𝑞-Advanced Models for Tsunami and Rogue Waves

Abstract

A wavelet 𝐾_𝑞(𝑡), that satisfies the q-advanced differential equation 𝐾^′_𝑞(𝑡)=𝐾_𝑞(𝑞𝑡) for 𝑞>1, is used to model N-wave oscillations observed in tsunamis. Although q-advanced ODEs may seem nonphysical, we present an application that model tsunamis, in particular the Japanese tsunami of March 11, 2011, by utilizing a one-dimensional wave equation that is forced by 𝐹_𝑞(𝑡,𝑥)=𝐾_𝑞(𝑡)_𝑞Sin(𝑥). The profile 𝐹_𝑞 is similar to tsunami models in present use. The function _𝑞Sin(𝑡) is a wavelet that satisfies a q-advanced harmonic oscillator equation. It is also shown that another wavelet, _𝑞Cos(𝑡), matches a rogue-wave profile. This is explained in terms of a resonance wherein two small amplitude forcing waves eventually lead to a large amplitude rogue. Since wavelets are used in the detection of tsunamis and rogues, the signal-analysis performance of 𝐾_𝑞 and _𝑞Cos is examined on actual data.

1. Introduction

Tsunami or maremoto waves occur in response to earthquakes or landslides on the seafloor of large bodies of water, as discussed in [1–4]. The consequential runup to the shore is such that the tide goes out, then returns as a large surge, only to be followed by several diminishing cycles of similar events [5]. An understanding of this behavior involves a consideration of the effects from the seafloor near the shore where the wave velocity decreases [6, 7]. We study a short-lived forcing that predominantly generates a traveling-wave profile 𝐾_𝑞(𝑥−𝑐𝑡) where 𝐾_𝑞 solves a multiplicatively advanced differential equation (MADE) whose profile resembles a typical tsunami (see Figure 1). In contrast to the 𝑁-wave profile proposed in [8], the 𝐾_𝑞-MADE profiles are asymmetric wavelets that are flat on a half-line.

(a)

(b)

(c)

Figure 1 

(a) Tsunami 52406 from DART, March 2011; (b) replica of tsunami-model profile adapted from [7]; (c) MADE solution profile 𝑦=𝐾𝑞(𝑡) for 𝑞=1.5.

In an apparently unrelated phenomena, rogue, freak, or monster waves are caused by small ripples or currents in layers near the water's surface [9, 10]. Various methods have been used to model these rare events, as in [11, 12]. Here, we demonstrate why rogue waves may be a type of resonance wherein an arbitrarily low amplitude forcing, for a sufficiently long period of time, can produce any size of localized wave. Our models use square-integrable versions of the sine and cosine functions that we call _𝑞Cos(𝑡) and _𝑞Sin(𝑡) (see Figure 2).

(a)

(b)

Figure 2 

(a) 𝑦=𝑞Cos(𝑡) for 𝑞=1.5; and (b) 𝑦=𝑞Sin(𝑡) for 𝑞=1.5.

Tsunami and rogue waves are perturbations of the water-surface elevation function 𝜂(𝑡,𝑥,𝑦). The surface velocity components 𝐔≡(𝑢,𝑣,𝑤) are small and the time-averaged mean-flow velocity ⟨𝐔⟩ is assumed to remain 0 everywhere. Subject to assumptions, the functions 𝐾_𝑞, _𝑞Cos and _𝑞Sin will be used in the construction of the forcing terms in the Matsuno equations [13] as given by 𝜕𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦−𝑓₀𝑣=−𝛼(𝑢,𝑣)𝑢−𝑔₀𝜕𝜂𝜕𝑥+𝑃_𝑥,𝜕𝑣𝜕𝑡+𝑢𝜕𝑣𝜕𝑥+𝑣𝜕𝑣𝜕𝑦+𝑓₀𝑢=−𝛼(𝑢,𝑣)𝑣−𝑔₀𝜕𝜂𝜕𝑦+𝑃_𝑦,𝜕𝑤+𝑢+𝜕𝜕𝑥𝐻+𝜂𝑣()𝜕𝑦𝐻+𝜂=−𝛼𝑢,𝑣𝜂+𝑄, (1.1) where 𝐻(𝑥,𝑦) is the depth of the water, 𝑤≡𝜕𝜂/𝜕𝑡 is the vertical velocity of the wave surface, 𝑃_𝑥 and 𝑃_𝑦 are variable external forcings, and 𝑄 is a mass source term. The acceleration due to gravity 𝑔₀ accounts for buoyancy, and the Coriolis parameter 𝑓₀ addresses the rotation of the earth (only required for very long waves over the earth [14]). The friction coefficient 𝛼≥0 acts as a generic sink that reduces the amplitude over time [6].

This work considers localized plane waves, or wavelets, on a flat sea propagating only in the 𝑥-direction. Thus, we set ()()()(𝑥)(𝑥)≡𝑢=𝑢𝑡,𝑥,𝑤=𝑤𝑡,𝑥,𝜂=𝜂𝑡,𝑥,𝑣𝑡,𝑥,𝑦=0,𝐻=𝐻,𝑐=𝑐𝑔₀𝐻(𝑥),𝑃_𝑥=𝑃_𝑥(𝑡,𝑥),𝛼=𝑓₀=𝑃_𝑦=𝑄=0. (1.2) In deep water 𝐻=𝐻₀>0 is a fixed constant and 𝑐₀=𝑔₀𝐻₀ is a constant surface-wave speed (called the celerity). Then, for small amplitudes, the elevation function 𝜂(𝑡,𝑥) satisfies generation∶𝜕²𝜂𝜕𝑡²−𝑐²₀𝜕²𝜂𝜕𝑥²=𝐹(𝑡,𝑥),where𝜂(𝑡,𝑥)≃0,for𝑡≪0,𝑥∈ℝ,propagation∶𝜕𝜂𝜕𝑡+𝑐₀𝜕𝛽𝜕𝑥𝜂+𝑐₀𝜂²+𝜈𝑐₀𝜕²𝜂𝜕𝑥²=0,for𝐻(𝑥)≃𝐻₀,runup∶𝜕𝜂𝜕𝑡+𝑐𝜕𝜂+𝜂𝜕𝑥2𝑑𝑐𝜕𝜂𝑑𝑥+𝛽²𝜕𝜕𝑥+𝜈³𝜂𝜕𝑥³=0,for𝐻(𝑥)>0. (1.3)(1.4)(1.5) These equations are found in [8, Equation (2)], [15, Equation (45)], and [16, Equation (13)], respectively. The new coefficients are defined as 𝛽(𝑥)≡3𝑐/(4𝐻) and 𝜈(𝑥)≡𝑐𝐻²/6. Equation (1.4) is referred to as the KdV-top model in [16]. Inclusion of an additional boundary-turbulence term in (1.5) would result in a loss of conservation properties for 𝜂 (considered in [17] but not here).

