levi’s Theorem we get

E σ_d−j

C^D_Q^ry lim

m→ ∞ lim

n→ ∞E F_X^n,my

≥

Q

E

det

∇Xx∇Xx^T_1/2

|Xx y

p_Xxydx. 2.30

Obtaining formula2.17, formula2.18follows by approximatingY uniformly by a nonde- creasing sequence of simple functions.

The following two propositions of Aza¨ıs and Wschebor7, pages 132–134 provide the arguments to improve Caba ˜na’s result. In the book, however, the hypothesis is a little different.

Proposition 2.6. LetX :U⊂R^d → R^mkbe a random field andIa subset ofU, and letu∈R^mk. One supposes thatXsatisfies the following conditions:

1the random field∇Xisα-H¨older continuous withm/mk< α≤1;

2for eachx∈U,the random vectorXxhas a density p_Xxysuch thatp_Xxy≤C, for x∈Iandy in some neighborhood of u;

3the Hausdorffdimension ofIis smaller than or equal tom.

Then, almost surely, there is no pointx∈Isuch thatXx u.

Proposition 2.7. LetX : U ⊂ R^d → R^j be a random field andUan open set ofR^d andy ∈ R^j. Suppose that∇X is a.s.α-H¨older continuous with 1−1/dj < α ≤ 1 and moreover for all x∈Uthe random vectorXx,∇Xxhas a bounded continuous densityp_Xx,∇Xxu,y, for˙ u in a neighborhood ofy andx,y˙ varying in a compact set ofU×R^d×j. Then

P

ω:∃x Xx y, rank ∇Xx< j

0. 2.31

Theorem 2.8. Under the hypotheses ofTheorem 2.5and that∇X·is a.sα-H¨older continuous with 1−1/dj< α≤1, one has the following.

1UnderA₁, for ally∈R^j,

E σ_d−j

CQy

Q

p_XxyE

det

∇Xx∇Xx^T_1/2

|Xx y

dx. 2.32

2UnderA2and ifY is an almost sure continuous function, for ally∈R^jone has

E

CQyYxdσd−jx

Q

p_XxyE

Yxdet

∇Xx∇Xx^T_1/2

|Xx y

dx.

2.33

In what follows we give two examples under which the hypothesesA1andA2hold.

1Suppose thatXxis a Gaussian field verifying the hypothesis ofTheorem 2.8and that VarXx>0 for eachx. By considering the regression model,

∇Xx αxXx Γxξx 2.34

with a Gaussianξx⊥Xx, where

αx E ∇XxXx^T

E XxXx^T₋₁ , ΓxΓx^TE ∇Xx∇Xx^T

−E ∇XxXx^T

E XxXx^T₋₁

E Xx∇Xx^T ,

2.35

the following equality in law is satisfied:

L∇Xx|Xx y y^Tαx Γxξx. 2.36

This result entails that the expression

E

det

∇Xx∇Xx^T_1/2

|Xx y

y^Tαxαx^Ty ΓxΓx^T 2.37

is a continuous function of variable y. Moreover, the hypothesis VarXx > 0 yields the continuity inyofp_Xxy.

1Finally let us consider the case of the real envelope of the stationary Gaussian fieldζt, x, y. As in16, we define the envelope of the Gaussian fieldζt, x, yas follows. Let us consider the random spectral measureMλ1, λ₂, ωrestricted to the Airy manifoldΛdefined below in3.1. In this manner if we restrict the stochastic integral to the setΛ {λ1, λ₂, ω:ω≥0,|k| ω²/g}.By using polar coordinates we can write

ζ t, x, y

2 _∞

0

_π

−πcoskcosΘxksinΘyωt

dcω,Θ. 2.38

We define the Hilbert transform ofζas the Gaussian field

"

ζ t, x, y

2 _∞

0

_π

−πsinkcosΘxksinΘyωt

dcω,Θ. 2.39

The real envelopeEt, x, yis defined as

E t, x, y

# ζ²

t, x, y ζ"²

t, x, y

. 2.40

It holds E∇E

t, x, yE t, x, y

u 1

uE ∇ζ0,0,0 ∇ζ0," 0,0

. 2.41

This expression is continuos wheneveru>0. Moreover the density ofE0,0,0, in the pointu, is the Rayleigh density1/σ_ζ²ue^−u2/2σ2ζ, that exists and is continuous if σ_ζ²Varζ0,0,0>0.

