# Level Sets of Random Fields and Applications: Specular Points and Wave Crests

Authors: Esteban Flores; José R. León R
Journal: International Journal of Stochastic Analysis
 (2010)

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## Abstract

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:ℝd→ℝj(d>j).


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## Body

## 1. Introduction

In 1944, Rice [1] proposed the model(1.1)ζ(t)=∑ncncos(σnt+ɛn),
to describe the noise in an electrical current. In this relation, σn/(2π) denotes the different frequencies, cn are Gaussian random variables, identically distributed and independent, and ɛn are random variables uniformly distributed in [0,2π].Later, in 1957, Longuet-Higgins [2] defined the following multidimensional generalization of Rice's model:(1.2)ζ(t,x,y)=∑ncncos(unx+vny+σnt+ɛn).
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 size (cardinal, length, area, and volume) and 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 in Section 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 (1.2) we are able to compute the density of the Palm distribution of the speed of the specular points in a 2D space (see [6] for 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 Azaïs and Wschebor [7, page 163] and Adler and Taylor's (page 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 in [8].All the expectations mentioned above can be rigorously computed by using the multidimensional Rice formula for Gaussian random fieldsX:ℝd→ℝd, recently proved in [7] and by using another Rice formula for random fields X:ℝd→ℝj (d>j) established by Cabaña in 1985 [9]. 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 to j.This work can thus be viewed as the implementation of several applications suggested in the book mentioned in [7] as well as a continuation of the second author's articles [10, 11].The paper is organized as follows. Section2 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. In Section 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 6 provides 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 7 establishes an application of Rice's formula to study the asymptotic distribution of the normal angle to the crests.In what followsλd and σd-m will denote, respectively, the Lebesgue measure in the space ℝd and the Hausdorff measure defined in the subspaces of dimension d-m, trivially by definition λd=σd.