The forcings 𝐹(𝑡,𝑥) in (1.3) will be constructed from bounded wavelets 𝜙(𝑢) that satisfy MADEs. As such they satisfy the following conditions: 𝜙∈ℒ¹(ℝ)∩ℒ²(ℝ)∩ℒ^∞(ℝ),^∞_−∞𝜙(𝑢)𝑑𝑢=0. (1.6) Physically, if 𝜙 represents the displacement of the water-level from equilibrium, conservation of mass necessitates that the conditions in (1.6) hold. Note that, as discussed in [15], solitons are not expected to occur in (1.4) since the total spatial integral of 𝜂 vanishes. We find that if 𝐹(𝑡_∗,𝑥) is a wavelet in the 𝑥 variable (for fixed 𝑡_∗∈ℝ), then so is 𝜂(𝑡_∗,𝑥) away from the shore (using 𝑥-integration of either (1.3) or (1.4)). During the runup, however, 𝜂(𝑡,𝑥) looses its wavelet properties due to the variability of the coefficients in (1.5).

It seems natural that wavelets should appear in the study of surface water waves. The wavelets presented here are particularly well suited for surface waves. In particular, we show that when the forcing 𝐹 is expressed in terms of 𝐾_𝑞, _𝑞Cos and _𝑞Sin in (2.2)-(2.3), the solution can be reexpressed in terms of these functions. An objective of this paper is to demonstrate that the modeling as well as the detection and analysis of an observed wave profile can be achieved efficiently in terms of the wavelets (2.2)-(2.3), see [18–20]. Hence, a brief discussion on the topics of signal analysis and recovery on real data is presented using 𝐾_𝑞 and _𝑞Cos. We apply these techniques to the Japanese tsunami of March 11, 2011. The paper is completed with the details of a perturbation analysis in 𝑞>1⁺ that is needed to establish the existence of a resonance for rogue waves.

2. Preliminaries on Special Functions

The first quantity that we introduce is the function 𝐾_𝑞(𝑡) that satisfies the MADE 𝐾^′_𝑞(𝑡)=𝐾_𝑞𝑞𝑡,∀𝑡∈ℝ,𝑞>1. (2.1) We set 𝐾_𝑞(𝑡)≡0 for 𝑡≤0, and define 𝐾_𝑞(𝑡)≡^∞_𝑗=−∞(−1)^𝑗𝑒^{−𝑞𝑗𝑡}𝑞^{𝑗(𝑗+1)/2},for𝑡>0,𝑞>1. (2.2) Note that 𝐾_𝑞(𝑡) satisfies the wavelet conditions in (1.6), see [18]. Furthermore, as in [20], the reproducing kernel of 𝐾_𝑞(𝑡) gives the functions Cos(_𝑞𝑡)=𝑁_𝑞⋅^∞₀𝐾_𝑞(𝑢)𝐾_𝑞(𝑢−𝑡)𝑑𝑢=𝑁_𝑞^∞_𝑗=−∞(−1)^𝑗𝑒^{−𝑞2𝑗|𝑡|}𝑞^𝑗2_𝑞,Sin(𝑡)=−𝑁_𝑞⋅^∞₀𝐾_𝑞(𝑢)𝐾_𝑞𝑡𝑞𝑢−𝑞𝑡𝑑𝑢=|𝑡|𝑁_𝑞^∞_𝑗=−∞(−1)^𝑗𝑒^{−𝑞2𝑗|𝑡|}𝑞^{𝑗(𝑗−1)}, (2.3) which are displayed in Figure 2 for 𝑞=1.5. The normalization constant 𝑁_𝑞 is chosen so that _𝑞Cos(0)=1. It is shown in [20] that the 𝑞-advanced harmonic oscillator equations, _𝑞Cos^′′(𝑡)=−𝑞⋅_𝑞Cos𝑞𝑡,_𝑞Sin^′′(𝑡)=−𝑞²⋅_𝑞Sin𝑞𝑡, (2.4) hold, which are second-order MADEs. Each of (2.2)-(2.3) lies in ℒ². We show as a consequence of Theorem 2.1 that _𝑞Cos(𝑡)→cos(𝑡) and _𝑞Sin(𝑡)→Sin(𝑡) pointwise for 𝑡∈ℝ as 𝑞→1⁺. Note, we also have uniform convergence on compact sets, which was obtained in [20]. Thus, _𝑞Cos(𝑡) and _𝑞Sin(𝑡) can be viewed as ℒ² approximations of cos(𝑡) and sin(𝑡), respectively, which are solutions of the limit of equations in (2.4) as 𝑞→1⁺.

2.1. Theta Functions

As part of our study, we employ the Jacobi theta function, defined for 𝜔∈ℂ⧵{−𝑞^𝑛}_𝑛∈ℤ, 𝜃𝑞,𝜔≡^∞_𝑘=−∞𝜔^𝑘𝑞^{𝑘(𝑘−1)/2}=𝜇_𝑞^∞_𝑛=0𝜔1+𝑞^𝑛11+𝜔𝑞^𝑛+1,𝜇_𝑞≡^∞_𝑛=011−𝑞^𝑛+1. (2.5) This function will only be used in regions where 𝜔>0 and 𝑞>1. Clearly the function 𝜃(𝑞²,𝜔²) is 𝐶^∞ for all 𝜔∈ℝ−{0} and grows faster than any rational polynomial at ±∞ and 0. As such, for each 𝑞>1 one has that 1/𝜃(𝑞²,𝜔²) is a Schwartz function that is flat at 𝜔=0. It also satisfies the algebraic identity 𝜃𝑞;𝑞^𝑛𝜔=𝑞^{𝑛(𝑛+1)/2}𝜔^𝑛𝜃𝑞;𝜔,𝑛∈ℤ, (2.6) which plays an important role in our analysis, [18].

Theorem 2.1. For 𝑞>1, let 𝛿_𝑞(𝜔)≡1ln𝑞𝜃𝑞²;𝜔²if0𝜔>0if𝜔≤0. (2.7) Then 𝛿_𝑞(𝜔) is a delta sequence in 𝑞>1⁺, at 𝜔=1 and has the following properties: (i)lim_𝑞→1+𝛿_𝑞(𝜔≠1)=0,(ii)lim_𝑞→1+𝛿_𝑞(1)=∞,(iii)lim_𝑞→1+^∞₀𝛿_𝑞(𝜔)𝑑𝜔=1. (2.8)