3. Representation with Integrals and the Directional Spectrum

In this section, we study a generalization of the Gaussian random fields defined in1.2that model the waves of the sea. We use its representation as a stochastic integral which also yields the spectral representation of a stationary mean zero Gaussian random field. The approach will be somewhat informal in order to make the reading easier. The interested reader can consult Kr´ee and Soize “Mecanique Aleatorie,” 17, pages 366–376, or the very readable article18in which.Lindgren gives a definitive treatment for this type of spectral stochastic integral models.

Another way of looking at Longuet-Higgins’ model is

ζ t, x, y

Λe^iλ1xλ2yωtdMλ1, λ2, ω, whereΛ is the Airy manifold

$

λ²₁λ²₂ ω⁴ g²

% , 3.1

gis the gravitational constant andMis a random Gaussian orthogonal measure defined onΛ.

Definingk λ1, λ₂and the following change of variable|k| ω²/gandλ₁ ω²/gcosΘ andλ₂ ω²/gsinΘ see17, we obtain

ζ t, x, y

_∞

−∞

_π

−πe^{i|k|cosΘx|k|sinΘyωt}dcω,Θ

_∞

−∞

_π

−πexp i

k·xωt

dcω,Θ,

3.2

wheredcω,Θis a random measure. The covariance function,Kτ, X, Y,is defined as Kτ, X, Y E ζ

t, x, y ζ

tτ, xX, yY

. 3.3

Then by using that

E dcω,Θdcω,Θ

Sω," Θδω−ωδΘ−Θdω dωdΘdΘ, 3.4 whereSω," Θis the two-dimensional spectrum of the wave surface andδrepresents Dirac’s delta function. Then,

Kτ, X, Y _∞

−∞

_π

−πexp i

k·Xωτ

"

Sω,Θdω dΘ. 3.5 This procedure is justified formally in17,18. If in3.5we letX 0 andY 0, then we obtain

Kτ:Kτ,0,0 _∞

−∞

_π

−π

"

Sω,Θe^iωτdω dΘ, 3.6

or equivalently Kτ _∞

−∞Sωe" ^iωτdω, where Sω " _π

−πSω," ΘdΘ. Function Sω"

represents the frequency spectrum of the sea surface. This spectrum contains the distribution of the wave energy in the frequency domain. The autocorrelation function Kτ for the elevation surfaceζt, in a fixed location, is a real even function.

Definition 3.1. The spectral moments of orderijkare defined as

mijk _∞

0

_π

−πuⁱv^jω^kSω,ΘdΘdω, 3.7

whereu ω²/gcosΘ−Θ0,v ω²/gsinΘ−Θ0,andgis the gravitational constant. If in3.7ij0, then

m_{00k ∞}

0

_π

−πω^kSω,ΘdΘdω, 3.8

and this can be rewritten asm00k _∞

0 ω^kSωdω.The previous relation corresponds to the one-dimensional moment of orderk,m_k∞

0 ω^kSωdω.Alsom_ij1andm_ij2will be denoted by

m_{ij ∞}

0

_π

−πωuⁱv^jSω,ΘdΘdω, m_{ij ∞}

0

_π

−πω²uⁱv^jSω,ΘdΘdω, 3.9

respectively.

4. Velocity of the Specular Points in an Arbitrary Direction

We are now able to study the dynamical behavior of the specular points. Thus letζt, x, y be the random field 3.2 representing the sea height and suppose that it belongs a.s. to C³R³,R. We observe the random field ∂ζ/∂xt, x, y in a fixed direction, y 0, for instance. The place where reflection occurs, when the surface ζt, x,0is illuminated by a light source, placed in0, h1and observed in0, h2, for each fixedtis the level curve

∂ζ

∂xt, x,0 ζxt, x kx, 4.1 wherek 1/21/h1 1/h2. This condition is approximately true, wheneverkζandζ_x are both small quantities, see4, page 845.

