Research Article | Open Access
Level Sets of Random Fields and Applications: Specular Points and Wave Crests
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 .
In 1944, Rice  proposed the model to describe the noise in an electrical current. In this relation, denotes the different frequencies, are Gaussian random variables, identically distributed and independent, and are random variables uniformly distributed in .
Later, in 1957, Longuet-Higgins  defined the following multidimensional generalization of Rice's model: 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 . 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 space (see  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 Azas 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 .
All the expectations mentioned above can be rigorously computed by using the multidimensional Rice formula for Gaussian random fields , recently proved in  and by using another Rice formula for random fields () established by Cabaña in 1985 . 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 .
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 and will denote, respectively, the Lebesgue measure in the space and the Hausdorff measure defined in the subspaces of dimension , trivially by definition .
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 be a continuously differentiable function. If we define then if , one has where is a continuous and bounded function. This formula was obtained by Banach in 1925 .
If is the Jacobian of in and is a continuous bounded function, then the version of Banach's formula for is This expression is usually called the area formula (cf. ).
Now, let be a Gaussian random field with continuously differentiable trajectories. The random number of times that takes the value in the set is defined as Let denote the marginal density of . By using formula (2.3), Fubini's Theorem and duality we get for a.s. The fact that this formula is true for all is not trivial. The book by Azas 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  and the references therein.
We will study below the more difficult case when function has a domain whose dimension is greater than the dimension of the rank, namely, () is a continuously differentiable function, with Jacobian defining the level set where is a compact set of . The following two results are well known as the Coarea formula (cf. Federer [12, pages 247–249] and Cabaňa ). The reader may consult the excellent set of lectures by Weizsäker and Geibler of the University of Kaiserslautern  for an up to data exposition.
Theorem 2.1. Let be a continuous and bounded function. Restricted to the set the following formula holds:
Corollary 2.2. Let be a continuous and bounded function under restriction (2.7), then
Remark 2.3. Formula (2.8) and (2.9) hold true without restriction (2.7). In fact it can be proved that for holds for almost all . This also implies that Let us define for a compact the functions The following lemma holds true.
Lemma 2.4. Under hypothesis of Theorem 2.1, functions and are continuous.
Let () be a stationary random field belonging to and suppose that for all , the density of exists (in the Gaussian case this holds whenever ). We have (i)For almost all , (ii)For almost all
Let us remark that formula (2.13) and (2.14) hold for almost all . However in applications, as we will see in the next sections, they are needed for a fixed . We will prove in what follows that the formulas hold for all .
Define the set 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 The following theorem was proved in 1985 by Cabaňa . 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 and of  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): for all , exists and is continuous. For a continuous function the following expression: is a continuous function in the variable.(ii): the expression is a continuous function in the variable. Let us note that if then is sufficient for to hold.
Theorem 2.5. Consider a random field belonging to . (i)Then under for all , (ii)If is an almost sure continuous function under , for all one has
Proof. We begin proving formula (2.17). Let the differentiable function be such that If and if . Let us define the function
For belonging to the complement of this set and defining the eigenvalues of , it holds and moreover, defining and its orthogonal subspace, we have Observe that the hypothesis of continuity of and the compactness of imply a uniform bound for the inverse. Lemma 2.4 implies that the following function: is a.s continuous, also the sequence is nondecreasing in both indexes. Moreover, the inequality yields that is a continuous function. Using formula (2.9) applied to the field and the function , we have From this we have that for almost all , Thus, Condition implies that the function in the right-hand side is continuous, hence the inequality holds for all . Taking limits as and and using Beppo-Levi's Theorem, we have that To prove the other inequality, let be such that for all equality (2.24) holds. This is possible because the equality is satisfied for almost all . Thus by applying Fatou's Lemma, we obtain By using Fatou's Lemma again and that is a nondecreasing sequence, we obtain Finally Moreover when . Clearly applying Beppo-Levi's Theorem we get Obtaining formula (2.17), formula (2.18) follows by approximating uniformly by a nondecreasing sequence of simple functions.
The following two propositions of Azas 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 be a random field and a subset of , and let . One supposes that satisfies the following conditions: (1)the random field is -Hölder continuous with ; (2)for each the random vector has a density such that , for and in some neighborhood of ;(3)the Hausdorff dimension of is smaller than or equal to .Then, almost surely, there is no point such that .
Proposition 2.7. Let be a random field and an open set of and . Suppose that is a.s. -Hölder continuous with and moreover for all the random vector has a bounded continuous density , for in a neighborhood of and varying in a compact set of . Then
Theorem 2.8. Under the hypotheses of Theorem 2.5 and that is a.s -Hölder continuous with , one has the following.(1)Under , for all , (2)Under and if is an almost sure continuous function, for all one has
In what follows we give two examples under which the hypotheses and hold. (1)Suppose that is a Gaussian field verifying the hypothesis of Theorem 2.8 and that for each . By considering the regression model, with a Gaussian , where the following equality in law is satisfied: This result entails that the expression is a continuous function of variable . Moreover, the hypothesis yields the continuity in of .(1)Finally let us consider the case of the real envelope of the stationary Gaussian field . As in , we define the envelope of the Gaussian field as follows. Let us consider the random spectral measure restricted to the Airy manifold defined below in (3.1). In this manner if we restrict the stochastic integral to the set By using polar coordinates we can write We define the Hilbert transform of as the Gaussian field The real envelope is defined as It holds This expression is continuos whenever . Moreover the density of , in the point , is the Rayleigh density , that exists and is continuous if .