## 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:ℝd→ℝd be a continuously differentiable function. If we define (2.1)NAg(y)=#{s∈A⊂ℝd:g(s)=y},
then if d=1, one has(2.2)∫Af(g(s))|g′(s)|ds=∫ℝdf(y)NAg(y)dy,
where f:ℝ→ℝ is a continuous and bounded function. This formula was obtained by Banach in 1925 [15].If∇g(x) is the Jacobian of g in x and f:ℝd→ℝ is a continuous bounded function, then the version of Banach's formula for d≥1 is(2.3)∫Af(g(s))|det∇g(s)|ds=∫ℝdf(y)NAg(y)dy.
This expression is usually called the area formula (cf. [12]).Now, letX:ℝd→ℝd be a Gaussian random field with continuously differentiable trajectories. The random number of times that X takes the value y in the set A is defined as (2.4)NAX(y)=#{s∈A⊂ℝd:X(s)=y}.
Let pX(s) denote the marginal density of X(s). By using formula (2.3), Fubini's Theorem and duality we get for a.s. y∈ℝd(2.5)𝔼[NAX(y)]=∫A𝔼[|det∇X(s)|X(s)=y]pX(s)(y)ds.
The fact that this formula is true for all y∈ℝd is not trivial. The book by Azaïs and Wschebor [7, page 163] contains 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 [14] and the references therein.We will study below the more difficult case when functiong has a domain whose dimension is greater than the dimension of the rank, namely, g:ℝd→ℝj (d>j) is a continuously differentiable function, with Jacobian ∇g(·) defining the level set (2.6)CQ(y)={x∈Q:g(x)=y}=g-1(y)∩Q,
where Q is a compact set of ℝd. The following two results are well known as the Coarea formula (cf. Federer [12, pages 247–249] and Cabaňa [9]). The reader may consult the excellent set of lectures by Weizsäker and Geibler of the University of Kaiserslautern [13] for an up to data exposition.Theorem 2.1.
Letf:ℝj→ℝ be a continuous and bounded function. Restricted to the set
(2.7){x∈ℝd:∇g(x)has  rankj},
the following formula holds:
(2.8)∫Qf(g(x))det(∇g(x)∇g(x)T)1/2dx=∫Rjf(y)σd-j(CQ(y))dy.Corollary 2.2.
LetY:ℝd→ℝ be a continuous and bounded function under restriction (2.7), then
(2.9)∫Rjf(y)[∫CQ(y)Y(x)dσd-j(x)]dy=∫ℝdf(g(x))Y(x)det(∇g(x)∇g(x)T)1/2dx.Remark 2.3.
Formula (2.8) and (2.9) hold true without restriction (2.7). In fact it can be proved that for
(2.10)A⊂{x∈ℝd:∇g(x)has  rank<j},σd-j(g-1(y)∩A)=0 holds for almost all y. This also implies that
(2.11)|∫A∩CQ(y)Ydσd-j|≤∥Y∥∞σd-j(g-1(y)∩A)=0.
Let us define for a compact Q the functions
(2.12)G(y)=σd-j(g-1(y)∩Q),F(y)=∫CQ(y)Y(x)dσd-j(x).
The following lemma holds true.Lemma 2.4.
Under hypothesis of Theorem2.1, functions G and F are continuous.The following results are simple consequences of Theorem2.1 and Corollary 2.2.LetX:Ω×ℝd→ℝj (d>j) be a stationary random field belonging to C1(ℝd,ℝj) and suppose that for all x∈ℝd, the density of X(x),pX(x)(·) exists (in the Gaussian case this holds whenever VarX(x)>0). We have (i)
For almost ally∈ℝj,
(2.13)𝔼[σd-j(𝒞Q(y))]=∫QpX(x)(y)𝔼[det(∇X(x)∇X(x)T)1/2∣X(x)=y]dx.(ii)
For almost ally∈ℝj,(2.14)𝔼[∫𝒞Q(y)Y(x)dσd-j(x)]=∫QpX(x)(y)𝔼[Y(x)det(∇X(x)∇X(x)T)1/2∣X(x)=y]dx.Let us remark that formula (2.13) and (2.14) hold for almost all y∈ℝj. However in applications, as we will see in the next sections, they are needed for a fixed y. We will prove in what follows that the formulas hold for all y.Define the setDr={x:∇X(x)hasrankj}. We will establish the continuity of the left-hand side term in formulas (2.13) and (2.14) restricting ourselves first to this set. Thus let us define 𝒞QDr(y)=𝒞Q(y)∩Dr. The following theorem was proved in 1985 by Cabaňa [9]. 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 of [7] yield 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.(i)
A1: for all x∈ℝd, pX(x)(·) exists and is continuous. For a continuous function H:ℝd×ℝj→ℝ the following expression:
(2.15)∫QpX(x)(y)𝔼[H(∇X(x))∣X(x)=y]dx
is a continuous function in the y variable.(ii)
A2: the expression
(2.16)∫QpX(x)(y)𝔼[Y(x)H(∇X(x))∣X(x)=y]dx
is a continuous function in the y variable. Let us note that if Y(x)=Y(∇X(x)), then A1 is sufficient for A2 to hold.Theorem 2.5.
ConsiderX:Ω×ℝd→ℝj(d>j) a random field belonging to C1(ℝd,ℝj). (i)
Then underA1 for all y∈ℝj,
(2.17)𝔼[σd-j(𝒞QDr(y))]=∫QpX(x)(y)𝔼[det(∇X(x)∇X(x)T)1/2∣X(x)=y]dx.(ii)
IfY is an almost sure continuous function under A2, for all y∈ℝj one has
(2.18)𝔼[∫𝒞QDr(y)Y(x)dσd-j(x)]=∫QpX(x)(y)𝔼[Y(x)det(∇X(x)∇X(x)T)1/2∣X(x)=y]dx.Proof.
We begin proving formula (2.17). Let the differentiable function ψ:ℝ+→[0,1] be such that ψ(t)=1 If 0≤t≤1 and ψ(t)=0 if t≥2. Let us define the function
(2.19)Ỹm(∇X(x))=ψ(1mdet(∇X(x)∇X(x)T))ψ(1mdet(∇X(x)∇X(x)T)),Ỹm(∇X(x))=0 if x∈{y:det(∇X(x)∇X(x))T>2m}∪{y:1/(det(∇X(x)∇X(x)T))>2m}.
Forx belonging to the complement of this set and defining λ1(x)≤λ2(x)≤⋯≤λj(x) the eigenvalues of ∇X(x)∇X(x)T, it holds(2.20)det(∇X(x)∇X(x))T≤2m,1det(∇X(x)∇X(x)T)≤2m,
and moreover, defining V=ker(∇X(x)) and V⊥ its orthogonal subspace, we have
(2.21)∥(∇X(x)∣V⊥)-1∥=1λ1(x)=(∏i=2jλi(x)det(∇X(x)∇X(x)T))1/2≤2m×∥∇X(x)∣V⊥∥(j/2).
Observe that the hypothesis of continuity of ∇X(x)|V⊥ and the compactness of Q imply a uniform bound for the inverse. Lemma 2.4 implies that the following function:
(2.22)F̃Xn,m(y):=ψ(1nσd-j(𝒞QDr(y)))∫𝒞QDr(y)Ỹm(∇X(x))dσd-j(x)
is a.s. continuous, also the sequence F̃Xn,m is nondecreasing in both indexes. Moreover, the inequality F̃Xn,m(y)≤n yields that E[F̃Xn,m(y)] is a continuous function. Using formula (2.9) applied to the field X and the function F̃Xn,m, we have
(2.23)∫ℝjψ(1nσd-j(𝒞QDr(y)))∫𝒞QDr(y)Ỹm(∇X(x))dσd-j(x)dy=∫Qψ(1nσd-j𝒞QDr(X(x)))Ỹm(∇X(x))det(∇X(x)∇X(x)T)1/2dx.
From this we have that for almost all y∈ℝj,
(2.24)𝔼[F̃Xn,m(y)]=∫QpX(x)(y)𝔼[ψ(1nσd-j𝒞QDr(X(x)))Ỹm(∇X(x))det(∇X(x)∇X(x)T)1/2∣X(x)=y(1nσd-j𝒞QDr(X(x)))]dx.
Thus,
(2.25)𝔼[F̃Xn,m(y)]≤∫Q𝔼[det(∇X(x)∇X(x)T)1/2∣X(x)=y]pX(x)(y)dx.
Condition A1 implies that the function in the right-hand side is continuous, hence the inequality holds for all y. Taking limits as n→∞ and m→∞ and using Beppo-Levi's Theorem, we have that
(2.26)𝔼[σd-j(𝒞QDr(y))]≤∫Q𝔼[det(∇X(x)∇X(x)T)1/2∣X(x)=y]pX(x)(y)dx.
To prove the other inequality, let yN→y be such that for all N equality (2.24) holds. This is possible because the equality is satisfied for almost all y. Thus by applying Fatou's Lemma, we obtain
(2.27)𝔼[F̃Xn,m(y)]=limN→∞𝔼[F̃Xn,m(yN)]=limN→∞∫Q𝔼[ψ(1nσd-j(𝒞QDr(yN)))×𝔼[Ỹm(∇X(x))det(∇X(x)∇X(x)T)1/2∣X(x)=yN](1nσd-j(𝒞QDr(yN)))]pX(x)(yN)dx≥∫Q𝔼[ψ(1nσd-j(𝒞QDr(y)))×𝔼[Ỹm(∇X(x))det(∇X(x)∇X(x)T)1/2∣X(x)=y](1nσd-j(𝒞QDr(yN)))]pX(x)(y)dx.
By using Fatou's Lemma again and that ψ(·/n) is a nondecreasing sequence, we obtain
(2.28)limm→∞limn→∞𝔼[F̃Xn,m(y)]=∫Q𝔼[det(∇X(x)∇X(x)T)1/2∣X(x)=y]pX(x)(y)dx.
Finally
(2.29)FXn,m(y)≤∫𝒞QDr(y)Ỹm(∇X(x))dσd-j(x)≤σd-j(𝒞QDr(y))<∞a.s.
Moreover FXn,m(y)↑∫𝒞QDr(y)Ỹm(∇X(x))dσd-j(x) when n→∞. Clearly applying Beppo-Levi's Theorem we get
(2.30)𝔼[σd-j(𝒞QDr(y))]=limm→∞limn→∞𝔼[F̃Xn,m(y)]≥∫Q𝔼[det(∇X(x)∇X(x)T)1/2∣X(x)=y]pX(x)(y)dx.
Obtaining formula (2.17), formula (2.18) follows by approximating Y uniformly by a nondecreasing sequence of simple functions.The following two propositions of Azaïs and Wschebor [7, pages 132–134] provide the arguments to improve Cabaña's result. In the book, however, the hypothesis is a little different.Proposition 2.6.
LetX:U⊂ℝd→ℝm+k be a random field and I a subset of U, and let u∈ℝm+k. One supposes that X satisfies the following conditions: (1)
the random field∇X is α-Hölder continuous with m/(m+k)<α≤1;(2)
for eachx∈U, the random vector X(x) has a density pX(x)(y) such that pX(x)(y)≤C, for x∈I and y in some neighborhood of u;(3)
the Hausdorff dimension ofI is smaller than or equal to m.Then, almost surely, there is no point x∈I such that X(x)=u.Proposition 2.7.
LetX:U⊂ℝd→ℝj be a random field and U an open set of ℝd and y∈ℝj. Suppose that ∇X is a.s. α-Hölder continuous with 1-(1/(d+j))<α≤1 and moreover for all x∈U the random vector (X(x),∇X(x)) has a bounded continuous density pX(x),∇X(x)(u,ẏ), for u in a neighborhood of y and (x,ẏ) varying in a compact set of U×ℝd×j. Then
(2.31)ℙ{ω:∃xX(x)=y,rank∇X(x)<j}=0.Theorem 2.8.
Under the hypotheses of Theorem2.5 and that ∇X(·) is a.s α-Hölder continuous with 1-(1/(d+j))<α≤1, one has the following.(1)
UnderA1, for all y∈ℝj,
(2.32)𝔼[σd-j(𝒞Q(y))]=∫QpX(x)(y)𝔼[det(∇X(x)∇X(x)T)1/2∣X(x)=y]dx.(2)
UnderA2 and if Y is an almost sure continuous function, for all y∈ℝj one has
(2.33)𝔼[∫𝒞Q(y)Y(x)dσd-j(x)]=∫QpX(x)(y)𝔼[Y(x)det(∇X(x)∇X(x)T)1/2∣X(x)=y]dx.In what follows we give two examples under which the hypothesesA1 and A2 hold. (1)
Suppose thatX(x) is a Gaussian field verifying the hypothesis of Theorem 2.8 and that Var(X(x))>0 for each x. By considering the regression model,
(2.34)∇X(x)=α(x)X(x)+Γ(x)ξ(x)
with a Gaussian ξ(x)⊥X(x), where(2.35)α(x)=𝔼[∇X(x)X(x)T](𝔼[X(x)X(x)T])-1,Γ(x)Γ(x)T=𝔼[∇X(x)∇X(x)T]-𝔼[∇X(x)X(x)T](𝔼[X(x)X(x)T])-1𝔼[X(x)∇X(x)T],
the following equality in law is satisfied:
(2.36)ℒ(∇X(x)∣X(x)=y)=yTα(x)+Γ(x)ξ(x).
This result entails that the expression
(2.37)𝔼[det(∇X(x)∇X(x)T)1/2∣X(x)=y]=yTα(x)α(x)Ty+Γ(x)Γ(x)T
is a continuous function of variable y. Moreover, the hypothesis Var(X(x))>0 yields the continuity in y of pX(x)(y).(1)
Finally let us consider the case of the real envelope of the stationary Gaussian fieldζ(t,x,y). As in [16], we define the envelope of the Gaussian field ζ(t,x,y) as follows. Let us consider the random spectral measure M(λ1,λ2,ω) restricted to the Airy manifold Λ defined below in (3.1). In this manner if we restrict the stochastic integral to the set Λ+={(λ1,λ2,ω):ω≥0,|k⃗|=(ω2/g)}. By using polar coordinates we can write
(2.38)ζ(t,x,y)=2∫0∞∫-ππcos(|k⃗|cosΘx+|k⃗|sinΘy+ωt)dc(ω,Θ).
We define the Hilbert transform of ζ as the Gaussian field
(2.39)ζ̂(t,x,y)=2∫0∞∫-ππsin(|k⃗|cosΘx+|k⃗|sinΘy+ωt)dc(ω,Θ).
The real envelope E(t,x,y) is defined as
(2.40)E(t,x,y)=ζ2(t,x,y)+ζ̂2(t,x,y).
It holds
(2.41)𝔼[∥∇E(t,x,y)∥E(t,x,y)=u]=1u𝔼[∥∇ζ(0,0,0)+∇ζ̂(0,0,0)∥].
This expression is continuos whenever u>0. Moreover the density of E(0,0,0), in the point u, is the Rayleigh density (1/σζ2)ue-u2/2σζ2, that exists and is continuous if σζ2=Var(ζ(0,0,0))>0.