Proof. From [21], for 1<𝑞<𝑒^𝜋 and for 𝜔>0 one has 0<𝑞^1/4|𝜔|𝑒^{(ln(|𝜔|))2/ln(𝑞)}𝜋𝑞𝑞ln−1≤𝜃²;𝜔²,0<𝜃𝑞²;𝜔²≤𝑞^1/4|𝜔|𝑒^{(ln(|𝜔|))2/ln(𝑞)}𝜋𝑞.ln+1 (2.9)(2.10) Multiplication of (2.9) and (2.10) by ln(𝑞), followed by reciprocation, gives that for 𝜔>0𝑞^−1/4|𝜔|⁻¹𝑒^{−(ln(|𝜔|))2/ln(𝑞)}ln𝑞√𝜋+ln𝑞≤𝛿_𝑞(𝜔)≤𝑞^−1/4|𝜔|⁻¹𝑒^{−(ln(|𝜔|))2/ln(𝑞)}ln𝑞√𝜋−𝑞.ln (2.11) Now, (i) follows from (2.11) and the fact that for 𝑘>0, lim_𝑞→1⁺[𝑒^{−𝑘/ln(𝑞)}/ln(𝑞)]=0. For 𝜔=1, (ii) also follows from (2.11). To obtain (iii), we handle some preliminaries. First, it was shown in [18] that 𝐾_𝑞²₂≡^∞_−∞𝐾_𝑞(𝑡)²𝑑𝑡=^∞_−∞𝑞²𝜇_𝑞⁴𝜇_𝑞22𝜋𝜃𝑞²;𝑞²𝜔²=𝑞𝜇𝑑𝜔_𝑞4𝜇_𝑞22𝜋^∞_−∞𝑑𝑣𝜃𝑞²;𝑣²=𝑞𝜇_𝑞⁴𝜇_𝑞2𝜋^∞₀𝑑𝑣𝜃𝑞²;𝑣². (2.12) The functional identity (𝜇_𝑞)⁴𝑞=2‖𝐾_𝑞‖²₂(𝜇_𝑞2)²𝑁_𝑞, established in [20], allows for the replacement of (‖𝐾_𝑞‖²₂/𝑞) by ((𝜇_𝑞)⁴/[2(𝜇_𝑞2)²𝑁_𝑞]) in (2.12) to obtain the following: ^∞₀𝑑𝑣𝜃𝑞²;𝑣²=𝜋𝐾_𝑞²₂𝑞𝜇_𝑞⁴𝜇_𝑞2=𝜋2𝑁_𝑞𝜇_𝑞2³. (2.13) This in turn gives that lim_𝑞→1+^∞₀𝛿_𝑞(𝜔)𝑑𝜔=lim_𝑞→1+^∞₀𝑑𝜔ln𝑞𝜃𝑞²;𝜔²=lim_𝑞→1+𝜋2ln𝑞𝑁_𝑞𝜇_𝑞2³=1, (2.14) where the last equality follows from the 𝑞-Wallis limit, lim_𝑞→1⁺[ln(𝑞)𝑁_𝑞(𝜇_𝑞2)³]=𝜋/2, from [21]. This proves (iii) and finishes the proof of the theorem.

2.2. 𝑞-Advanced Wavelets That Solve MADEs

The appearance of theta functions is a consequence of the Laplace transform of (2.2), 𝐾_𝑞(𝑡)=ℒ⁻¹−𝜇³_𝑞(𝑡),𝑠𝜃𝑞,𝑠whereℒ𝑓(𝑡)(𝑠)≡^∞_𝑡=0𝑒^−𝑠𝑡𝑓(𝑡)𝑑𝑡. (2.15) In [20], we use the inverse Fourier transform to compute the expressions _𝑞(𝑡)=𝜇Cos^𝑞23𝑁_𝑞𝜋^∞_−∞𝑒^𝑖𝜔𝑡𝑑𝜔𝜃𝑞²,𝜔²_𝑞,(𝑡)=𝜇Sin−𝑖_𝑞2³𝑁_𝑞𝜋^∞_−∞𝑒^𝑖𝜔𝑡𝜔𝑑𝜔𝜃𝑞²,𝜔², (2.16) and these expressions will be used in the proof of a resonance for rogue waves. The representation in (2.15) verifies the MADE in (2.1), and (2.16) implies the identities _𝑞Cos^′(𝑡)=−_𝑞Sin(𝑡),_𝑞Sin^′(𝑡)=𝑞⋅_𝑞Cos𝑞𝑡, (2.17) which verify the MADEs in (2.4). Now Theorem 2.1 gives that both 𝛿_𝑞(𝜔) and 𝛿_𝑞(−𝜔) are delta sequences in 𝑞>1⁺ at 𝜔=±1, respectively. These applied to (2.16), in combination with the 𝑞-Wallis limit, give that pointwise lim_{𝑞→1+𝑞}Cos(𝑡)=cos(𝑡),lim_{𝑞→1+𝑞}Sin(𝑡)=sin(𝑡),for𝑡∈ℝ. (2.18) For 𝑞>1, the functions in (2.15) and (2.16) are in the Schwartz space. In particular, their rates of decay for large |𝑡| are typically slower than exponential but faster than any reciprocal polynomial. It can be shown that ∃𝐶>0 so that for 𝑡∈ℝ and 𝑞>1,max𝐾_𝑞(𝑡),_𝑞Cos(𝑡),_𝑞(𝑡,1Sin)𝜃𝑞;𝑡≤𝐶𝑞^{−log2𝑞(|𝑡|)}. (2.19)

Remark 2.2. The use of theta functions in frequency space provides decaying versions of cosine and sine while preserving many differential properties of these functions.

3. Tsunami Modeling Using MADEs

A tsunami wave is the consequence of a spontaneous change in elevation on the seafloor, which creates a variable pressure field throughout the volume of water. This sets up forces that extend to the surface of the ocean causing it to be moved up and down, locally. The perturbed wave height then propagates away from this disturbance. In still water, the surface wave speed 𝑐₀=𝑔₀𝐻₀ mainly depends on the average depth 𝐻₀ of the ocean. For tsunamis, the wavelength is much longer than 𝐻₀ and so the shallow-water wave equation applies (see [22, page 195]). Near the shore nonlinear effects need to be introduced to account for the sloping of the shoreline [4, 12, 23, 24].

Suppose that the faultline on the seafloor is parallel to the 𝑦 coordinate. Then the 𝑦-dependence can be ignored even after the tsunami-causing event has taken place. In this situation, the wave front will travel in the 𝑥 direction only.

