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

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 fields 𝑋∶ℝ^𝑑→ℝ^𝑗(𝑑>𝑗).

1. Introduction

In 1944, Rice [1] proposed the model𝜁(𝑡)=_𝑛𝑐_𝑛cos(𝜎_𝑛𝑡+𝜀_𝑛), (1.1) to describe the noise in an electrical current. In this relation, 𝜎_𝑛/(2𝜋) denotes the different frequencies, 𝑐_𝑛 are Gaussian random variables, identically distributed and independent, and 𝜀_𝑛 are random variables uniformly distributed in [0,2𝜋].

Later, in 1957, Longuet-Higgins [2] defined the following multidimensional generalization of Rice's model:𝜁(𝑡,𝑥,𝑦)=_𝑛𝑐_𝑛cos(𝑢_𝑛𝑥+𝑣_𝑛𝑦+𝜎_𝑛𝑡+𝜀_𝑛). (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 field 𝜁. 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 fields 𝑋∶ℝ^𝑑→ℝ^𝑑, recently proved in [7] and by using another Rice formula for random fields 𝑋∶ℝ^𝑑→ℝ^𝑗 (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. 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. 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 follows 𝜆_𝑑 and 𝜎_𝑑−𝑚 will denote, respectively, the Lebesgue measure in the space ℝ^𝑑 and the Hausdorff measure defined in the subspaces of dimension 𝑑−𝑚, trivially by definition 𝜆_𝑑=𝜎_𝑑.

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. Let 𝑔∶ℝ^𝑑→ℝ^𝑑 be a continuously differentiable function. If we define 𝑁^𝑔_𝐴(𝐲)=#{𝐬∈𝐴⊂ℝ^𝑑∶𝑔(𝐬)=𝐲}, (2.1) then if 𝑑=1, one has_𝐴𝑓(𝑔(𝐬))|𝑔^′(𝐬)|𝑑𝐬=_ℝ𝑑𝑓(𝐲)𝑁^𝑔_𝐴(𝐲)𝑑𝐲, (2.2) 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∶ℝ^𝑑→ℝ is a continuous bounded function, then the version of Banach's formula for 𝑑≥1 is_𝐴𝑓(𝑔(𝐬))|det∇𝑔(𝐬)|𝑑𝐬=_ℝ𝑑𝑓(𝐲)𝑁^𝑔_𝐴(𝐲)𝑑𝐲. (2.3) This expression is usually called the area formula (cf. [12]).

Now, let 𝑋∶ℝ^𝑑→ℝ^𝑑 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 𝑁^𝑋_𝐴(𝐲)=#{𝐬∈𝐴⊂ℝ^𝑑∶𝑋(𝐬)=𝐲}. (2.4) Let 𝑝_𝑋(𝐬) denote the marginal density of 𝑋(𝐬). By using formula (2.3), Fubini's Theorem and duality we get for a.s. 𝐲∈ℝ^𝑑𝔼[𝑁^𝑋_𝐴(𝐲)]=_𝐴𝔼[|det∇𝑋(𝐬)|𝑋(𝐬)=𝐲]𝑝_𝑋(𝐬)(𝐲)𝑑𝐬. (2.5) The fact that this formula is true for all 𝐲∈ℝ^𝑑 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 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 𝐶_𝑄(𝐲)={𝐱∈𝑄∶𝑔(𝐱)=𝐲}=𝑔⁻¹(𝐲)∩𝑄, (2.6) where 𝑄 is a compact set of ℝ^𝑑. 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. Let 𝑓∶ℝ^𝑗→ℝ be a continuous and bounded function. Restricted to the set {𝐱∈ℝ^𝑑∶∇𝑔(𝐱)hasrank𝑗}, (2.7) the following formula holds: _𝑄𝑓(𝑔(𝐱))det(∇𝑔(𝐱)∇𝑔(𝐱)^𝑇)^1/2𝑑𝐱=_𝑅𝑗𝑓(𝐲)𝜎_𝑑−𝑗(𝐶_𝑄(𝐲))𝑑𝐲. (2.8)

Corollary 2.2. Let 𝑌∶ℝ^𝑑→ℝ be a continuous and bounded function under restriction (2.7), then _𝑅𝑗𝑓(𝐲)[_{𝐶𝑄(𝐲)}𝑌(𝐱)𝑑𝜎_𝑑−𝑗(𝐱)]𝑑𝑦=_ℝ𝑑𝑓(𝑔(𝐱))𝑌(𝐱)det(∇𝑔(𝐱)∇𝑔(𝐱)^𝑇)^1/2𝑑𝐱. (2.9)

Remark 2.3. Formula (2.8) and (2.9) hold true without restriction (2.7). In fact it can be proved that for 𝐴⊂{𝐱∈ℝ^𝑑∶∇𝑔(𝐱)hasrank<𝑗}, (2.10)𝜎_𝑑−𝑗(𝑔⁻¹(𝐲)∩𝐴)=0 holds for almost all 𝐲. This also implies that |_{𝐴∩𝐶𝑄(𝐲)}𝑌𝑑𝜎_𝑑−𝑗|≤‖𝑌‖_∞𝜎_𝑑−𝑗(𝑔⁻¹(𝐲)∩𝐴)=0. (2.11) Let us define for a compact 𝑄 the functions 𝐺(𝐲)=𝜎_𝑑−𝑗(𝑔⁻¹(𝐲)∩𝑄),𝐹(𝐲)=_{𝐶𝑄(𝐲)}𝑌(𝐱)𝑑𝜎_𝑑−𝑗(𝐱). (2.12) The following lemma holds true.

Lemma 2.4. Under hypothesis of Theorem 2.1, functions 𝐺 and 𝐹 are continuous.

The following results are simple consequences of Theorem 2.1 and Corollary 2.2.

Let 𝑋∶Ω×ℝ^𝑑→ℝ^𝑗 (𝑑>𝑗) be a stationary random field belonging to 𝐂¹(ℝ^𝑑,ℝ^𝑗) and suppose that for all 𝐱∈ℝ^𝑑, the density of 𝑋(𝐱),𝑝_𝑋(𝐱)(⋅) exists (in the Gaussian case this holds whenever Var𝑋(𝐱)>0). We have (i)For almost all 𝐲∈ℝ^𝑗, 𝔼[𝜎_𝑑−𝑗(𝒞_𝑄(𝐲))]=_𝑄𝑝_𝑋(𝐱)(𝐲)𝔼[det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑑𝐱. (2.13)(ii)For almost all𝐲∈ℝ^𝑗,𝔼[_{𝒞𝑄(𝐲)}𝑌(𝐱)𝑑𝜎_𝑑−𝑗(𝐱)]=_𝑄𝑝_𝑋(𝐱)(𝐲)𝔼[𝑌(𝐱)det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=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 fixed 𝐲. We will prove in what follows that the formulas hold for all 𝐲.