## 3. Representation with Integrals and the Directional Spectrum

In this section, we study a generalization of the Gaussian random fields defined in (1.2) that 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 Krée and Soize “Mecanique Aleatorie,” [17, pages 366–376], or the very readable article [18] in which. Lindgren gives a definitive treatment for this type of spectral stochastic integral models.Another way of looking at Longuet-Higgins' model is(3.1)ζ(t,x,y)=∫Λei(λ1x+λ2y+ωt)dM(λ1,λ2,ω),whereΛistheAirymanifold{λ12+λ22=ω4g2},g is the gravitational constant and M is a random Gaussian orthogonal measure defined on Λ. Defining k⃗=(λ1,λ2) and the following change of variable |k⃗|=(ω2/g) and λ1=(ω2/g)cosΘ and λ2=(ω2/g)sinΘ (see [17]), we obtain(3.2)ζ(t,x,y)=∫-∞∞∫-ππei(|k⃗|cosΘx+|k⃗|sinΘy+ωt)dc(ω,Θ)=∫-∞∞∫-ππexp[i(k⃗·x⃗+ωt)]dc(ω,Θ),
where dc(ω,Θ) is a random measure. The covariance function, K(τ,X,Y), is defined as(3.3)K(τ,X,Y)=E[ζ(t,x,y)ζ¯(t+τ,x+X,y+Y)].
Then by using that(3.4)𝔼[dc(ω,Θ)dc¯(ω′,Θ′)]=Ŝ(ω,Θ)δ(ω-ω′)δ(Θ-Θ′)dωdω′dΘdΘ′,
where Ŝ(ω,Θ) is the two-dimensional spectrum of the wave surface and δ represents Dirac's delta function. Then,(3.5)K(τ,X,Y)=∫-∞∞∫-ππexp[i(k⃗·X⃗+ωτ)]Ŝ(ω,Θ)dωdΘ.
This procedure is justified formally in [17, 18]. If in (3.5) we let X=0 and Y=0, then we obtain(3.6)K(τ):=K(τ,0,0)=∫-∞∞∫-ππŜ(ω,Θ)eiωτdωdΘ,
or equivalently K(τ)=∫-∞∞Ŝ(ω)eiωτdω, where Ŝ(ω)=∫-ππŜ(ω,Θ)dΘ. Function Ŝ(ω) represents the frequency spectrum of the sea surface. This spectrum contains the distribution of the wave energy in the frequency domain. The autocorrelation function K(τ) for the elevation surface ζ(t), in a fixed location, is a real even function.Definition 3.1.
The spectral moments of orderijk are defined as
(3.7)mijk=∫0∞∫-ππuivjωkS(ω,Θ)dΘdω,
where u=(ω2/g)cos(Θ-Θ0), v=(ω2/g)sin(Θ-Θ0), and g is the gravitational constant. If in (3.7) i=j=0, then
(3.8)m00k=∫0∞∫-ππωkS(ω,Θ)dΘdω,
and this can be rewritten as m00k=∫0∞ωkS(ω)dω. The previous relation corresponds to the one-dimensional moment of order k, mk=∫0∞ωkS(ω)dω. Also mij1 and mij2 will be denoted by
(3.9)mij′=∫0∞∫-ππωuivjS(ω,Θ)dΘdω,mij′′=∫0∞∫-ππω2uivjS(ω,Θ)dΘdω,
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 letζ(t,x,y) be the random field (3.2) representing the sea height and suppose that it belongs a.s. to C3(ℝ3,ℝ). We observe the random field (∂ζ/∂x)(t,x,y) in a fixed direction, y=0, for instance. The place where reflection occurs, when the surface ζ(t,x,0) is illuminated by a light source, placed in (0,h1) and observed in (0,h2), for each fixed t is the level curve (4.1)∂ζ∂x(t,x,0)=ζx(t,x)=kx,
where k=(1/2)[(1/h1)+(1/h2)]. This condition is approximately true, whenever kζ and ζx are both small quantities, see [4, page 845].A consequence of the implicit function theorem is(4.2)(ζxx-k)dx+ζxtdt=0,
that is, (4.3)c=dxdt=-ζxt(ζxx-k).
This expression defines the velocity of the specular points. Thus let us define the number of specular points in [0,M] having a speed in [α1,α2] as Ñsp(s,0,α1,α2), where (4.4)Ñsp(s,u,α1,α2):=#{x≤M:ζx(s,x)=kx+u;α1≤ζxt(ζxx-k)≤α2}for0≤s≤t.
Now, define the latter number per unit time as (4.5)Nsp(u,α1,α2,t):=1t∫0tÑsp(s,u,α1,α2)ds.
Notice that the process 𝒵(t)=Ñsp(t,u,α1,α2) is stationary, has finite mean, and it is Riemann integrable, as a function of t. Define 𝒜t=σ{ζ(τ,x,0):τ>t,x∈[0,M]} and the σ-algebra of invariant events 𝒜=∩𝒜t. Under the hypothesis that for each x∈[0,M], K(t,x,0)→0 whenever t→0 the σ-algebra 𝒜 is trivial. By the Birkoff-Khintchine Ergodic Theorem, we have (4.6)∫0t𝒵(s)dst→𝔼ℬ[𝒵(0)],
where ℬ is the σ-algebra of t-invariants associated to 𝒵. Since for each t, ℬt=σ{𝒵(τ):τ>t}⊂𝒜t, it follows that ℬ⊂𝒜, so that 𝔼ℬ[𝒵(0)]=𝔼[𝒵(0)]=𝔼[Ñsp(0,u,α1,α2)] (for references, see [19, page 151]).Our interest here is to compute the Palm distribution of the number of specular points having speed between[α1,α2] defined as (4.7)F(α2)-F(α1)=limt→∞Nsp(0,α1,α2,t)Nsp(0,-∞,∞,t)=𝔼[Ñsp(0,0,α1,α2)]𝔼[Ñsp(0,0,-∞,∞)].
The last equality, as we have seem, is a consequence of the Ergodic Theorem. We will show the following result.Proposition 4.1.
Letζ(t,x,y) be a Gaussian random field (3.2) and assume that it belongs to C3(ℝ3,ℝ) and for each pair (x,y), K(t,x,y)→0 whenever t→0. The Palm distribution F defined above satisfies
(4.8)F(α2)-F(α1)=𝔼[1[α1,α2](ζxt(0,0)/(ζxx(0,0)-k))|ζxx(0,0)-k|]m400ℜ(k),
where ℜ(k):=(2/π)(e-k2/2m400+(k/m400)∫0k/m400e-v2/2dv).Proof.
For a continuous and bounded functionh, we have
(4.9)∫-∞∞h(u)Nsp(u,α1,α2,t)du=1t∫0t∫-∞∞h(u)Ñsp(s,u,α1,α2)duds,
and by the area formula (2.3), we have
(4.10)∫-∞∞h(u)Nsp(u,α1,α2,t)du=1t∫0t∫0Mh(ζx(s,x)-kx)1[α1,α2](ζxt(s,x)(ζxx(s,x)-k))|(ζxx(s,x)-k)|dxds.
Taking expectations, by stationarity and duality, for almost all u, it follows that
(4.11)𝔼[Nsp(u,α1,α2,t)]=∫0Mpζx(0,0)(u+kx)𝔼[1[α1,α2](ζxt(0,0)(ζxx(0,0)-k))|ζxx(0,0)-k|∣ζx(0,0)=u+kx(ζxt(0,0)(ζxx(0,0)-k))]dx.
This may be written, in the Gaussian case, by independence, as
(4.12)𝔼[Nsp(u,α1,α2,t)]=∫0Mexp{-(u+kx)2/2m200}2πm200dx×𝔼[1[α1,α2](ζxt(0,0)(ζxx(0,0)-k))|ζxx(0,0)-k|].
The formula is true for all u as it follows analogously to the result shown by Azaïs and Wschebor [7, page 163].
For the specular points, the interesting level isu=0, thus we obtain that the expectation of the number of specular points having speed between α1 and α2 is
(4.13)∫0Mpζx(0,0)(kx)𝔼[1[α1,α2](ζxt(0,0)(ζxx(0,0)-k))|ζxx(0,0)-k|∣ζx(0,0)=kx]dx,
which in the Gaussian case may be written as
(4.14)𝔼[Nsp(0,α1,α2,t)]=𝔼[Ñsp(0,0,α1,α2)]=∫0Mexp{-k2x2/2m200}2πm200dx×𝔼[1[α1,α2](ζxt(0,0)(ζxx(0,0)-k))|ζxx(0,0)-k|].
Moreover, the expectation of the number of specular points per unit of time Nsp(0,-∞,∞,t) is easily computed yielding formula (2.14) of [4, page 846]
(4.15)𝔼[Nsp(0,-∞,∞,t)]=∫0Mexp{-k2x2/2m200}2πm200dx𝔼|ζxx(0,0)-k|=ℜ(k)m4002πm200∫0Mexp{-k2x22m200}dx,
where ℜ(k):=2/π(e-k2/2m400+(k/m400)∫0k/m400e-v2/2dv).
As the processζ satisfies that K(t,x,0)→0, we obtain (4.8) by simple division.
Let us now definepx,t(ξ1,ξ2) the Gaussian density of the random vector (ζxt(0,0),ζxx(0,0)). We may write (4.8) as
(4.16)F(α2)-F(α1)=1m400ℜ(k)∫α1α2∫-∞∞(ξ2-k)2px,t(c(ξ2-k),ξ2)dcdξ2.Remark 4.2.
Differentiating the previous expression one obtains the density of the velocity of the specular points:(4.17)p̂k(c)=1m400ℜ(k)∫-∞∞(ξ2-k)2px,t(c(ξ2-k),ξ2)dξ2.
If k=0 we recover formula (2-5-19) of [2] (modified in order to consider the case of specular points)
(4.18)p̂0(c)=Δ22m0042((c-c¯)2+Δ2m400-2)-3/2,
where
(4.19)Δ2=[det(m200m110m110m020)]-1,c¯=-m301m400.