3.1. 𝑞-Advanced Tsunami Model

Consider the forced one-dimensional wave equation for the water-level function 𝜓 [14, 22], 𝜕²_𝑡𝜓(𝑡,𝑥)−𝑐²₀𝜕²_𝑥𝜓(𝑡,𝑥)=𝐹(𝑡,𝑥),∀𝑡∈ℝ,𝑥∈ℝ,𝜓(𝑡,𝑥)=0,∀𝑡<0,𝑥∈ℝ, (3.1) where the condition on the right in (3.1) constitutes the boundary and initial conditions. The forcing is expected to satisfy the conditions, for fixed 𝑡_∗, 𝑥_∗∈ℝ, lim_|𝑡|→∞𝐹𝑡,𝑥_∗=0,lim_|𝑥|→∞𝐹𝑡_∗,𝑥=0,^∞_−∞𝐹𝑡_∗,𝑥𝑑𝑥=0, (3.2) and is related to the depth of the ocean floor by 𝜕²_𝑡𝐻(𝑡,𝑥)=𝐹(𝑡,𝑥). The models in [4, 8] start with ground-motion profiles 𝐻=𝐻_TS or 𝐻=𝐻_ZWL where 𝐻_TS≡𝐴𝛾𝑡tanh𝛾𝑥−𝜏,𝐻_ZWL≡2𝐴𝛾sech²𝛾𝑥tanh𝛾𝑥sin𝜋𝑡,2𝜏 (3.3) respectively. In [8], it is noted that using the first model 𝐻=𝐻_TS gives a force consistent with a landslide that continues for all time. In [4], the second model 𝐻=𝐻_ZWL suggests an earthquake that continues for all time. In these settings, (3.2) does not hold. However, these models are still used as part of the initial forcings for the wave equation since they lead to integrable solutions whose evolutions resemble that of actual tsunamis. As a comparison, we propose the following 𝑞-advanced model: 𝐻_𝑞(𝑡,𝑥)≡𝐴⋅𝐾_𝑞𝑡𝜏^𝑞,𝐹Sin𝛾⋅𝑥_𝑞()≡𝜕𝑡,𝑥²𝐻_𝑞𝜕𝑡²=𝐴⋅𝑞𝜏²𝐾_𝑞𝑞²𝑡𝜏^𝑞Sin𝛾⋅𝑥, (3.4) where 𝐹_𝑞(𝑡,𝑥) now satisfies the conditions in (3.2). When (3.4) is substituted into (3.1), solved and simplified, one obtains a unique solution 𝜓(𝑡,𝑥) to the forced wave equation. To express the solution, define the two phase functions, corresponding to right and left propagation, 𝜑⁺_𝜇()≡𝑡𝑡,𝑥,𝜉𝜏−𝛾⋅𝑥−𝜉𝑞^𝜇,𝜑⁻_𝜇()≡𝑡𝑡,𝑥,𝜉𝜏+𝛾⋅𝑥−𝜉𝑞^𝜇,∀𝑡,𝑥,𝜉∈ℝ, (3.5) where the parameters are related by 𝑐₀𝛾𝜏=𝑞^𝜇. Define, for any 𝛼,𝛽,𝜇∈ℝ and 𝑞>1, 𝒯^±_{𝛼,𝛽,𝜇}(𝑡,𝑥)≡_𝑞^𝛾⋅𝑥_𝜉=0Sin𝑞^𝛼⋅𝜉⋅𝐾_𝑞𝑞^𝛽⋅𝜑^±_𝜇(𝑡,𝑥,𝜉)𝑑𝜉. (3.6) Then, for 𝑡>0 and 𝑥∈ℝ, the reader can verify that 𝜓(𝑡,𝑥)=𝜓_particular(𝑡,𝑥)+𝜓_homogeneous(𝑡,𝑥), (3.7) where a particular solution to (3.1), with the forcing in (3.4), is 𝜓_particular(𝑡,𝑥)=−𝐴⋅𝑞^−𝜇2⋅𝒯⁻_0,1,𝜇(𝑡,𝑥)−𝒯⁺_0,1,𝜇(𝑡,𝑥). (3.8) A smooth solution that satisfies 𝜓(𝑡,𝑥)=0, for 𝑡≤0, is obtained by choosing 𝜓_hom.(𝑡,𝑥)=−12𝜓_par.0,𝑥+𝑐₀𝑡+𝜓_par.0,𝑥−𝑐₀𝑡+^{𝑥+𝑐0𝑡}_{𝑥−𝑐0𝑡}𝜕𝜓_par.(0,𝑢)𝑐₀.⋅𝜕𝑡𝑑𝑢 (3.9) One can see from (3.5)–(3.9) that the solution has the basic wavelet properties 𝜓⋅,𝑥_∗,𝜓𝑡_∗,⋅∈ℒ¹(ℝ)∩ℒ²(ℝ)∩ℒ^∞(ℝ),_ℝ𝜓𝑡,𝑥_∗𝑑𝑡=_ℝ𝜓𝑡_∗,𝑥𝑑𝑥=0, (3.10) for each 𝑡_∗,𝑥_∗∈ℝ. To analyze the long-term behavior of the solution 𝜓, we use (2.1), (2.17), and integration by parts, twice, to obtain Proposition 3.1.

Proposition 3.1. For any 𝛼,𝛽,𝜇∈ℝ and 𝑞>1 the following identity holds: 𝒯^±_{𝛼,𝛽,𝜇}(𝑡,𝑥)=𝑞^−𝛼⋅𝐾_𝑞𝑞^𝛽⋅𝑡𝜏∓𝛾𝑥𝑞^𝜇−𝑞^{−2𝛼+2𝛽−2𝜇+1}⋅𝒯^±_{𝛼−1,𝛽+2,𝜇}(𝑡,𝑥)−𝑞^−𝛼⋅_𝑞Cos𝑞^𝛼𝛾𝑥⋅𝐾_𝑞𝑞^𝛽𝑡𝜏±𝑞_𝑞^{−2𝛼+𝛽−𝜇}Sin𝑞^𝛼−1𝛾𝑥⋅𝐾_𝑞𝑞^𝛽+1𝑡𝜏. (3.11)

The expression in (3.11) demonstrates that 𝒯^±_{𝛼,𝛽,𝜇}, as used in (3.8), can be written as a series of localized (bound) states and traveling (free) states. These different terms appear as translated and scaled versions of the wavelets 𝐾_𝑞, _𝑞Cos and _𝑞Sin. Thus, such functions provide a good match for a wave profile 𝜓 that was generated by the family of forcings 𝐹_𝑞 in (3.4). Furthermore, for 𝑡≫0, 𝑥≫0 and 𝜇≫0 (reasonable for tsunamis), 𝜓_particular(𝑡,𝑥)≃𝐴𝑞^−3𝜇−12⋅𝐾_𝑞𝛾⋅𝑞^−𝜇𝑐₀𝑡−𝑥. (3.12) Finally, observe that 𝐹_𝑞(𝑡,𝑥) in (3.4) is 𝑞-advanced in time compared to the ground-motion profile 𝐻_𝑞(𝑡,𝑥). We also find that the forcing precedes the response 𝜓 for the 𝑞-advanced models. Figure 4(a) illustrates a case where a forcing profile generates a similarly shaped tsunami in Figure 4(b).

Remark 3.2. Similar results hold for height functions 𝐻_𝑞(𝑡,𝑥)≡𝐴⋅𝐾_𝑞(𝑡/𝜏)_𝑞Cos(𝛾⋅𝑥).