Define the set 𝐷^𝑟={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 𝒞^𝐷𝑟_𝑄(𝐲)=𝒞_𝑄(𝐲)∩𝐷^𝑟. 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∈ℝ^𝑑, 𝑝_𝑋(𝐱)(⋅) exists and is continuous. For a continuous function 𝐻∶ℝ^𝑑×ℝ^𝑗→ℝ the following expression: _𝑄𝑝_𝑋(𝐱)(𝐲)𝔼[𝐻(∇𝑋(𝐱))∣𝑋(𝐱)=𝐲]𝑑𝐱 (2.15) is a continuous function in the 𝐲 variable.(ii)𝐴₂: the expression _𝑄𝑝_𝑋(𝐱)(𝐲)𝔼[𝑌(𝐱)𝐻(∇𝑋(𝐱))∣𝑋(𝐱)=𝐲]𝑑𝐱 (2.16) 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. Consider 𝑋∶Ω×ℝ^𝑑→ℝ^𝑗(𝑑>𝑗) a random field belonging to 𝐂¹(ℝ^𝑑,ℝ^𝑗). (i)Then under 𝐴₁ for all 𝐲∈ℝ^𝑗, 𝔼[𝜎_𝑑−𝑗(𝒞^𝐷𝑟_𝑄(𝐲))]=_𝑄𝑝_𝑋(𝐱)(𝐲)𝔼[det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑑𝐱. (2.17)(ii)If 𝑌 is an almost sure continuous function under 𝐴₂, for all 𝐲∈ℝ^𝑗 one has 𝔼[∫_𝒞𝐷^r_𝑄(𝐲)𝑌(𝐱)𝑑𝜎_𝑑−𝑗(𝐱)]=_𝑄𝑝_𝑋(𝐱)(𝐲)𝔼[𝑌(𝐱)det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑑𝐱. (2.18)

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 𝑌_𝑚1(∇𝑋(𝐱))=𝜓(𝑚det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇1))𝜓(𝑚det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)), (2.19)𝑌_𝑚(∇𝑋(𝐱))=0 if 𝐱∈{𝐲∶det(∇𝑋(𝐱)∇𝑋(𝐱))^𝑇>2𝑚}∪{𝐲∶1/(det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇))>2𝑚}.
For 𝐱 belonging to the complement of this set and defining 𝜆₁(𝐱)≤𝜆₂(𝐱)≤⋯≤𝜆_𝑗(𝐱) the eigenvalues of ∇𝑋(𝐱)∇𝑋(𝐱)^𝑇, it holdsdet(∇𝑋(𝐱)∇𝑋(𝐱))^𝑇1≤2𝑚,det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)≤2𝑚, (2.20) and moreover, defining 𝑉=ker(∇𝑋(𝑥)) and 𝑉^⟂ its orthogonal subspace, we have ‖(∇𝑋(𝐱)∣_𝑉⟂)⁻¹1‖=𝜆₁∏(𝐱)=(^𝑗𝑖=2𝜆_𝑖(𝐱)det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇))^1/2≤√2𝑚×‖∇𝑋(𝐱)∣_𝑉⟂‖^(𝑗/2). (2.21) 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: 𝐹_𝑋^𝑛,𝑚1(𝐲)∶=𝜓(𝑛𝜎_𝑑−𝑗(𝒞^𝐷𝑟_𝑄(𝐲)))_{𝒞𝐷𝑟𝑄(𝐲)}𝑌_𝑚(∇𝑋(𝐱))𝑑𝜎_𝑑−𝑗(𝐱) (2.22) is a.s. continuous, also the sequence 𝐹_𝑋^𝑛,𝑚 is nondecreasing in both indexes. Moreover, the inequality 𝐹_𝑋^𝑛,𝑚(𝐲)≤𝑛 yields that 𝐸[𝐹_𝑋^𝑛,𝑚(𝐲)] is a continuous function. Using formula (2.9) applied to the field 𝑋 and the function 𝐹_𝑋^𝑛,𝑚, we have _ℝ𝑗1𝜓(𝑛𝜎_𝑑−𝑗(𝒞^𝐷𝑟_𝑄(𝐲)))_{𝒞𝐷𝑟𝑄(𝐲)}𝑌_𝑚(∇𝑋(𝐱))𝑑𝜎_𝑑−𝑗=(𝐱)𝑑𝐲_𝑄𝜓1𝑛𝜎_𝑑−𝑗𝒞^𝐷𝑟_𝑄(𝑋(𝐱))𝑌_𝑚(∇𝑋(𝐱))det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2𝑑𝐱. (2.23) From this we have that for almost all 𝐲∈ℝ^𝑗, 𝔼[𝐹_𝑋^𝑛,𝑚=(𝐲)]_𝑄𝑝_𝑋(𝐱)1(𝐲)𝔼[𝜓(𝑛𝜎_𝑑−𝑗𝒞^𝐷𝑟_𝑄(𝑋(𝐱)))𝑌_𝑚(∇𝑋(𝐱))det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑑𝐱. (2.24) Thus, 𝔼[𝐹_𝑋^𝑛,𝑚(𝐲)]≤_𝑄𝔼[det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑝_𝑋(𝐱)(𝐲)𝑑𝐱. (2.25) 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 𝔼[𝜎_𝑑−𝑗(𝒞^𝐷𝑟_𝑄(𝐲))]≤_𝑄𝔼[det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑝_𝑋(𝐱)(𝐲)𝑑𝐱. (2.26) To prove the other inequality, let 𝐲_𝑁→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 𝔼[𝐹_𝑋^𝑛,𝑚(𝐲)]=lim_𝑁→∞𝔼[𝐹_𝑋^𝑛,𝑚(𝐲_𝑁)]=lim_{𝑁→∞𝑄}1𝔼[𝜓(𝑛𝜎_𝑑−𝑗(𝒞^𝐷𝑟_𝑄(𝐲_𝑁𝑌)))×𝔼[_𝑚(∇𝑋(𝐱))det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲_𝑁]]𝑝_𝑋(𝐱)(𝐲_𝑁≥)𝑑𝐱_𝑄1𝔼[𝜓(𝑛𝜎_𝑑−𝑗(𝒞^𝐷𝑟_𝑄𝑌(𝐲)))×𝔼[_𝑚(∇𝑋(𝐱))det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]]𝑝_𝑋(𝐱)(𝐲)𝑑𝐱. (2.27) By using Fatou's Lemma again and that 𝜓(⋅/𝑛) is a nondecreasing sequence, we obtain lim_𝑚→∞lim_𝑛→∞𝔼[𝐹_𝑋^𝑛,𝑚(𝐲)]=_𝑄𝔼[det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑝_𝑋(𝐱)(𝐲)𝑑𝐱. (2.28) Finally 𝐹_𝑋^𝑛,𝑚(𝐲)≤_{𝒞𝐷𝑟𝑄(𝐲)}𝑌_𝑚(∇𝑋(𝐱))𝑑𝜎_𝑑−𝑗(𝐱)≤𝜎_𝑑−𝑗(𝒞^𝐷𝑟_𝑄(𝐲))<∞a.s. (2.29) Moreover 𝐹_𝑋^𝑛,𝑚(𝐲)↑∫_{𝒞𝐷𝑟𝑄(𝐲)}𝑌_𝑚(∇𝑋(𝐱))𝑑𝜎_𝑑−𝑗(𝐱) when 𝑛→∞. Clearly applying Beppo-Levi's Theorem we get 𝔼[𝜎_𝑑−𝑗(𝒞^𝐷𝑟_𝑄(𝐲))]=lim_𝑚→∞lim_𝑛→∞𝔼[𝐹_𝑋^𝑛,𝑚≥(𝐲)]_𝑄𝔼[det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑝_𝑋(𝐱)(𝐲)𝑑𝐱. (2.30) 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. Let 𝑋∶𝑈⊂ℝ^𝑑→ℝ^𝑚+𝑘 be a random field and 𝐼 a subset of 𝑈, and let 𝐮∈ℝ^𝑚+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 each 𝐱∈𝑈, the random vector 𝑋(𝐱) has a density p_𝑋(𝐱)(𝐲) such that 𝑝_𝑋(𝐱)(𝐲)≤𝐶, for 𝐱∈𝐼 and 𝐲 in some neighborhood of 𝐮;(3)the Hausdorff dimension of 𝐼 is smaller than or equal to 𝑚.Then, almost surely, there is no point 𝐱∈𝐼 such that 𝑋(𝐱)=𝐮.