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  in which Lindgren gives a definitive treatment for this type of spectral stochastic integral models.
Another way of looking at Longuet-Higgins' model is is the gravitational constant and is a random Gaussian orthogonal measure defined on . Defining and the following change of variable and and (see ), we obtain where is a random measure. The covariance function, is defined as Then by using that where is the two-dimensional spectrum of the wave surface and represents Dirac's delta function. Then, This procedure is justified formally in [17, 18]. If in (3.5) we let and , then we obtain or equivalently where Function represents the frequency spectrum of the sea surface. This spectrum contains the distribution of the wave energy in the frequency domain. The autocorrelation function for the elevation surface , in a fixed location, is a real even function.
Definition 3.1. The spectral moments of order are defined as where , and is the gravitational constant. If in (3.7) , then and this can be rewritten as The previous relation corresponds to the one-dimensional moment of order , Also and will be denoted by 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 be the random field (3.2) representing the sea height and suppose that it belongs a.s. to . We observe the random field in a fixed direction, for instance. The place where reflection occurs, when the surface is illuminated by a light source, placed in and observed in , for each fixed is the level curve where . This condition is approximately true, whenever and are both small quantities, see [4, page 845].
A consequence of the implicit function theorem is that is, This expression defines the velocity of the specular points. Thus let us define the number of specular points in having a speed in as , where Now, define the latter number per unit time as Notice that the process is stationary, has finite mean, and it is Riemann integrable, as a function of . Define and the -algebra of invariant events . Under the hypothesis that for each , whenever the -algebra is trivial. By the Birkoff-Khintchine Ergodic Theorem, we have where is the -algebra of -invariants associated to . Since for each , , it follows that , so that (for references, see [19, page 151]).
Our interest here is to compute the Palm distribution of the number of specular points having speed between defined as The last equality, as we have seem, is a consequence of the Ergodic Theorem. We will show the following result.
Proposition 4.1. Let be a Gaussian random field (3.2) and assume that it belongs to and for each pair , whenever . The Palm distribution defined above satisfies where
Proof. For a continuous and bounded function , we have
and by the area formula (2.3), we have
Taking expectations, by stationarity and duality, for almost all , it follows that
This may be written, in the Gaussian case, by independence, as
The formula is true for all as it follows analogously to the result shown by Azas and Wschebor [7, page 163].
For the specular points, the interesting level is , thus we obtain that the expectation of the number of specular points having speed between and is which in the Gaussian case may be written as Moreover, the expectation of the number of specular points per unit of time is easily computed yielding formula (2.14) of [4, page 846] where
As the process satisfies that , we obtain (4.8) by simple division.
Let us now define the Gaussian density of the random vector . We may write (4.8) as
Remark 4.2. Differentiating the previous expression one obtains the density of the velocity of the specular points: If we recover formula (2-5-19) of  (modified in order to consider the case of specular points) where
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 at point and for a fixed time .
Defining the vectorial process we say that we have a specular point if and the number of such points in a fixed time and in a region will be We denote as in formula (2.5) and . Then applying this formula to the process , we get for almost all where .
This formula turns out to be valid for all under the hypotheses of Theorem in  (in our case and will be enough) and in particular for the specular points, that is when , The independence property allows writing obtaining finally the following result.
Proposition 5.1. Let the stationary mean zero Gaussian random field a.s. and . Then,
Remark 5.2. The Li and Wei formula (cf. ) provides a way to compute the expectation of the absolute value of the determinant in the above formula, see Azaïs et al. , which one will not pursue. Instead one will apply a Monte Carlo method. Let us consider the regression model where and . Therefore it yields This last expression can be evaluated readily by using Monte Carlo. Indeed let , be a sample of standard Gaussian vectors in . We have
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 ; the number of specular points of the field , in a fixed time and in a region , was defined in (5.2) and denoted as . We have already computed the expectation of the number of specular points The condition satisfied for the specular points (i.e., ) and the implicit function theorem entails Let us define as Longuet-Higgins and . The objective is to find the Palm distribution associated to the velocity field . The following computations, in the case , are essentially contained in the very original and seminal work of Longuet-Higgins .
Now define for : for and a compact set in
Let be a continuous bounded function. On the one hand, using (2.3), we have Let us denote by the density of the Gaussian random vector It follows that is the density function of the random vector Taking expectations in (6.5), using duality and putting (under the hypothesis and the formula holds for all levels (cf. [7, page 163]) we obtain 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 and is