3.2. Numerical Solution of a 𝑞-Advanced Tsunami Wave Event

Here, we model the Japan tsunami of March 11, 2011, using (1.3), (1.4), and (1.5) with a forcing term 𝐹_𝑞 in (3.4) with environmental parameters chosen to be, in mks units, 𝑔₀=10ms⁻²,𝐻₀=4000m,𝑐₀=200ms⁻¹,𝛾=0.01m⁻¹,𝜏=100s,𝐴=6×10⁷m, (3.13) which are based on past experience [4]. The sea-depth function is chosen to be 𝐻(𝑥)=𝐻₀,for𝑥≤0,𝐻(𝑥)=𝐻₀⋅𝑥1−5km²,for𝑥∈(0,5km), (3.14) which models the sea-floor near Wake Island. This is a required modification of the model used in [7]. For stability, we employ a Lax-Wendroff correction term in the numerical method. A good match with typical tsunami profiles and run-up profiles was obtained by using 𝑞=1.25. In this case, 𝜇≃23.74, which justifies the approximation in (3.12).

Figure 3 represents data from the Japan tsunami of March 11, 2011, at the first oceanic observation site DART 21418. This data includes a preliminary forcing profile from the (local) times 5 min to 20 min, along with the actual tsunami profile from the (local) times 42 min to 72 min.

Figure 3 

DART 21418: Data showing earthquake ( forcing, on the left) and resulting tsunami (on the right) observed on March 11, 2011, from 5:43 am to 6:51 am (horizontal axis is time in minutes, vertical axis is in meters).

(a)

(b)

Figure 4 

(a) Comparison of forcing from data (dashed line) and model (solid line) results in 𝑞=1.25; (b) comparing tsunami at DART 21418 and propagation model results in 𝐴=6×107m.

Figure 4(a) again shows the Dart 21418 forcing profile from the times 5 min to 20 min, along with a 𝑞-advanced forcing profile as in (3.4), with 𝐴 and 𝑞 chosen to be 𝐴=6×10⁷ and 𝑞=1.25 to effect comparable forcing profiles.

Figure 4(b) shows the Dart 21418 actual tsunami profile from the times 42 min to 72 min along with the numerically propagated theoretical tsunami for the same time interval that was generated by (1.3) and (1.4) for the forcing 𝐹_𝑞 in (3.4) as above (with parameters 𝐴=6×10⁷ and 𝑞=1.25).

Figure 5 shows the actual tsunami profile at a later time and greater distance at oceanic observation DART 21413 from the (local) times 5 min to 65 min. Also shown for comparison is the numerically propagated theoretical tsunami for the same time interval that was generated by (1.3) and (1.4) for the forcing 𝐹_𝑞 in (3.4) with parameters 𝐴=6×10⁷ and 𝑞=1.25.

Figure 5 

DART 21413: Data showing tsunami about 90 minutes after the earthquake and propagation model.

Figure 6 shows the actual run-up profile taken from the tide gauge at Wake Island at an even later time and greater distance. This is compared with our predicted run-up profile generated by (1.3) and (1.4) together with (1.5) for the forcing 𝐹_𝑞 in (3.4) with parameters 𝐴=6×10⁷ and 𝑞=1.25. The run-up data and theoretical profile have similar profiles initially, for the first few oscillations.

Figure 6 

Wake Island: Data showing tsunami about 125 minutes after the earthquake and run-up model for Wake Island shoal.

4. Rogue-Wave Modeling Using Solutions of MADEs

A debate continues on the physical cause of rogue waves, [9]. One possible mechanism is a natural outcome of a constructive interference between rippling surface waves that propagate in different directions [22, page 191]. We construct a localized plane wave using a MADE-generated solution of the wave equation, for 𝐴>0, Ψ_𝑞(𝑡,𝑥)=𝐴⋅_𝑞𝑡Cos𝜏⋅_𝑞Cos𝛾⋅𝑥. (4.1) Substituting Ψ_𝑞 into (3.1) results in a small but persistent forcing. The model in (4.1) ignores possible translational behavior, which will be discussed at the end of this section. Here, 𝛾 and 𝜏 are parameters that satisfy 𝑐𝛾=1/𝜏 for a constant celerity 𝑐>0. The forcing required to obtain Ψ_𝑞 from a calm distant past consists of two terms 𝐹_𝑞(𝑡,𝑥)=𝐴𝑞𝛾²𝑐²⋅_𝑞Cos𝑞⋅𝑡𝜏⋅_𝑞Cos𝛾⋅𝑥−_𝑞𝑡Cos𝜏⋅_𝑞Cos𝑞⋅𝛾⋅𝑥, (4.2) and it is easily seen that ‖𝐹_𝑞‖_∞≤𝐴/𝜏². We show that ‖𝐹_𝑞‖_∞→0 as 𝑞→1⁺ while from (4.1) it follows that ‖𝜓_𝑞‖_∞=𝐴 remains constant for all 𝑞>1. This is a type of triad resonance with peak at (𝑡,𝑥)=(0,0). To proceed with our analysis, estimates on the differences between 𝑞-advanced functions, which are solutions of MADEs, are needed.

4.1. Analysis of Forcing Terms for _𝑞Cos and _𝑞Sin-Type Rogue Waves

In this section, we show that small amplitude forces, over long periods of time, can naturally produce large rogue waves. This demonstrates the existence of a resonance for the system externally forced by (4.2).

Proposition 4.1. Let 𝜖>0 be given. Then there exists 𝑄^𝜖_𝐶>1 such that for all 𝑞 with 1<𝑞<𝑄^𝜖_𝐶 one has Cos𝑞𝑡−_𝑞𝑞Cos(𝑡)<𝜖, (4.3) for all 𝑡∈ℝ; also there exists 𝑄^𝜖_𝑆>1 such that for all 𝑞 with 1<𝑞<𝑄^𝜖_𝑆 one has Sin𝑞𝑡−_𝑞𝑞Sin(𝑡)<𝜖, (4.4) for all 𝑡∈ℝ.

The proof of this result is given in the last section. It is used here to show that small amplitude forces, over long periods of time, can naturally produce rogue waves. That is, for 𝑞 sufficiently near 1, we now apply Proposition 4.1 to show that for arbitrarily small forcing terms 𝐹_𝑞(𝑡,𝑥) there are large rogue solutions Ψ_𝑞(𝑡,𝑥) of the forced wave equation (4.5) (and of (4.10)).

Theorem 4.2. Let 𝐴,𝑐>0 and define Ψ_𝑞 as in (4.1) and 𝐹_𝑞 as in (4.2). Let 𝜖>0 be given, and let 𝑞>1 be sufficiently close to 1. Then Ψ_𝑞(𝑡,𝑥) satisfies the forced wave equation 𝜕²_𝑡−𝑐²𝜕²_𝑥Ψ_𝑞(𝑡,𝑥)=𝐹_𝑞(𝑡,𝑥), (4.5) where Ψ_𝑞(0,0)=𝐴 is fixed, but |𝐹_𝑞(𝑡,𝑥)|<𝜖 for all 𝑡,𝑥∈ℝ.