Proposition 2.7. Let 𝑋∶𝑈⊂ℝ^𝑑→ℝ^𝑗 be a random field and 𝑈 an open set of ℝ^𝑑 and 𝐲∈ℝ^𝑗. 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 𝑝_{𝑋(𝐱),∇𝑋(𝐱)}(𝐮,𝐲̇), for 𝐮 in a neighborhood of 𝐲 and (𝐱,𝐲̇) varying in a compact set of 𝑈×ℝ^𝑑×𝑗. Then ℙ{𝜔∶∃𝐱𝑋(𝐱)=𝐲,rank∇𝑋(𝐱)<𝑗}=0. (2.31)

Theorem 2.8. Under the hypotheses of Theorem 2.5 and that ∇𝑋(⋅) is a.s 𝛼-Hölder continuous with 1−(1/(𝑑+𝑗))<𝛼≤1, one has the following.(1)Under 𝐴₁, for all 𝐲∈ℝ^𝑗, 𝔼[𝜎_𝑑−𝑗(𝒞_𝑄(𝐲))]=_𝑄𝑝_𝑋(𝐱)(𝐲)𝔼[det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑑𝐱. (2.32)(2)Under 𝐴₂ and if 𝑌 is an almost sure continuous function, for all 𝐲∈ℝ^𝑗 one has 𝔼[_{𝒞𝑄(𝐲)}𝑌(𝐱)𝑑𝜎_𝑑−𝑗(𝐱)]=_𝑄𝑝_𝑋(𝐱)(𝐲)𝔼[𝑌(𝐱)det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]𝑑𝐱. (2.33)