## 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 condition(ζx(t,x,y),ζy(t,x,y))=(kx,ky) at point (x,y) and for a fixed time t.Defining the vectorial process(5.1)Z(t,x,y)=(ζx(t,x,y)-kx,ζy(t,x,y)-ky),
we say that we have a specular point if Z=0 and the number of such points in a fixed time t and in a region Ω⊂ℝ2 will be(5.2)NΩZ(0,0)=#{(x,y)∈Ω:Z(t,x,y)=(0,0)}.
We denote as in formula (2.5) x=(x,y) and w=(w1,w2). Then applying this formula to the process Z, we get for almost all (w1,w2)(5.3)𝔼[NΩZ(w1,w2)]=∫ΩpZ(0,x,y)(w1,w2)𝔼[|Δ(t,x,y)|/Z(0,x,y)=(w1,w2)]dxdy,
where Δ(t,x,y)=(ζxx(t,x,y)-k)(ζyy(t,x,y)-k)-ζxy2(t,x,y).This formula turns out to be valid for all(w1,w2) under the hypotheses of Theorem 6.2 in [7] (in our case ζ∈C3(ℝ3) and VarZ>0 will be enough) and in particular for the specular points, that is when (w1,w2)=(0,0), (5.4)𝔼[NΩZ(0,0)]=∫ΩpZ(0,x,y)(0,0)𝔼[|Δ(0,0,0)|/Z(x,y)=(0,0)]dxdy.
The independence property allows writing (5.5)𝔼[NΩZ(0,0)]=𝔼[|Δ(0,0,0)|]∫ΩpZ(0,x,y)(0,0)dxdy,
obtaining finally the following result.Proposition 5.1.
Let the stationary mean zero Gaussian random fieldζ(t,x,y)∈C3(ℝ3) a.s. and VarZ>0. Then,
(5.6)𝔼[NΩZ(0,0)]=𝔼[|(ζxx(0,0,0)-k)(ζyy(0,0,0)-k)-(ζxy(0,0,0))2|]∫ΩpZ(0,x,y)(0,0)dxdy.Remark 5.2.
The Li and Wei formula (cf. [20]) provides a way to compute the expectation of the absolute value of the determinant in the above formula, see Azaïs et al. [10], which one will not pursue. Instead one will apply a Monte Carlo method. Let us consider the regression model
(5.7)ζyy(0,0,0)=αζxx(0,0,0)+βζxy(0,0,0)+σ1ɛ1,
where
(5.8)α=m2202-m310m130m400m220-m3102β=m400m130-m310m220m400m220-m3102,ɛ1=N(0,1)⊥(ζxx(0,0,0),ζxy(0,0,0)) and σ2=-m400+α2m400+β2m220+2αβm220. Therefore it yields
(5.9)𝔼[|Δ(0,0,0)|]=𝔼[|(ζxx(0,0,0)-k)((ζxx(0,0,0)+βζxy(0,0,0)+σɛ1)-k)-(ζxy(0,0,0))2|].
This last expression can be evaluated readily by using Monte Carlo. Indeed let ɛ⃗i=(ɛ1i,ɛ2i,ɛ3i)t,  i=1,…,N, be a sample of standard Gaussian vectors in ℝ3. We have
(5.10)limN→∞1N∑i=1N|((m220-m3102m400)1/2ɛ2i+m310m400ɛ3i)2(m400ɛ3i-k)×(((αm400ɛ3i+β((m220-m3102m400)1/2ɛ2i+m310m400ɛ3i)+σɛ1i))-k)-((m220-m3102m400)1/2ɛ2i+m310m400ɛ3i)2|=𝔼[|Δ(0,0,0)|].