A consequence of the implicit function theorem is

ζxx−kdxζ_xtdt0, 4.2

that is,

c dx

dt − ζxt

ζxx−k. 4.3 This expression defines the velocity of the specular points. Thus let us define the number of specular points in0, Mhaving a speed inα1, α2asN&sps,0, α1, α2, where

&

N_sps, u, α1, α₂:# '

x≤M:ζ_xs, x kxu;α₁≤ ζ_xt

ζxx−k ≤α₂ (

for 0≤s≤t. 4.4

Now, define the latter number per unit time as

N_spu, α1, α₂, t: 1 t

_t

0

&

N_sps, u, α1, α₂ds. 4.5

Notice that the processZt N&spt, u, α1, α2is stationary, has finite mean, and it is Riemann integrable, as a function oft. DefineAtσ{ζτ, x,0 : τ > t, x ∈0, M}and theσ-algebra

of invariant eventsA ∩At. Under the hypothesis that for eachx ∈0, M,Kt, x,0 → 0 whenevert → 0 theσ-algebraA is trivial. By the Birkoff-Khintchine Ergodic Theorem, we have

_t

0Zsds

t −→E_BZ0, 4.6

where B is the σ-algebra of t-invariants associated to Z. Since for each t, Bt σ{Zτ : τ > t} ⊂ At, it follows thatB ⊂ A, so thatE_BZ0 EZ0 EN&_sp0, u, α1, α₂ for references, see19, page 151.

Our interest here is to compute the Palm distribution of the number of specular points having speed betweenα1, α₂defined as

Fα2−Fα1 lim

t→ ∞

Nsp0, α1, α2, t

Nsp0,−∞,∞, t E N&sp0,0, α1, α2

E N&_sp0,0,−∞,∞. 4.7

The last equality, as we have seem, is a consequence of the Ergodic Theorem. We will show the following result.

Proposition 4.1. Let ζt, x, y be a Gaussian random field 3.2 and assume that it belongs to C³R³,Rand for each pair x, y, Kt, x, y → 0 whenevert → 0. The Palm distribution F defined above satisfies

Fα2−Fα1

E

1_α1,α2ζxt0,0/ζxx0,0−k|ζxx0,0−k|

√m₄₀₀Rk , 4.8

whereRk: 2/πe^−k2/2m400 k/√

m_400k/^√_m400

0 e^−v2/2dv.

Proof. For a continuous and bounded functionh, we have _∞

−∞huNspu, α1, α₂, tdu 1 t

_t

0

_∞

−∞huN&_sps, u, α1, α₂du ds, 4.9

and by the area formula2.3, we have _∞

−∞huNspu, α1, α2, tdu 1

t _t

0

_M

0

hζxs, x−kx1_α1,α2

ζ_xts, x ζxxs, x−k

|ζxxs, x−k|dx ds.

4.10

Taking expectations, by stationarity and duality, for almost allu, it follows that E

N_spu, α1, α₂, t

_M

0

pζx0,0ukxE

1_α1,α2

ζ_xt0,0 ζxx0,0−k

|ζxx0,0−k| |ζx0,0 ukx

dx.

4.11 This may be written, in the Gaussian case, by independence, as

E

Nspu, α1, α2, t

_M

0

exp

−ukx²/2m200

2πm200

dx

×E

1α1,α2

ζxt0,0 ζxx0,0−k

|ζxx0,0−k|

.

4.12

The formula is true for all u as it follows analogously to the result shown by Aza¨ıs and Wschebor7, page 163.