In what follows we give two examples under which the hypotheses 𝐴₁ and 𝐴₂ hold. (1)Suppose that 𝑋(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, ∇𝑋(𝐱)=𝛼(𝐱)𝑋(𝐱)+Γ(𝐱)𝜉(𝐱) (2.34) with a Gaussian 𝜉(𝐱)⟂𝑋(𝐱), where𝛼(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], (2.35) the following equality in law is satisfied: ℒ(∇𝑋(𝐱)∣𝑋(𝐱)=𝐲)=𝐲^𝑇𝛼(𝐱)+Γ(𝐱)𝜉(𝐱). (2.36) This result entails that the expression 𝔼[det(∇𝑋(𝐱)∇𝑋(𝐱)^𝑇)^1/2∣𝑋(𝐱)=𝐲]=𝐲^𝑇𝛼(𝐱)𝛼(𝐱)^𝑇𝐲+Γ(𝐱)Γ(𝐱)^𝑇 (2.37) is a continuous function of variable 𝐲. Moreover, the hypothesis Var(𝑋(𝐱))>0 yields the continuity in 𝑦 of 𝑝_𝑋(𝐱)(𝐲).(1)Finally let us consider the case of the real envelope of the stationary Gaussian field 𝜁(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,|𝑘|=(𝜔²/g)}. By using polar coordinates we can write 𝜁(𝑡,𝑥,𝑦)=2^∞₀^𝜋_−𝜋cos(|𝑘|cosΘ𝑥+|𝑘|sinΘ𝑦+𝜔𝑡)𝑑𝑐(𝜔,Θ). (2.38) We define the Hilbert transform of 𝜁 as the Gaussian field 𝜁̂(𝑡,𝑥,𝑦)=2^∞₀^𝜋_−𝜋sin(|𝑘|cosΘ𝑥+|𝑘|sinΘ𝑦+𝜔𝑡)𝑑𝑐(𝜔,Θ). (2.39) The real envelope 𝐸(𝑡,𝑥,𝑦) is defined as 𝐸(𝑡,𝑥,𝑦)=𝜁²(𝑡,𝑥,𝑦)+𝜁̂²(𝑡,𝑥,𝑦). (2.40) It holds 1𝔼[‖∇𝐸(𝑡,𝑥,𝑦)‖𝐸(𝑡,𝑥,𝑦)=𝐮]=𝐮𝔼[‖∇𝜁(0,0,0)+∇𝜁̂(0,0,0)‖]. (2.41) This expression is continuos whenever 𝐮>0. Moreover the density of 𝐸(0,0,0), in the point 𝐮, is the Rayleigh density (1/𝜎²_𝜁)𝐮𝑒^{−𝐮2/2𝜎2𝜁}, that exists and is continuous if 𝜎²_𝜁=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 is𝜁(𝑡,𝑥,𝑦)=_Λ𝑒^{𝑖(𝜆1𝑥+𝜆2𝑦+𝜔𝑡)}𝑑𝑀(𝜆₁,𝜆₂,𝜔),whereΛistheAirymanifold{𝜆²₁+𝜆²₂=𝜔⁴g²}, (3.1)𝑔 is the gravitational constant and 𝑀 is a random Gaussian orthogonal measure defined on Λ. Defining 𝑘=(𝜆₁,𝜆₂) and the following change of variable |𝑘|=(𝜔²/g) and 𝜆₁=(𝜔²/g)cosΘ and 𝜆₂=(𝜔²/g)sinΘ (see [17]), we obtain𝜁(𝑡,𝑥,𝑦)=^∞_−∞^𝜋_−𝜋𝑒^{𝑖(|𝑘|cosΘ𝑥+|𝑘|sinΘ𝑦+𝜔𝑡)}=𝑑𝑐(𝜔,Θ)_{∞−∞𝜋−𝜋}exp[𝑖(𝑘⋅𝑥+𝜔𝑡)]𝑑𝑐(𝜔,Θ), (3.2) where 𝑑𝑐(𝜔,Θ) is a random measure. The covariance function, 𝐾(𝜏,𝑋,𝑌), is defined as𝐾(𝜏,𝑋,𝑌)=𝐸[𝜁(𝑡,𝑥,𝑦)𝜁(𝑡+𝜏,𝑥+𝑋,𝑦+𝑌)]. (3.3) Then by using that𝔼[𝑑𝑐(𝜔,Θ)𝑑𝑐(𝜔^′,Θ^′)]=𝑆(𝜔,Θ)𝛿(𝜔−𝜔^′)𝛿(Θ−Θ^′)𝑑𝜔𝑑𝜔^′𝑑Θ𝑑Θ^′, (3.4) where 𝑆(𝜔,Θ) is the two-dimensional spectrum of the wave surface and 𝛿 represents Dirac's delta function. Then,𝐾(𝜏,𝑋,𝑌)=^∞_−∞^𝜋_−𝜋exp[𝑖(𝑘⋅𝑋+𝜔𝜏)]𝑆(𝜔,Θ)𝑑𝜔𝑑Θ. (3.5) This procedure is justified formally in [17, 18]. If in (3.5) we let 𝑋=0 and 𝑌=0, then we obtain𝐾(𝜏)∶=𝐾(𝜏,0,0)=^∞_−∞^𝜋_−𝜋𝑆(𝜔,Θ)𝑒^𝑖𝜔𝜏𝑑𝜔𝑑Θ, (3.6) or equivalently 𝐾(𝜏)=∫^∞_−∞𝑆(𝜔)𝑒^𝑖𝜔𝜏𝑑𝜔, where 𝑆(𝜔)=∫^𝜋_−𝜋𝑆(𝜔,Θ)𝑑Θ. Function 𝑆(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 order 𝑖𝑗𝑘 are defined as 𝑚_𝑖𝑗𝑘=^∞₀^𝜋_−𝜋𝑢^𝑖𝑣^𝑗𝜔^𝑘𝑆(𝜔,Θ)𝑑Θ𝑑𝜔, (3.7) 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 𝑚_00𝑘=^∞₀^𝜋_−𝜋𝜔^𝑘𝑆(𝜔,Θ)𝑑Θ𝑑𝜔, (3.8) and this can be rewritten as 𝑚_00𝑘=∫^∞₀𝜔^𝑘𝑆(𝜔)𝑑𝜔. The previous relation corresponds to the one-dimensional moment of order 𝑘, 𝑚_𝑘=∫^∞₀𝜔^𝑘𝑆(𝜔)𝑑𝜔. Also 𝑚_𝑖𝑗1 and 𝑚_𝑖𝑗2 will be denoted by 𝑚^′_𝑖𝑗=^∞₀^𝜋_−𝜋𝜔𝑢^𝑖𝑣^𝑗𝑆(𝜔,Θ)𝑑Θ𝑑𝜔,𝑚^′′_𝑖𝑗=^∞₀^𝜋_−𝜋𝜔²𝑢^𝑖𝑣^𝑗𝑆(𝜔,Θ)𝑑Θ𝑑𝜔, (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 let 𝜁(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 𝜕𝜁𝜕𝑥(𝑡,𝑥,0)=𝜁_𝑥(𝑡,𝑥)=𝑘𝑥, (4.1) where 𝑘=(1/2)[(1/ℎ₁)+(1/ℎ₂)]. This condition is approximately true, whenever 𝑘𝜁 and 𝜁_𝑥 are both small quantities, see [4, page 845].

A consequence of the implicit function theorem is (𝜁_𝑥𝑥−𝑘)𝑑𝑥+𝜁_𝑥𝑡𝑑𝑡=0, (4.2) that is, 𝑐=𝑑𝑥𝜁𝑑𝑡=−^𝑥𝑡(𝜁_𝑥𝑥.−𝑘) (4.3) 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 𝑁_𝑠𝑝(𝑠,0,𝛼₁,𝛼₂), where 𝑁_𝑠𝑝(𝑠,𝑢,𝛼₁,𝛼₂)∶=#{𝑥≤𝑀∶𝜁_𝑥(𝑠,𝑥)=𝑘𝑥+𝑢;𝛼₁≤𝜁_𝑥𝑡(𝜁_𝑥𝑥−𝑘)≤𝛼₂}for0≤𝑠≤𝑡. (4.4) Now, define the latter number per unit time as 𝑁_𝑠𝑝(𝑢,𝛼₁,𝛼₂1,𝑡)∶=𝑡^𝑡₀𝑁_𝑠𝑝(𝑠,𝑢,𝛼₁,𝛼₂)𝑑𝑠. (4.5) Notice that the process 𝒵(𝑡)=𝑁_𝑠𝑝(𝑡,𝑢,𝛼₁,𝛼₂) is stationary, has finite mean, and it is Riemann integrable, as a function of 𝑡. Define 𝒜_𝑡=𝜎{𝜁(𝜏,𝑥,0)∶𝜏>𝑡,𝑥∈[0,𝑀]} and the 𝜎-algebra of invariant events 𝒜=∩𝒜_𝑡. 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 ∫^𝑡₀𝒵(𝑠)𝑑𝑠𝑡→𝔼_ℬ[𝒵(0)], (4.6) where ℬ is the 𝜎-algebra of 𝑡-invariants associated to 𝒵. Since for each 𝑡, ℬ_𝑡=𝜎{𝒵(𝜏)∶𝜏>𝑡}⊂𝒜_𝑡, it follows that ℬ⊂𝒜, so that 𝔼_ℬ[𝒵(0)]=𝔼[𝒵(0)]=𝔼[𝑁_𝑠𝑝(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 𝐹(𝛼₂)−𝐹(𝛼₁)=lim_𝑡→∞𝑁_𝑠𝑝(0,𝛼₁,𝛼₂,𝑡)N_𝑠𝑝=𝑁(0,−∞,∞,𝑡)𝔼[_𝑠𝑝(0,0,𝛼₁,𝛼₂)]𝔼[𝑁_𝑠𝑝(0,0,−∞,∞)]. (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 𝜁(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 𝐹(𝛼₂)−𝐹(𝛼₁)=𝔼[𝟏_{[𝛼1,𝛼2]}(𝜁_𝑥𝑡(0,0)/(𝜁_𝑥𝑥(0,0)−𝑘))|𝜁_𝑥𝑥(0,0)−𝑘|]√𝑚₄₀₀,ℜ(𝑘) (4.8) where ℜ(𝑘)∶=(2/𝜋)(𝑒^{−𝑘2/2𝑚400}+(𝑘/√𝑚₄₀₀)∫^{√𝑘/𝑚400}₀𝑒^−𝑣2/2𝑑𝑣).