## 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 fieldZ(t,x,y)=(ζx(t,x,y)-kx,ζy(t,x,y)-ky); the number of specular points of the field ζ(t,x,y), in a fixed time t and in a region Ω⊂ℝ2, was defined in (5.2) and denoted as NΩZ(0,0). We have already computed the expectation of the number of specular points (6.1)𝔼[NΩZ(0,0)]=𝔼[|Δ(0,0,0)|]∫ΩpZ(0,x,y)(0,0)dxdy.
The condition satisfied for the specular points (i.e., (ζx(t,x,y),ζy(t,x,y))=(kx,ky)) and the implicit function theorem entails (6.2)ζxt=-(ζxx-k)dxdt-ζxydydt,ζyt=-ζxydxdt-(ζyy-k)dydt.
Let us define as Longuet-Higgins cx=dx/dtand cy=dy/dt. The objective is to find the Palm distribution associated to the velocity field (cx,cy)=(dx/dt),(dy/dt). The following computations, in the case k=0, are essentially contained in the very original and seminal work of Longuet-Higgins [2].Now define foru=(u1,u2): (6.3)Ñsp(s,u,v1,v2,v3,v4)=#{(x,y)∈Ω:Z(s,x,y)=u;v1≤cx≤v2;v3≤cy≤v4}
for 0≤s≤t and Ω a compact set in ℝ2.Consider(6.4)Nsp(u,v1,v2,v3,v4,t):=1t∫0tÑsp(s,u,v1,v2,v3,v4)ds.Letg be a continuous bounded function. On the one hand, using (2.3), we have(6.5)∫ℝ2g(u)Nsp(u,v1,v2,v3,v4,t)du=1t∫0t∫ℝ2g(u)Ñsp(s,u,v1,v2,v3,v4)duds=1t∫0t∫Ωg(Z(s,x,y)-u)1[v1,v2](cx)1[v3,v4](cy)|Δ(t,x,y)|dxdyds.
Let us denote by p(ξ4,ξ5,ξ6,ξ7,ξ8) the density of the Gaussian random vector (6.6)(ζxx(0),ζxy(0),ζyy(0),ζxt(0),ζyt(0)).
It follows that (6.7)p(ξ4,ξ5,ξ6,cx,cy):=p(ξ4,ξ5,ξ6,-(ξ4-k)cx-ξ5cy,-ξ5cx-(ξ6-k)cy)
is the density function of the random vector (ζxx(0),ζxy(0)ζyy(0),cx(0),cy(0)). Taking expectations in (6.5), using duality and putting u=0 (under the hypothesis ζ∈C3(ℝ3,ℝ) and Varζ>0 the formula holds for all levels u (cf. [7, page 163]) we obtain (6.8)𝔼[Nsp(0,v1,v2,v3,v4,t)]=∫ΩpZ(0,0,0)(-kx,-ky)dxdy×∫v1v2∫v3v4∫R3p(ξ4,ξ5,ξ6,cx,cy)|(ξ4-k)(ξ6-k)-ξ52|dξ4dξ5dξ6dcxdcy.
Hence, analogously as in Section 4 and by using the same arguments that lead to apply the Ergodic Theorem, the Palm distribution of a specular point having the components of its velocity cx∈[υ1,υ2] and cy∈[υ3,υ4] is (6.9)limt→∞Nsp(0,v1,v2,v3,v4,t)Nsp(0,-∞,∞,-∞,∞,t)=∫v1v2∫v3v4∫R3p(ξ4,ξ5,ξ6,cx,cy)|(ξ4-k)(ξ6-k)-ξ52|dξ4dξ5dξ6dcxdcy𝔼|Δ(0,0,0)|.
We can summarize the above computations in the following result.Proposition 6.1.
Letζ(t,x,y) be a mean stationary Gaussian field which is a.s. three times continuously differentiable and Varζ>0. Assume also that its covariance function satisfies K(t,x,y)→0 for each (x,y) whenever t→0. Hence the Palm distribution of a specular point having the components of its velocity cx∈[υ1,υ2] and cy∈[υ3,υ4] is
(6.10)F(υ3,υ4)-F(υ1,υ2)=∫v1v2∫v3v4∫R3p(ξ4,ξ5,ξ6,cx,cy)|(ξ4-k)(ξ6-k)-ξ52|dξ4dξ5dξ6dcxdcy𝔼|Δ(0,0,0)|.Remark 6.2.
Taking derivatives one gets the density of the speed of the two-dimensional specular points(6.11)p̂(x,y,k)(cx,cy)=∫R3p(ξ4,ξ5,ξ6,cx,cy)|(ξ4-k)(ξ6-k)-ξ52|dξ4dξ5dξ6𝔼|Δ(0,0,0)|.
In the particular case (infinite distance), where k=0, we obtain the Longuet-Higgins formula (see [2, pages 362–365]). Nevertheless, formula (6.11) is well suited for numerical computations, for k≠0.