Proof. Without loss of generality, set 𝛾=1 and 𝑐=1/𝜏. Then one has, for all 𝑞>1, that 𝜕²_𝑡−𝑐²𝜕²_𝑥Ψ_𝑞(𝑡,𝑥)=𝜕²_𝑡−𝑐²𝜕²_𝑥𝐴⋅_𝑞Cos(𝑐𝑡)⋅_𝑞(𝑥⋅Cos)=𝐴−𝑞_𝑞Cos𝑞𝑐𝑡𝑐²⋅_𝑞Cos(𝑥)−𝑐²𝐴⋅_𝑞Cos(𝑐𝑡)−𝑞⋅_𝑞Cos𝑞𝑥=−𝐴𝑞𝑐²_𝑞Cos𝑞𝑐𝑡⋅_𝑞Cos(𝑥)−_𝑞Cos(𝑐𝑡)⋅_𝑞Cos𝑞𝑥=𝐹_𝑞(𝑡,𝑥), (4.6) giving (4.5). Choose 𝑞₀>1. Now let 𝜖̃≡𝜖/[2𝐴𝑞₀𝑐²], and let 𝑄𝜖_𝐶̃>1 be chosen so that for 1<𝑞<𝑄𝜖_𝐶̃ one has |_𝑞Cos(𝑞𝑥)−_𝑞Cos(𝑥)|<𝜖̃ by Proposition 4.1. Then, for all 1<𝑞<min{𝑄𝜖_𝐶̃,𝑞₀}, 𝐹_𝑞(𝑡,𝑥)=−𝐴𝑞𝑐²_𝑞Cos𝑞𝑐𝑡⋅_𝑞Cos(𝑥)−_𝑞Cos(𝑐𝑡)⋅_𝑞Cos(𝑥)−𝐴𝑞𝑐²_𝑞Cos(𝑐𝑡)⋅_𝑞Cos(𝑥)−_𝑞Cos(𝑐𝑡)⋅_𝑞Cos𝑞𝑥≤𝐴𝑞𝑐²_𝑞Cos(𝑥)_𝑞Cos𝑞𝑐𝑡−_𝑞Cos(𝑐𝑡)+𝐴𝑞𝑐²_𝑞Cos(𝑐𝑡)_𝑞Cos(𝑥)−_𝑞Cos𝑞𝑥≤𝐴𝑞𝑐²1_𝑞Cos𝑞𝑐𝑡−_𝑞Cos(𝑐𝑡)+𝐴𝑞𝑐²1_𝑞Cos𝑞𝑥−_𝑞Cos(𝑥)≤𝐴𝑞𝑐²𝜖̃+𝐴𝑞𝑐²̃𝜖𝜖<2+𝜖2=𝜖, (4.7)(4.8) where (4.8) follows from the fact that ‖_𝑞Cos‖_∞=1. Now, since _𝑞Cos(0)=1, we have Ψ_𝑞(0,0)≡𝐴⋅_𝑞Cos(0)⋅_𝑞Cos(0)=𝐴.

Theorem 4.3. Let 𝐴,𝑐>0. Let 𝜖>0 be given, and let 𝑞>1 be sufficiently close to 1. Define Φ_𝑞(𝑡,𝑥)≡𝐴⋅_𝑞Sin(𝑐𝑡)⋅_𝑞(𝑥),𝐺Sin_𝑞(𝑡,𝑥)≡−𝐴𝑞²𝑐²_𝑞Sin𝑞𝑐𝑡⋅_𝑞Sin(𝑥)−_𝑞Sin(𝑐𝑡)⋅_𝑞Sin𝑞𝑥. (4.9) Then Φ_𝑞(𝑡,𝑥) satisfies the forced wave equation 𝜕²_𝑡−𝑐²𝜕²_𝑥Φ_𝑞(𝑡,𝑥)=𝐺_𝑞(𝑡,𝑥), (4.10) where sup_{𝑡,𝑥∈ℝ}Φ_𝑞≈𝐴≫𝜖, but |𝐺_𝑞(𝑡,𝑥)|<𝜖 for all 𝑡,𝑥∈ℝ.

Proof. From (4.9), one has for all 𝑞>1 that 𝜕²_𝑡−𝑐²𝜕²_𝑥Φ_𝑞(𝑡,𝑥)=𝜕²_𝑡−𝑐²𝜕²_𝑥𝐴⋅_𝑞Sin(𝑐𝑡)⋅_𝑞Sin(𝑥)=𝐴−𝑞²⋅_𝑞Sin𝑞𝑐𝑡𝑐²⋅_𝑞Sin(𝑥)−𝑐²𝐴⋅_𝑞Sin(𝑐𝑡)−𝑞²⋅_𝑞Sin𝑞𝑥=−𝐴𝑞²𝑐²_𝑞Sin𝑞𝑐𝑡⋅_𝑞Sin(𝑥)−_𝑞Sin(𝑐𝑡)⋅_𝑞Sin𝑞𝑥=𝐺_𝑞(𝑡,𝑥), (4.11) giving (4.10). Choose 𝑞₀>1. Now let 𝜖̃≡𝜖/[2𝐴𝑞₀²𝑐²√𝑞₀], and let 𝑄𝜖_𝑆̃>1 be chosen so that for 1<𝑞<𝑄𝜖_𝑆̃ one has |_𝑞Sin(𝑞𝑥)−_𝑞Sin(𝑥)|<𝜖̃ by Proposition 4.1. Then, for all 1<𝑞<min{𝑄𝜖_𝑆̃,𝑞₀}, 𝐺_𝑞(𝑡,𝑥)=−𝐴𝑞²𝑐²_𝑞Sin𝑞𝑐𝑡⋅_𝑞Sin(𝑥)−_𝑞Sin(𝑐𝑡)⋅_𝑞Sin(𝑥)−𝐴𝑞²𝑐²_𝑞Sin(𝑐𝑡)⋅_𝑞Sin(𝑥)−_𝑞Sin(𝑐𝑡)⋅_𝑞Sin𝑞𝑥≤𝐴𝑞²𝑐²_𝑞Sin(𝑥)_𝑞Sin𝑞𝑐𝑡−_𝑞Sin(𝑐𝑡)+𝐴𝑞²𝑐²_𝑞Sin(𝑐𝑡)_𝑞Sin(𝑥)−_𝑞Sin𝑞𝑥≤𝐴𝑞²𝑐²√𝑞_𝑞Sin𝑞𝑐𝑡−_𝑞Sin(𝑐𝑡)+𝐴𝑞²𝑐²√𝑞_𝑞Sin𝑞𝑥−_𝑞Sin(𝑥)≤𝐴𝑞²𝑐²√𝑞𝜖̃+𝐴𝑞𝑐²√𝑞̃𝜖𝜖<2+𝜖2=𝜖, (4.12)(4.13)(4.14)(4.15) where (4.14) follows from the fact that ‖_𝑞Sin‖_∞≤√𝑞, as in [20]. Now, since sup_𝑞Sin≈1 for 𝑞>1 sufficiently close to 1, we have that supΦ_𝑞≈𝐴.