Proof. For a continuous and bounded function ℎ, we have ^∞_−∞ℎ(𝑢)𝑁_𝑠𝑝(𝑢,𝛼₁,𝛼₂1,𝑡)𝑑𝑢=𝑡^𝑡₀^∞_−∞ℎ(𝑢)𝑁_𝑠𝑝(𝑠,𝑢,𝛼₁,𝛼₂)𝑑𝑢𝑑𝑠, (4.9) and by the area formula (2.3), we have ^∞_−∞ℎ(𝑢)𝑁_𝑠𝑝(𝑢,𝛼₁,𝛼₂=1,𝑡)𝑑𝑢𝑡^𝑡₀^𝑀₀ℎ(𝜁_𝑥(𝑠,𝑥)−𝑘𝑥)𝟏_{[𝛼1,𝛼2]}(𝜁_𝑥𝑡(𝑠,𝑥)(𝜁_𝑥𝑥(𝑠,𝑥)−𝑘))|(𝜁_𝑥𝑥(𝑠,𝑥)−𝑘)|𝑑𝑥𝑑𝑠. (4.10) Taking expectations, by stationarity and duality, for almost all 𝑢, it follows that 𝔼[𝑁_𝑠𝑝(𝑢,𝛼₁,𝛼₂=,𝑡)]_𝑀0𝑝_{𝜁𝑥(0,0)}(𝑢+𝑘𝑥)𝔼[𝟏_{[𝛼1,𝛼2]}(𝜁_𝑥𝑡(0,0)(𝜁_𝑥𝑥(0,0)−𝑘))|𝜁_𝑥𝑥(0,0)−𝑘|∣𝜁_𝑥(0,0)=𝑢+𝑘𝑥]𝑑𝑥. (4.11) This may be written, in the Gaussian case, by independence, as 𝔼[𝑁_𝑠𝑝(𝑢,𝛼₁,𝛼₂,𝑡)]=^𝑀₀exp{−(𝑢+𝑘𝑥)²/2𝑚₂₀₀}2𝜋𝑚₂₀₀𝑑𝑥×𝔼[𝟏_{[𝛼1,𝛼2]}(𝜁_𝑥𝑡(0,0)(𝜁_𝑥𝑥(0,0)−𝑘))|𝜁_𝑥𝑥(0,0)−𝑘|]. (4.12) 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 is 𝑢=0, thus we obtain that the expectation of the number of specular points having speed between 𝛼₁ and 𝛼₂ is ^𝑀₀𝑝_{𝜁𝑥(0,0)}(𝑘𝑥)𝔼[𝟏_{[𝛼1,𝛼2]}(𝜁_𝑥𝑡(0,0)(𝜁_𝑥𝑥(0,0)−𝑘))|𝜁_𝑥𝑥(0,0)−𝑘|∣𝜁_𝑥(0,0)=𝑘𝑥]𝑑𝑥, (4.13) which in the Gaussian case may be written as 𝔼[𝑁_𝑠𝑝(0,𝛼₁,𝛼₂,𝑡)]=𝔼[𝑁_𝑠𝑝(0,0,𝛼₁,𝛼₂)]=^𝑀₀exp{−𝑘²𝑥²/2𝑚₂₀₀}2𝜋𝑚₂₀₀𝑑𝑥×𝔼[𝟏_{[𝛼1,𝛼2]}(𝜁_𝑥𝑡(0,0)(𝜁_𝑥𝑥(0,0)−𝑘))|𝜁_𝑥𝑥(0,0)−𝑘|]. (4.14) Moreover, the expectation of the number of specular points per unit of time 𝑁_𝑠𝑝(0,−∞,∞,𝑡) is easily computed yielding formula (2.14) of [4, page 846] 𝔼[𝑁_𝑠𝑝(0,−∞,∞,𝑡)]=^𝑀₀exp{−𝑘²𝑥²/2𝑚₂₀₀}2𝜋𝑚₂₀₀𝑑𝑥𝔼|𝜁_𝑥𝑥(0,0)−𝑘|=ℜ(𝑘)𝑚₄₀₀2𝜋𝑚₂₀₀^𝑀₀𝑘exp{−²𝑥²2𝑚₂₀₀}𝑑𝑥, (4.15) where ℜ(𝑘)∶=2/𝜋(𝑒^{−𝑘2/2𝑚400}+(𝑘/√𝑚₄₀₀)∫^{√𝑘/𝑚400}₀𝑒^−𝑣2/2𝑑𝑣).
As the process 𝜁 satisfies that 𝐾(𝑡,𝑥,0)→0, we obtain (4.8) by simple division.
Let us now define 𝑝_𝑥,𝑡(𝜉₁,𝜉₂) the Gaussian density of the random vector (𝜁_𝑥𝑡(0,0),𝜁_𝑥𝑥(0,0)). We may write (4.8) as 𝐹(𝛼₂)−𝐹(𝛼₁1)=√𝑚₄₀₀ℜ(𝑘)^𝛼2_𝛼1^∞_−∞(𝜉₂−𝑘)²𝑝_𝑥,𝑡(𝑐(𝜉₂−𝑘),𝜉₂)𝑑𝑐𝑑𝜉₂. (4.16)