## 7. Another Application of Rice Formula

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

Letζ(t,x):=ζ(t,x,y) be again a stationary zero mean Gaussian random field modeling the height of the sea waves, here t∈ℝ+ and x=(x,y)∈ℝ2. Let us recall that such a field has the spectral representation given in (3.1). Also in (3.5), we give an expression for its covariance function. In this expression, the function Ŝ(ω,Θ) is known as the directional spectral function and if it does not depend on Θ the random field ζ is called isotropic.In what follows, we will get information about the crest of the waves in a directionθ. Let us define, as in [8], the crest of the waves ζ in direction θ at time s as the level set (7.1)ℭQ(s,θ)={(x,y)∈Q:ζ′θ(s,x,y)=0;ζθθ′′(s,x,y)<0},
where ζ′θ and ζθθ′′ denote the first and second derivatives in the direction θ, respectively. This set is the zero level set 𝒞QZθ(s,0) of the field (7.2)Zθ(s,x,y)=ζx(s,x,y)cosθ+ζy(s,x,y)sinθ,
under the additional condition that Zθθ′′(s,x,y)<0. If θ̂ is the direction orthogonal to θ, we can express the gradient of Zθ′′(s,x,y) with respect to θ and its orthogonal, denoted as ∇θ, as (7.3)∇θZθ(s,x,y)=(∂θZθ(s,x,y),∂θ̂Zθ(s,x,y))=∥∇θZθ(s,x,y)∥(cosΦ(s,x,y),sinΦ(s,x,y)),
where Φ(s,x,y)=arctan∂θ̂Zθ(s,x,y)/∂θZθ(s,x,y). Thus taking into account that in the crest Φ∈[π/2,3π/2], if H:[π/2,3π/2]→ℝ is a continuous function by using Theorem 2.8 we get (7.4)1t∫0t∫ℭQ(s,θ)H(Φ(s,x))dσ1(x)ds=1t∫0t∫𝒞QZθ(s,0)H(Φ(s,x))1[-1,0)(cosΦ(s,x))dσ1(x)ds→𝔼[∫𝒞QZθ(0,0)H(Φ(0,x))1[-1,0)(cosΦ(0,x))dσ1(x)].
To obtain the above result we assume, as in the precedent sections, that the covariance K(t,x,y) of the stationary field ζ satisfies that for each (x,y)(7.5)K(t,x,y)→0whenevert→∞.
This hypothesis allows us to place ourselves in the framework of the Ergodic Theorem. Defining the Palm distribution νθ of the normal angle at the crest in the direction θ as the following integral: (7.6)∫π/23π/2H(φ)dνθ(φ):=limt→∞∫0t∫ℭQ(s,θ)H(Φ(s,x))dσ1(x)∫0tσ1(ℭQ(s,θ))ds.
Let us denote E(k) as the elliptic integral of the first kind. Also let us define γ2(θ)=λ-(θ)/λ+(θ), where λ-(θ)≤λ+(θ) are the eigenvalues of the covariance matrix of the Gaussian vector (∂θZθ(0,0,0),∂θ̂Zθ(0,0,0)) and κ(θ) the angle that turn diagonal this matrix. We get the following result.Proposition 7.1.
If the mean zero and stationary random fieldζ is three times continuously differentiable and hypothesis (7.5) holds, the Palm distribution νθ of the normal angle at the crest in the direction θ satisfies
(7.7)∫π/23π/2H(φ)dνθ(φ)=γ2(θ)2E(1-γ2(θ))∫π/23π/2H(φ)(1-(1-γ2(θ))sin2(φ-κ(θ)))3/2dφ.Proof.
By using the Ergodic Theorem, we get(7.8)∫π/23π/2H(φ)dνθ(φ)=𝔼[∫𝒞QZθ(0,0)H(Φ(0,x))1[-1,0)(cosΦ(0,x))dσ1(x)]𝔼σ1(ℭQ(0,θ))=𝔼[∫𝒞QZθ(0,0)H(Φ(0,x))1[-1,0)(cosΦ(0,x))dσ1(x)](1/2)𝔼σ1(𝒞QZθ(0,0)).
To get (7.7) it is enough to compute the two expectations in the last equality. Let us define a(θ)=𝔼(Zθ2(0,0,0)). First we consider the numerator. We have changing to polar coordinates
(7.9)𝔼[∫𝒞QZθ(0,0)H(Φ(0,x))1[-1,0)(cosΦ(0,x))dσ1(x)]=λ-(θ)(2π)2/3a(θ)(λ+(θ)λ-(θ))1/2×∫0∞∫π/23π/2ρ2H(φ)e-(ρ2/2λ+(θ)λ-(θ))(λ+cos2(φ-κ(θ))+λ-sin2(φ-κ(θ))dρdφ=λ-(θ)a(θ)λ+(θ)4π∫π/23π/2H(φ)1(1-(1-γ2(θ))sin2(φ-κ(θ))3/2)dφ.
For the denominator, we have H=1 obtaining, (1/2)𝔼σ1(𝒞QZθ(0,0))=(1/2π)(λ+(θ)/a(θ))E(1-γ2(θ)). Thus (7.7) follows by a simple division.Remark 7.2.
Formula (7.7) can be developed further for the model where the directional spectrum has the following representation Ŝ(ω,Θ)=f(ω)Ω(Θ), function Ω is usually called the spreading function. This matter needs further research and we will not pursue this study in this work.