For the specular points, the interesting level is u 0, thus we obtain that the expectation of the number of specular points having speed betweenα₁andα₂is

_M

0

pζx0,0kxE

1α1,α2

ζxt0,0 ζxx0,0−k

|ζxx0,0−k| |ζx0,0 kx

dx, 4.13

which in the Gaussian case may be written as

E

N_sp0, α1, α₂, t

E N&_sp0,0, α₁, α₂

_M

0

exp

−k²x²/2m200

2πm₂₀₀ dx

×E

1_α1,α2

ζ_xt0,0 ζxx0,0−k

|ζxx0,0−k|

.

4.14

Moreover, the expectation of the number of specular points per unit of timeNsp0,−∞,∞, t is easily computed yielding formula2.14of4, page 846

E

N_sp0,−∞,∞, t

_M

0

exp

−k²x²/2m200

2πm₂₀₀ dxE|ζxx0,0−k|

Rk

) m₄₀₀ 2πm200

_M

0

exp

$

−k²x² 2m200

% dx,

4.15

whereRk: 2/πe^−k2/2m400 k/√

m400_k/^√_m400

0 e^−v2/2dv.

As the processζsatisfies thatKt, x,0 → 0, we obtain4.8by simple division.

Let us now define px,tξ1, ξ2 the Gaussian density of the random vector ζxt0,0, ζ_xx0,0. We may write4.8as

Fα2−Fα1

√ 1

m400Rk _α2

α1

_∞

−∞ξ2−k²px,tcξ2−k, ξ2dc dξ2. 4.16

Remark 4.2. Differentiating the previous expression one obtains the density of the velocity of the specular points:

"

pkc 1

√m₄₀₀Rk _∞

−∞ξ2−k²px,tcξ2−k, ξ2dξ2. 4.17

Ifk0 we recover formula2-5-19of2 modified in order to consider the case of specular points

"

p₀c Δ2

2m²₀₀₄

c−c² Δ2m⁻²_400−3/2

, 4.18

where

Δ2

det

*m200 m110

m₁₁₀ m₀₂₀ +−1

,

c −m301

m₄₀₀.

4.19

5. Number of Specular Points in Two Dimensions

The specular points in two dimensions are described, as we have seen in the last section in the one-dimensional case, by the conditionζxt, x, y, ζyt, x, y kx, kyat pointx, y and for a fixed timet.

Defining the vectorial process

Z t, x, y

ζx

t, x, y

−kx, ζy

t, x, y

−ky

, 5.1

we say that we have a specular point ifZ0 and the number of such points in a fixed timet and in a regionΩ⊂R²will be

N_Ω^Z0,0 # x, y

∈Ω :Z t, x, y

0,0

. 5.2

We denote as in formula2.5x x, yandw w1, w2. Then applying this formula to the processZ, we get for almost allw1, w₂

E N_Ω^Zw1, w₂

Ωp_Z0,x,yw1, w₂EΔ

t, x, y/Z 0, x, y

w₁, w₂

dx dy, 5.3

whereΔt, x, y ζxxt, x, y−kζyyt, x, y−k−ζ²_xyt, x, y.

This formula turns out to be valid for allw1, w₂under the hypotheses of Theorem 6.2 in7 in our caseζ∈C³R³and VarZ >0 will be enoughand in particular for the specular points, that is whenw1, w2 0,0,

E N^Z_Ω0,0

Ωp_Z0,x,y0,0E

|Δ0,0,0|/Z x, y

0,0

dx dy. 5.4

The independence property allows writing E N_Ω^Z0,0

E|Δ0,0,0|

Ωp_Z0,x,y0,0dx dy, 5.5

obtaining finally the following result.

Proposition 5.1. Let the stationary mean zero Gaussian random fieldζt, x, y ∈ C³R³a.s. and VarZ >0. Then,

E N_Ω^Z0,0

E ζxx0,0,0−k

ζ_yy0,0,0−k

−

ζ_xy0,0,02

Ωp_Z0,x,y0,0dx dy.