Remark 4.2. Differentiating the previous expression one obtains the density of the velocity of the specular points: 𝑝_𝑘1(𝑐)=√𝑚₄₀₀ℜ(𝑘)^∞_−∞(𝜉₂−𝑘)²𝑝_𝑥,𝑡(𝑐(𝜉₂−𝑘),𝜉₂)𝑑𝜉₂. (4.17) If 𝑘=0 we recover formula (2-5-19) of [2] (modified in order to consider the case of specular points) 𝑝₀Δ(𝑐)=²2𝑚²₀₀₄((𝑐−𝑐)²+Δ₂𝑚⁻²₄₀₀)^−3/2, (4.18) where Δ₂𝑚=[det²⁰⁰𝑚₁₁₀𝑚₁₁₀𝑚₀₂₀]⁻¹,𝑐𝑚=−³⁰¹𝑚₄₀₀. (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 condition (𝜁_𝑥(𝑡,𝑥,𝑦),𝜁_𝑦(𝑡,𝑥,𝑦))=(𝑘𝑥,𝑘𝑦) at point (𝑥,𝑦) and for a fixed time 𝑡.

Defining the vectorial process 𝑍(𝑡,𝑥,𝑦)=(𝜁_𝑥(𝑡,𝑥,𝑦)−𝑘𝑥,𝜁_𝑦(𝑡,𝑥,𝑦)−𝑘𝑦), (5.1) 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 be𝑁^𝑍_Ω(0,0)=#{(𝑥,𝑦)∈Ω∶𝑍(𝑡,𝑥,𝑦)=(0,0)}. (5.2) 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)𝔼[𝑁^𝑍_Ω(𝑤₁,𝑤₂)]=_Ω𝑝_{𝑍(0,𝑥,𝑦)}(𝑤₁,𝑤₂)𝔼[|Δ(𝑡,𝑥,𝑦)|/𝑍(0,𝑥,𝑦)=(𝑤₁,𝑤₂)]𝑑𝑥𝑑𝑦, (5.3) where Δ(𝑡,𝑥,𝑦)=(𝜁_𝑥𝑥(𝑡,𝑥,𝑦)−𝑘)(𝜁_𝑦𝑦(𝑡,𝑥,𝑦)−𝑘)−𝜁²_𝑥𝑦(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), 𝔼[𝑁^𝑍_Ω(0,0)]=_Ω𝑝_{𝑍(0,𝑥,𝑦)}(0,0)𝔼[|Δ(0,0,0)|/𝑍(𝑥,𝑦)=(0,0)]𝑑𝑥𝑑𝑦. (5.4) The independence property allows writing 𝔼[𝑁^𝑍_Ω(0,0)]=𝔼[|Δ(0,0,0)|]_Ω𝑝_{𝑍(0,𝑥,𝑦)}(0,0)𝑑𝑥𝑑𝑦, (5.5) obtaining finally the following result.

Proposition 5.1. Let the stationary mean zero Gaussian random field 𝜁(𝑡,𝑥,𝑦)∈𝐶³(ℝ³) a.s. and Var𝑍>0. Then, 𝔼[𝑁^𝑍_Ω(0,0)]=𝔼[|(𝜁_𝑥𝑥(0,0,0)−𝑘)(𝜁_𝑦𝑦(0,0,0)−𝑘)−(𝜁_𝑥𝑦(0,0,0))²|]_Ω𝑝_{𝑍(0,𝑥,𝑦)}(0,0)𝑑𝑥𝑑𝑦. (5.6)

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 𝜁_𝑦𝑦(0,0,0)=𝛼𝜁_𝑥𝑥(0,0,0)+𝛽𝜁_𝑥𝑦(0,0,0)+𝜎₁𝜀₁, (5.7) where 𝑚𝛼=²²²⁰−𝑚₃₁₀𝑚₁₃₀𝑚₄₀₀𝑚₂₂₀−𝑚²₃₁₀𝑚𝛽=⁴⁰⁰𝑚₁₃₀−𝑚₃₁₀𝑚₂₂₀𝑚₄₀₀𝑚₂₂₀−𝑚²₃₁₀, (5.8)𝜀₁=𝑁(0,1)⟂(𝜁_𝑥𝑥(0,0,0),𝜁_𝑥𝑦(0,0,0)) and 𝜎²=−𝑚₄₀₀+𝛼²𝑚₄₀₀+𝛽²𝑚₂₂₀+2𝛼𝛽𝑚₂₂₀. Therefore it yields 𝔼[|Δ(0,0,0)|]=𝔼[|(𝜁_𝑥𝑥(0,0,0)−𝑘)((𝜁_𝑥𝑥(0,0,0)+𝛽𝜁_𝑥𝑦(0,0,0)+𝜎𝜀₁)−𝑘)−(𝜁_𝑥𝑦(0,0,0))²|]. (5.9) This last expression can be evaluated readily by using Monte Carlo. Indeed let 𝜀^𝑖⃗=(𝜀^𝑖₁,𝜀^𝑖₂,𝜀^𝑖₃)^𝑡, 𝑖=1,…,𝑁, be a sample of standard Gaussian vectors in ℝ³. We have lim_𝑁→∞1𝑁^𝑁_𝑖=1|(√𝑚₄₀₀𝜀^𝑖₃√−𝑘)×(((𝛼𝑚₄₀₀𝜀^𝑖₃+𝛽((𝑚₂₂₀−𝑚²₃₁₀𝑚₄₀₀)^1/2𝜀^𝑖₂+𝑚₃₁₀√𝑚₄₀₀𝜀^𝑖₃)+𝜎𝜀^𝑖₁))−𝑘)−((𝑚₂₂₀−𝑚²₃₁₀𝑚₄₀₀)^1/2𝜀^𝑖₂+𝑚₃₁₀√𝑚₄₀₀𝜀^𝑖₃)²|=𝔼[|Δ(0,0,0)|]. (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 field 𝑍(𝑡,𝑥,𝑦)=(𝜁_𝑥(𝑡,𝑥,𝑦)−𝑘𝑥,𝜁_𝑦(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 𝑁^𝑍_Ω(0,0). We have already computed the expectation of the number of specular points 𝔼[𝑁^𝑍_Ω(0,0)]=𝔼[|Δ(0,0,0)|]_Ω𝑝_{𝑍(0,𝑥,𝑦)}(0,0)𝑑𝑥𝑑𝑦. (6.1) The condition satisfied for the specular points (i.e., (𝜁_𝑥(𝑡,𝑥,𝑦),𝜁_𝑦(𝑡,𝑥,𝑦))=(𝑘𝑥,𝑘𝑦)) and the implicit function theorem entails 𝜁_𝑥𝑡=−(𝜁_𝑥𝑥−𝑘)𝑑𝑥𝑑𝑡−𝜁_𝑥𝑦𝑑𝑦,𝜁𝑑𝑡_𝑦𝑡=−𝜁_𝑥𝑦𝑑𝑥𝑑𝑡−(𝜁_𝑦𝑦−𝑘)𝑑𝑦.𝑑𝑡 (6.2) Let us define as Longuet-Higgins 𝑐_𝑥=𝑑𝑥/𝑑𝑡and 𝑐_𝑦=𝑑𝑦/𝑑𝑡. The objective is to find the Palm distribution associated to the velocity field (𝑐_𝑥,𝑐_𝑦)=(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 for 𝐮=(𝑢₁,𝑢₂): 𝑁_𝑠𝑝(𝑠,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≤𝑡 and Ω a compact set in ℝ².