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

Letζ(t,x):=ζ(t,x,y) be again a stationary zero mean Gaussian random field modeling the height of the sea waves, here t∈ℝ+ and x=(x,y)∈ℝ2. Let us recall that such a field has the spectral representation given in (3.1). Also in (3.5), we give an expression for its covariance function. In this expression, the function Ŝ(ω,Θ) is known as the directional spectral function and if it does not depend on Θ the random field ζ is called isotropic.In what follows, we will get information about the crest of the waves in a directionθ. Let us define, as in [8], the crest of the waves ζ in direction θ at time s as the level set (7.1)ℭQ(s,θ)={(x,y)∈Q:ζ′θ(s,x,y)=0;ζθθ′′(s,x,y)<0},
where ζ′θ and ζθθ′′ denote the first and second derivatives in the direction θ, respectively. This set is the zero level set 𝒞QZθ(s,0) of the field (7.2)Zθ(s,x,y)=ζx(s,x,y)cosθ+ζy(s,x,y)sinθ,
under the additional condition that Zθθ′′(s,x,y)<0. If θ̂ is the direction orthogonal to θ, we can express the gradient of Zθ′′(s,x,y) with respect to θ and its orthogonal, denoted as ∇θ, as (7.3)∇θZθ(s,x,y)=(∂θZθ(s,x,y),∂θ̂Zθ(s,x,y))=∥∇θZθ(s,x,y)∥(cosΦ(s,x,y),sinΦ(s,x,y)),
where Φ(s,x,y)=arctan∂θ̂Zθ(s,x,y)/∂θZθ(s,x,y). Thus taking into account that in the crest Φ∈[π/2,3π/2], if H:[π/2,3π/2]→ℝ is a continuous function by using Theorem 2.8 we get (7.4)1t∫0t∫ℭQ(s,θ)H(Φ(s,x))dσ1(x)ds=1t∫0t∫𝒞QZθ(s,0)H(Φ(s,x))1[-1,0)(cosΦ(s,x))dσ1(x)ds→𝔼[∫𝒞QZθ(0,0)H(Φ(0,x))1[-1,0)(cosΦ(0,x))dσ1(x)].
To obtain the above result we assume, as in the precedent sections, that the covariance K(t,x,y) of the stationary field ζ satisfies that for each (x,y)(7.5)K(t,x,y)→0whenevert→∞.
This hypothesis allows us to place ourselves in the framework of the Ergodic Theorem. Defining the Palm distribution νθ of the normal angle at the crest in the direction θ as the following integral: (7.6)∫π/23π/2H(φ)dνθ(φ):=limt→∞∫0t∫ℭQ(s,θ)H(Φ(s,x))dσ1(x)∫0tσ1(ℭQ(s,θ))ds.
Let us denote E(k) as the elliptic integral of the first kind. Also let us define γ2(θ)=λ-(θ)/λ+(θ), where λ-(θ)≤λ+(θ) are the eigenvalues of the covariance matrix of the Gaussian vector (∂θZθ(0,0,0),∂θ̂Zθ(0,0,0)) and κ(θ) the angle that turn diagonal this matrix. We get the following result.Proposition 7.1.
If the mean zero and stationary random fieldζ is three times continuously differentiable and hypothesis (7.5) holds, the Palm distribution νθ of the normal angle at the crest in the direction θ satisfies
(7.7)∫π/23π/2H(φ)dνθ(φ)=γ2(θ)2E(1-γ2(θ))∫π/23π/2H(φ)(1-(1-γ2(θ))sin2(φ-κ(θ)))3/2dφ.Proof.
By using the Ergodic Theorem, we get(7.8)∫π/23π/2H(φ)dνθ(φ)=𝔼[∫𝒞QZθ(0,0)H(Φ(0,x))1[-1,0)(cosΦ(0,x))dσ1(x)]𝔼σ1(ℭQ(0,θ))=𝔼[∫𝒞QZθ(0,0)H(Φ(0,x))1[-1,0)(cosΦ(0,x))dσ1(x)](1/2)𝔼σ1(𝒞QZθ(0,0)).
To get (7.7) it is enough to compute the two expectations in the last equality. Let us define a(θ)=𝔼(Zθ2(0,0,0)). First we consider the numerator. We have changing to polar coordinates
(7.9)𝔼[∫𝒞QZθ(0,0)H(Φ(0,x))1[-1,0)(cosΦ(0,x))dσ1(x)]=λ-(θ)(2π)2/3a(θ)(λ+(θ)λ-(θ))1/2×∫0∞∫π/23π/2ρ2H(φ)e-(ρ2/2λ+(θ)λ-(θ))(λ+cos2(φ-κ(θ))+λ-sin2(φ-κ(θ))dρdφ=λ-(θ)a(θ)λ+(θ)4π∫π/23π/2H(φ)1(1-(1-γ2(θ))sin2(φ-κ(θ))3/2)dφ.
For the denominator, we have H=1 obtaining, (1/2)𝔼σ1(𝒞QZθ(0,0))=(1/2π)(λ+(θ)/a(θ))E(1-γ2(θ)). Thus (7.7) follows by a simple division.Remark 7.2.
Formula (7.7) can be developed further for the model where the directional spectrum has the following representation Ŝ(ω,Θ)=f(ω)Ω(Θ), function Ω is usually called the spreading function. This matter needs further research and we will not pursue this study in this work.



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*Source: 621038-2010-05-20.xml*