5.6 Remark 5.2. The Li and Wei formulacf.20provides a way to compute the expectation of the absolute value of the determinant in the above formula, see Aza¨ıs et al.10, which one will not pursue. Instead one will apply a Monte Carlo method. Let us consider the regression model

ζ_yy0,0,0 αζ_xx0,0,0 βζ_xy0,0,0 σ₁ε₁, 5.7 where

α m²₂₂₀−m₃₁₀m₁₃₀

m₄₀₀m₂₂₀−m²₃₁₀ β m₄₀₀m₁₃₀−m₃₁₀m₂₂₀

m₄₀₀m₂₂₀−m²₃₁₀ , 5.8 ε₁N0,1⊥ζxx0,0,0, ζxy0,0,0andσ²−m400α²m₄₀₀β²m₂₂₀2αβm₂₂₀. Therefore it yields

E|Δ0,0,0|E ζxx0,0,0−k

ζ_xx0,0,0βζxy0,0,0σε1

−k

−

ζ_xy0,0,02 . 5.9

This last expression can be evaluated readily by using Monte Carlo. Indeed letεⁱ εⁱ₁, εⁱ₂, εⁱ₃^t, i1, . . . , N,be a sample of standard Gaussian vectors inR³. We have

Nlim→ ∞

1 N

N i1

√

m400εⁱ₃−k

×

⎛

⎝

⎛

⎝

⎛

⎝α√

m400εⁱ₃β

⎛

⎝

*

m220−m²₃₁₀ m₄₀₀

+_1/2

εⁱ₂ √m310

m₄₀₀εⁱ₃

⎞

⎠σεⁱ₁

⎞

⎠

⎞

⎠−k

⎞

⎠

−

⎛

⎝

*

m220−m²₃₁₀ m₄₀₀

+_1/2

ε₂ⁱ √m310

m₄₀₀εⁱ₃

⎞

⎠

2

E|Δ0,0,0|.

5.10

6. Movement and Velocity of the Specular Points

In this section we will compute the density of the velocity of the specular points in two spatial dimensions. Let us consider the random fieldZt, x, y ζxt, x, y−kx, ζyt, x, y−ky; the number of specular points of the fieldζt, x, y, in a fixed timetand in a regionΩ⊂R², was defined in5.2and denoted asN_Ω^Z0,0. We have already computed the expectation of the number of specular points

E N_Ω^Z0,0

E|Δ0,0,0|

Ωp_Z0,x,y0,0dx dy. 6.1

The condition satisfied for the specular pointsi.e.,ζxt, x, y, ζyt, x, y kx, kyand the implicit function theorem entails

ζ_xt −ζxx−kdx

dt −ζ_xydy dt, ζyt −ζxydx

dt −

ζyy−kdy dt.

6.2

Let us define as Longuet-Higginscx dx/dtand cy dy/dt. The objective is to find the Palm distribution associated to the velocity fieldcx, c_y dx/dt,dy/dt. The following computations, in the casek 0, are essentially contained in the very original and seminal work of Longuet-Higgins2.

Now define foru u1, u₂:

&

N_sps,u, v1, v₂, v₃, v₄ # x, y

∈Ω:Z s, x, y

u;v₁≤c_x≤v₂;v₃≤c_y≤v₄

6.3 for 0≤s≤tandΩa compact set inR².

Consider

N_spu, v1, v₂, v₃, v₄, t: 1 t

_t

0

&

N_sps,u, v1, v₂, v₃, v₄ds. 6.4

Letgbe a continuous bounded function. On the one hand, using2.3, we have

R²guNspu, v1, v₂, v₃, v₄, tdu 1 t

_t

0

R²guN&_sps,u, v1, v₂, v₃, v₄duds 1

t _t

0

Ωg Z

s, x, y

−u

1_v1,v2cx1_v3,v4 c_yΔ

t, x, ydx dy ds.