4.2. Slowly Moving Rogue Waves

When a slight drift in the rogue-generating current is present, there may be a speed 𝑣≪𝑐 to the wave-height profile. A model for such a rogue can be given by 𝜓_𝑞,𝑣(𝑡,𝑥)=𝐴⋅_𝑞CosΓ_𝑣⋅𝛾⋅𝑐𝑡⋅_𝑞Cos𝛾⋅(𝑥−𝑣𝑡), (4.16) where Γ_𝑣≡1−(𝑣/𝑐)². The peak of this wave still occurs at (𝑡,𝑥)=(0,0) but moves to the right at speed 𝑣. The techniques used in the previous section show that by choice of parameters, a small amplitude forcing, over a long period, will create a moving rogue of arbitrary size.

Theorem 4.4. Let 𝐴,𝑐,𝛾>0 and 𝑐>𝑣>0. Define Γ_𝑣≡1−(𝑣/𝑐)². Let 𝜖>0 be given, and let 𝑞>1 be sufficiently close to 1. Define the surface-height function Ψ_𝑞,𝑣(𝑡,𝑥)≡𝐴⋅_𝑞CosΓ_𝑣⋅𝛾⋅𝑐𝑡⋅_𝑞Cos𝛾⋅(𝑥−𝑣𝑡), (4.17) and let the forcing be given by 𝐹_𝑞(𝑡,𝑥)≡−2𝐴𝛾²𝑣𝑐Γ_𝑣⋅_𝑞SinΓ_𝑣𝛾𝑐𝑡_𝑞Sin𝛾(𝑥−𝑣𝑐𝑡)−𝐴𝛾²𝑞𝑐²−𝑣²⋅_𝑞Cos𝑞Γ_𝑣𝛾𝑐𝑡_𝑞𝛾(+Cos𝑥−𝑣𝑡)_𝑞CosΓ_𝑣𝛾𝑐𝑡_𝑞Cos𝑞𝛾(𝑥−𝑣𝑡). (4.18) Then Ψ_𝑞𝑣(𝑡,𝑥) satisfies the forced wave equation 𝜕²_𝑡−𝑐²𝜕²_𝑥Ψ_𝑞,𝑣(𝑡,𝑥)=𝐹_𝑞(𝑡,𝑥), (4.19) where Ψ_𝑞,𝑣(0,0)=𝐴 is fixed, but |𝐹_𝑞(𝑡,𝑥)|<𝜖+(2𝐴𝑞Γ_𝑣𝛾²𝑐)⋅𝑣 for all 𝑡,𝑥∈ℝ.

Proof. Applying the operator (𝜕²_𝑡−𝑐²𝜕²_𝑥) to (4.17) yields (4.19) for 𝐹_𝑞 in (4.18). The magnitude of the first term in (4.18) is handled as follows: 2𝐴𝛾²𝑣𝑐Γ_𝑣𝑞SinΓ_𝑣𝛾𝑐𝑡_𝑞𝛾()≤Sin𝑥−𝑣𝑡2𝐴𝛾²𝑣𝑐Γ_𝑣𝑞SinΓ_𝑣𝛾𝑐𝑡⋅_𝑞𝛾()≤Sin𝑥−𝑣𝑡2𝐴𝛾²𝑣𝑐Γ_𝑣√𝑞√𝑞=2𝐴𝛾²𝑞𝑐Γ_𝑣⋅𝑣, (4.20) where the last inequality follows from the fact that ‖_𝑞Sin‖_∞≤√𝑞. The remaining expressions in (4.18) are controlled by noticing that _𝑞Cos𝑞⋅Γ_𝑣𝛾𝑐𝑡⋅_𝑞Cos𝛾(𝑥−𝑣𝑡)−_𝑞CosΓ_𝑣𝛾𝑐𝑡⋅_𝑞Cos𝑞⋅𝛾(𝑥−𝑣𝑡) (4.21) can be brought below 𝜖 for 𝑞 sufficiently close to 1⁺ by paralleling steps (4.7) through (4.8) in Theorem 4.2 and applying Proposition 4.1. The result is now proven.

Remark 4.5. For 𝑣 sufficiently small (as well as 𝑞 sufficiently near 1⁺), the term (2𝐴𝑞Γ_𝑣𝛾²𝑐)⋅𝑣 in Theorem 4.4 is small, and one then has that 𝐹_𝑞(𝑡,𝑥) can be made arbitrarily small compared with the rogue amplitude 𝐴, independently of 𝑡,𝑥∈ℝ. For smaller values of 𝛾, the moving rogue wave maintains a large amplitude near 𝐴 for a longer period of time.

Remark 4.6. There is a corresponding theorem for the moving _𝑞Sin rogue wave given by Φ_𝑞,𝑣(𝑡,𝑥)≡𝐴⋅_𝑞Sin(Γ_𝑣𝛾𝑐𝑡)⋅_𝑞Sin(𝛾(𝑥−𝑣𝑡)).

Remark 4.7. Note that, as is demonstrated in Figure 7, classic rogue wave profiles emerge from smaller forcings even for 𝑞 relatively far from 1.

(a)

(b)

(c)

Figure 7 

Parameters: 𝑐=1.0, 𝜆=1.0, 𝜏=𝑞=3/2, and 𝛾=1/𝑞=2/3. (a) Forcing profile 𝐹𝑞 converges toward 𝑥=0 as 𝑡→0−; (b) rogue solution Ψ𝑞 in a neighborhood of 𝑥=0 that oscillates over time; (c) forcing with ‖𝐹𝑞‖∞≤1/𝑞2=4/9 and solution with ‖Ψ𝑞‖∞=1.