Consider 𝑁_𝑠𝑝(𝐮,𝑣₁,𝑣₂,𝑣₃,𝑣₄1,𝑡)∶=𝑡^𝑡₀𝑁_𝑠𝑝(𝑠,𝐮,𝑣₁,𝑣₂,𝑣₃,𝑣₄)𝑑𝑠. (6.4)

Let 𝑔 be a continuous bounded function. On the one hand, using (2.3), we have_ℝ2𝑔(𝐮)𝑁_𝑠𝑝(𝐮,𝑣₁,𝑣₂,𝑣₃,𝑣₄1,𝑡)𝑑𝐮=𝑡^𝑡_0ℝ²𝑔(𝐮)𝑁_𝑠𝑝(𝑠,𝐮,𝑣₁,𝑣₂,𝑣₃,𝑣₄=1)𝑑𝐮𝑑𝑠𝑡^𝑡_0Ω𝑔(𝑍(𝑠,𝑥,𝑦)−𝐮)𝟏_{[𝑣1,𝑣2]}(𝑐_𝑥)𝟏_{[𝑣3,𝑣4]}(𝑐_𝑦)|Δ(𝑡,𝑥,𝑦)|𝑑𝑥𝑑𝑦𝑑𝑠. (6.5) Let us denote by 𝑝(𝜉₄,𝜉₅,𝜉₆,𝜉₇,𝜉₈) the density of the Gaussian random vector (𝜁_𝑥𝑥(𝟎),𝜁_𝑥𝑦(𝟎),𝜁_𝑦𝑦(𝟎),𝜁_𝑥𝑡(𝟎),𝜁_𝑦𝑡(𝟎)). (6.6) It follows that 𝑝(𝜉₄,𝜉₅,𝜉₆,𝑐_𝑥,𝑐_𝑦)∶=𝑝(𝜉₄,𝜉₅,𝜉₆,−(𝜉₄−𝑘)𝑐_𝑥−𝜉₅𝑐_𝑦,−𝜉₅𝑐_𝑥−(𝜉₆−𝑘)𝑐_𝑦) (6.7) is the density function of the random vector (𝜁_𝑥x(𝟎),𝜁_𝑥𝑦(𝟎)𝜁_𝑦𝑦(𝟎),𝑐_𝑥(𝟎),𝑐_𝑦(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 𝔼[𝑁_𝑠𝑝(𝟎,𝑣₁,𝑣₂,𝑣₃,𝑣₄=,𝑡)]_Ω𝑝_𝑍(0,0,0)×(−𝑘𝑥,−𝑘𝑦)𝑑𝑥𝑑𝑦_{𝑣2𝑣1𝑣4𝑣3𝐑3}𝑝(𝜉₄,𝜉₅,𝜉₆,𝑐_𝑥,𝑐_𝑦)|(𝜉₄−𝑘)(𝜉₆−𝑘)−𝜉²₅|𝑑𝜉₄𝑑𝜉₅𝑑𝜉₆𝑑𝑐_𝑥𝑑𝑐_𝑦. (6.8) 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 𝑐_𝑥∈[𝜐₁,𝜐₂] and 𝑐_𝑦∈[𝜐₃,𝜐₄] is lim_𝑡→∞𝑁_𝑠𝑝(𝟎,𝑣₁,𝑣₂,𝑣₃,𝑣₄,𝑡)𝑁_𝑠𝑝=∫(𝟎,−∞,∞,−∞,∞,𝑡)_𝑣2𝑣1∫^𝑣4_𝑣3∫_𝐑3𝑝(𝜉₄,𝜉₅,𝜉₆,𝑐_𝑥,𝑐_𝑦)|(𝜉₄−𝑘)(𝜉₆−𝑘)−𝜉²₅|𝑑𝜉₄𝑑𝜉₅𝑑𝜉₆𝑑𝑐_𝑥𝑑𝑐_𝑦.𝔼|Δ(0,0,0)| (6.9) We can summarize the above computations in the following result.

Proposition 6.1. Let 𝜁(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 𝑐_𝑥∈[𝜐₁,𝜐₂] and 𝑐_𝑦∈[𝜐₃,𝜐₄] is 𝐹(𝜐₃,𝜐₄)−𝐹(𝜐₁,𝜐₂∫)=^𝑣2𝑣1∫^𝑣4_𝑣3∫_𝐑3𝑝(𝜉₄,𝜉₅,𝜉₆,𝑐_𝑥,𝑐_𝑦)|(𝜉₄−𝑘)(𝜉₆−𝑘)−𝜉²₅|𝑑𝜉₄𝑑𝜉₅𝑑𝜉₆𝑑𝑐_𝑥𝑑𝑐_𝑦.𝔼|Δ(0,0,0)| (6.10)

Remark 6.2. Taking derivatives one gets the density of the speed of the two-dimensional specular points 𝑝_{(𝑥,𝑦,𝑘)}(𝑐_𝑥,𝑐_𝑦)=∫_𝐑3𝑝(𝜉₄,𝜉₅,𝜉₆,𝑐_𝑥,𝑐_𝑦)|(𝜉₄−𝑘)(𝜉₆−𝑘)−𝜉²₅|𝑑𝜉₄𝑑𝜉₅𝑑𝜉₆.𝔼|Δ(0,0,0)| (6.11) 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 Direction 𝜃