6.5

Let us denote bypξ4, ξ₅, ξ₆, ξ₇, ξ₈the density of the Gaussian random vector ζ_xx0, ζxy0, ζyy0, ζxt0, ζyt0

. 6.6

It follows that p

ξ4, ξ5, ξ6, cx, cy

:p

ξ4, ξ5, ξ6,−ξ4−kcx−ξ5cy,−ξ5cx−ξ6−kcy

6.7

is the density function of the random vector ζxx0, ζxy0ζyy0, cx0, cy0. Taking expectations in6.5, using duality and puttingu 0 under the hypothesisζ ∈ C³R³,R and Varζ >0 the formula holds for all levelsucf.7, page 163we obtain

E

N_sp0, v1, v₂, v₃, v₄, t

Ωp_Z0,0,0

−kx,−ky dx dy

× _v2

v1

_v4

v3

R³p

ξ₄, ξ₅, ξ₆, c_x, c_yξ4−kξ6−k−ξ²₅dξ₄dξ₅dξ₆dc_xdc_y.

6.8

Hence, analogously as inSection 4and by using the same arguments that lead to apply the Ergodic Theorem, the Palm distribution of a specular point having the components of its velocityc_x∈υ1, υ₂andc_y ∈υ3, υ₄is

tlim→ ∞

N_sp0, v1, v₂, v₃, v₄, t N_sp0,−∞,∞,−∞,∞, t

_v2

v1

_v4

v3

R³p

ξ₄, ξ₅, ξ₆, c_x, c_yξ4−kξ6−k−ξ²₅dξ₄dξ₅dξ₆dc_xdc_y

E|Δ0,0,0| .

6.9

We can summarize the above computations in the following result.

Proposition 6.1. Letζt, x, ybe a mean stationary Gaussian field which is a.s. three times continu- ously differentiable and Varζ >0. Assume also that its covariance function satisfiesKt, x, y → 0 for each x, y whenever t → 0. Hence the Palm distribution of a specular point having the components of its velocitycx∈υ1, υ2andcy ∈υ3, υ4is

Fυ3, υ₄−Fυ1, υ_{2 v2}

v1

_v4

v3

R³p

ξ₄, ξ₅, ξ₆, c_x, c_yξ4−kξ6−k−ξ₅²dξ₄dξ₅dξ₆dc_xdc_y

E|Δ0,0,0| .

6.10 Remark 6.2. Taking derivatives one gets the density of the speed of the two-dimensional specular points

"

p_x,y,k cx, cy

R³p

ξ4, ξ5, ξ6, cx, cyξ4−kξ6−k−ξ₅²dξ4dξ5dξ6

E|Δ0,0,0| . 6.11 In the particular case infinite distance, where k 0, we obtain the Longuet-Higgins formulasee2, pages 362–365. Nevertheless, formula6.11is well suited for numerical computations, fork /0.

7. Another Application of Rice Formula

7.1. Angle between the Normal and the Level Curves Defining a Crest in Directionθ

Letζt,x :ζt, x, ybe again a stationary zero mean Gaussian random field modeling the height of the sea waves, heret∈Randx x, y∈R². Let us recall that such a field has the spectral representation given in3.1. Also in3.5, we give an expression for its covariance function. In this expression, the functionSω," Θis known as the directional spectral function and if it does not depend onΘthe random fieldζis called isotropic.

In what follows, we will get information about the crest of the waves in a directionθ.

Let us define, as in8, the crest of the wavesζin directionθat timesas the level set CQs, θ

x, y

∈Q: ζ_θ s, x, y

0; ζ_θθ s, x, y

<0

, 7.1

whereζ_θandζ_θθdenote the first and second derivatives in the directionθ, respectively. This set is the zero level setC^Z_Q^θs,0of the field

Z_θ s, x, y

ζ_x s, x, y

cosθζ_y s, x, y

sinθ, 7.2

under the additional condition thatZ_θθ s, x, y< 0. Ifθ"is the direction orthogonal toθ, we can express the gradient ofZ_θs, x, ywith respect toθand its orthogonal, denoted as∇θ, as

∇θZ_θ s, x, y

∂_θZ_θ s, x, y

, ∂_θ"Z_θ

s, x, y ∇θZθ

s, x, ycosΦ s, x, y

,sinΦ

s, x, y

, 7.3