Let 𝜁(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 𝑆(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 direction 𝜃. Let us define, as in [8], the crest of the waves 𝜁 in direction 𝜃 at time 𝑠 as the level set ℭ_𝑄(𝑠,𝜃)={(𝑥,𝑦)∈𝑄∶𝜁^′_𝜃(𝑠,𝑥,𝑦)=0;𝜁^′′_𝜃𝜃(𝑠,𝑥,𝑦)<0}, (7.1) where 𝜁^′_𝜃 and 𝜁^′′_𝜃𝜃 denote the first and second derivatives in the direction 𝜃, respectively. This set is the zero level set 𝒞^𝑍𝜃_𝑄(𝑠,0) of the field 𝑍_𝜃(𝑠,𝑥,𝑦)=𝜁_𝑥(𝑠,𝑥,𝑦)cos𝜃+𝜁_𝑦(𝑠,𝑥,𝑦)sin𝜃, (7.2) under the additional condition that 𝑍^′′_𝜃𝜃(𝑠,𝑥,𝑦)<0. If 𝜃 is the direction orthogonal to 𝜃, we can express the gradient of 𝑍_𝜃^′′(𝑠,𝑥,𝑦) with respect to 𝜃 and its orthogonal, denoted as ∇_𝜃, as ∇_𝜃𝑍_𝜃(𝑠,𝑥,𝑦)=(𝜕_𝜃𝑍_𝜃𝜃(𝑠,𝑥,𝑦),𝜕𝑍_𝜃(𝑠,𝑥,𝑦))=‖∇_𝜃𝑍_𝜃(𝑠,𝑥,𝑦)‖(cosΦ(𝑠,𝑥,𝑦),sinΦ(𝑠,𝑥,𝑦)), (7.3) where 𝜃Φ(𝑠,𝑥,𝑦)=arctan𝜕𝑍_𝜃(𝑠,𝑥,𝑦)/𝜕_𝜃𝑍_𝜃(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 1𝑡^𝑡_{0ℭ𝑄(𝑠,𝜃)}𝐻(Φ(𝑠,𝐱))𝑑𝜎₁1(𝐱)𝑑𝑠=𝑡^𝑡_0𝒞^𝑍𝜃_𝑄(𝑠,0)𝐻(Φ(𝑠,𝐱))1_[−1,0)(cosΦ(𝑠,𝐱))𝑑𝜎₁(𝐱)𝑑𝑠→𝔼[_{𝒞𝑍𝜃𝑄(0,0)}𝐻(Φ(0,𝐱))1_[−1,0)(cosΦ(0,𝐱))𝑑𝜎₁(𝐱)]. (7.4) 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)𝐾(𝑡,𝑥,𝑦)→0whenever𝑡→∞. (7.5) This hypothesis allows us to place ourselves in the framework of the Ergodic Theorem. Defining the Palm distribution 𝜈_𝜃 of the normal angle at the crest in the direction 𝜃 as the following integral: ^3𝜋/2_𝜋/2𝐻(𝜑)𝑑𝜈_𝜃(𝜑)∶=lim_𝑡→∞∫^𝑡₀∫_{ℭ𝑄(𝑠,𝜃)}𝐻(Φ(𝑠,𝐱))𝑑𝜎₁(𝐱)∫^𝑡₀𝜎₁(ℭ_𝑄.(𝑠,𝜃))𝑑𝑠 (7.6) 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 (𝜕_𝜃𝑍_𝜃𝜃(0,0,0),𝜕𝑍_𝜃(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 field 𝜁 is three times continuously differentiable and hypothesis (7.5) holds, the Palm distribution 𝜈_𝜃 of the normal angle at the crest in the direction 𝜃 satisfies ^3𝜋/2_𝜋/2𝐻(𝜑)𝑑𝜈_𝜃𝛾(𝜑)=²(𝜃)2𝐄(1−𝛾²(𝜃))^{3𝜋/2𝜋/2}𝐻(𝜑)(1−(1−𝛾²(𝜃))sin²(𝜑−𝜅(𝜃)))^3/2𝑑𝜑. (7.7)

Proof. By using the Ergodic Theorem, we get ^3𝜋/2_𝜋/2𝐻(𝜑)𝑑𝜈_𝜃(𝜑)=𝔼[∫_𝒞^𝑍_{𝜃𝑄(0,0)}𝐻(Φ(0,𝐱))1_[−1,0)(cosΦ(0,𝐱))𝑑𝜎₁(𝐱)]𝔼𝜎₁(ℭ_𝑄=(0,𝜃))𝔼[∫_𝒞^𝑍_{𝜃𝑄(0,0)}𝐻(Φ(0,𝐱))1_[−1,0)(cosΦ(0,𝐱))𝑑𝜎₁(𝐱)](1/2)𝔼𝜎₁(𝒞^𝑍𝜃_𝑄.(0,0)) (7.8) To get (7.7) it is enough to compute the two expectations in the last equality. Let us define 𝑎(𝜃)=𝔼(𝑍²_𝜃(0,0,0)). First we consider the numerator. We have changing to polar coordinates 𝔼[_{𝒞Z𝜃Q(0,0)}𝐻(Φ(0,𝐱))1_[−1,0)(cosΦ(0,𝐱))𝑑𝜎₁=𝜆(𝐱)]₋(𝜃)(2𝜋)^2/3𝑎(𝜃)(𝜆₊(𝜃)𝜆₋(𝜃))^1/2×^∞₀^3𝜋/2_𝜋/2𝜌²𝐻(𝜑)𝑒^{−(𝜌2/2𝜆+(𝜃)𝜆−(𝜃))(𝜆+cos2(𝜑−𝜅(𝜃))+𝜆−sin2(𝜑−𝜅(𝜃))}=𝜆𝑑𝜌𝑑𝜑₋(𝜃)𝑎(𝜃)𝜆₊(𝜃)4𝜋^{3𝜋/2𝜋/2}1𝐻(𝜑)(1−(1−𝛾²(𝜃))sin²(𝜑−𝜅(𝜃))^3/2)𝑑𝜑. (7.9) For the denominator, we have 𝐻=1 obtaining, (1/2)𝔼𝜎₁(𝒞^𝑍𝜃_𝑄(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 𝑆(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.

Acknowledgments

The second author's research was partially funded by Total Oil, through the project LOCTI “Transporte de contaminantes en el Lago de Valencia.“ The authors also thank two anonymous referees whose comments significantly improved the first version of the paper.