5. Wavelet Signal Analysis, Inversion, and the Frame Operator

We now have a collection of solutions to differential equations that give the qualitative behavior of a physical phenomena. Next, to detect, analyze, store, and recover a tsunami waveform, it is common to use a wavelet analysis [25]. The process begins by identifying a discrete set of functions, called an affine frame, Λ_𝜓≡𝜓_𝑗,𝑘(𝑡)≡𝑞^𝑗/2𝜓𝑞^𝑗𝑡−𝑘𝑏𝜓𝑗,𝑘∈ℤ. (5.1) With some effort, it can be shown that spanΛ_𝜓=ℒ²(ℝ), for appropriate 𝜓 and 𝑏>0 sufficiently small [26]. This leads a wavelet transform 𝑊_𝜓∶ℒ²(ℝ)→ℓ²(ℤ²) where 𝑊_𝜓𝑓𝑗,𝑘≡𝜓_𝑗,𝑘=𝑞,𝑓^𝑗/2𝜓_ℝ𝜓𝑞^𝑗𝑡−𝑘𝑏𝑓(𝑡)𝑑𝑡. (5.2) The range of 𝑊 is a subspace of ℓ²(ℤ²) and has an adjoint 𝑊^∗ defined to be 𝑊^∗∶𝑊ℒ²(ℝ)⊆ℓ²ℤ²→ℒ²(ℝ),𝑊^∗𝑐_{𝑗,𝑘ℤ2}=𝑐_𝑗,𝑘𝜓^∗_𝑗,𝑘(𝑡), (5.3) where 𝜓^∗_𝑗,𝑘 are the elements of the dual frame to Λ_𝜓. The frame operator 𝑆≡𝑊^∗𝑊∶ℒ²(ℝ)→ℒ²(ℝ) is invertible for 𝑏>0 sufficiently small. If Λ_𝜓 is an orthonormal basis for ℒ²(ℝ), then one can use 𝜓^∗_𝑗,𝑘=𝜓_𝑗,𝑘. The wavelets 𝐾_𝑞, _𝑞Cos, and _𝑞Sin generate frames whose inner product structures are nearly orthogonal [20]. Consequently, as will be shown below, a reasonable analysis of the different waveforms is obtained without computing the dual to Λ_𝜓.

The results of a signal analysis and synthesis, briefly presented here, consist of (1) choosing 256 equallyspaced points from the data, (2) computing the inner products with elements of Λ_𝜓, (3) sorting and identifying inner products with the largest magnitudes, (4) reconstructing the wavelet-based waveform, (5) normalizing in ℒ², and (6) computing the normalized RMS error.

By [19], the sizes chosen for the parameters are 𝑞=1.5 and 𝑏∼2, with slight adjustments being made to improve the result. The reconstructed signal has an amplitude about 𝑏/(2𝑞) of the size of the data profile. The need for this scaling was explained in [20] and is due to the fact that the ℒ² operator norm of 𝑆⁻¹ is estimated to be 𝑏/(2𝑞).

5.1. Tsunami Wavelet Analysis

On March 11, 2011, an earthquake of magnitude 9.0 occurred off the coast of Japan causing a tsunami no less than 10m high. Surface wave levels were detected by buoys operated by DART. To detect, analyze, store, and recover a tsunami waveform, we choose the wavelet frame Λ_𝐾𝑞=𝑞^𝑗𝐶_𝐾⋅𝐾_𝑞𝑞^𝑗𝑡−𝑘𝑏𝑗,𝑘∈ℤ,𝐶_𝐾≡^∞₀𝐾²_𝑞(𝑡)𝑑𝑡, (5.4) with 𝑞=1.5 and 𝑏=1.95. Then the application of 𝑊_𝐾𝑞 on tsunami data gives coefficients {𝑐_𝑗,𝑘} which can be plotted in a 𝑘 versus 𝑗 diagram. The largest values of |𝑐_𝑗,𝑘| indicate position (due to 𝑘) and narrowness (due to 𝑗). Only the largest 20 coefficients in magnitude, for the ranges −1≤𝑗≤3 and 1≤𝑘≤10, were used to obtain the fit on the bottom right of Figure 8. The normalized-RMS error NRMSE≡1max𝑦^wavelet_𝑖−min𝑦^wavelet_𝑖⋅1𝑁^𝑁_𝑖=1𝑦^data_𝑖−𝑦^wavelet_𝑖^21/2 (5.5) was computed to be NRMSE=16%. The bubble plots of the relative sizes of the coefficients are shown with the discrete time translation variable 𝑘 displayed horizontally, and the discrete frequency exponent 𝑗 displayed vertically.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8 

(a) Tsunami 46411 from DART, March 2011; (b) relative magnitude of coefficients for 3 scales {𝑞−1,1,𝑞} for 𝑞=1.5; (c) relative magnitude of coefficients for 5 scales {𝑞𝑗}3𝑗=−1; (d) 𝐾𝑞(𝑡) wavelet approximation with 20 largest coefficients; (e) largest magnitude of 𝐾𝑞 (𝑞=1.5, scale = 𝑞0=1, shift = 2⋅𝑏 = 3.9); (f) comparison between DART data and 𝐾𝑞 wavelet reconstruction.

5.2. Rogue Wavelet Analysis

On January 1, 1995, a rogue wave was detected on the Draupner platform in the North Sea. Surface wave heights were recorded using a laser-detection method [10]. Here we use the wavelet frameΛ_𝑞Cos=𝑞^𝑗𝐶_𝐶⋅_𝑞Cos𝑞^𝑗𝑡−𝑘𝑏∣𝑗,𝑘∈ℤ,𝐶_𝐶≡^∞_0𝑞Cos²(𝑡)𝑑𝑡, (5.6) with 𝑞=1.5 and 𝑏=2.44. Here, we used 30 coefficients, for the ranges 1≤𝑗≤4 and 1≤𝑘≤10, to obtain the fit on the bottom right. The normalized RMS error is 18%. See Figure 9 for the results of this analysis.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9 

(a) Rogue 1520 from Draupner, January 1995; (b) relative magnitude of coefficients for 3 scales {1, 𝑞, 𝑞2} for 𝑞=1.5; (c) relative magnitude of coefficients for 5 scales {𝑞𝑗}4𝑗=0; (d) 𝑞Cos wavelet approximation using largest 30 coefficients; (e) largest magnitude of 𝑞Cos (𝑞=1.5, scale = 𝑞2, shift = 5⋅𝑏 =12.2); (f) comparison between Draupner data and 𝑞Cos wavelet reconstruction.

6. Estimates for |_𝑞Cos(𝑞𝑡)−_𝑞Cos(𝑡)| and |_𝑞Sin(𝑞𝑡)−_𝑞Sin(𝑡)|

In this section, the estimates for the differences of 𝑞-advanced trigonometric functions are proven. First we record useful Fourier transform expressions. From (2.3) we have the Fourier transforms of _𝑞Cos(𝑡),_𝑞Cos(𝑞𝑡),_𝑞Sin(𝑡), and _𝑞Sin(𝑞𝑡), respectively, are given by ℱ_𝑞(𝑡𝜔)=2𝜇Cos)(^𝑞23𝑁_𝑞√12𝜋𝜃𝑞²;𝜔²,ℱ_𝑞𝜔)=1Cos𝑞𝑡(𝑞2𝜇_𝑞2³𝑁_𝑞√12𝜋𝜃𝑞²;𝑞²⁻¹𝜔²=𝜔²𝑞ℱ_𝑞Cos(𝑡)(𝜔),ℱ_𝑞(𝑡𝜔)=2𝜇Sin)(^𝑞23𝑁_𝑞√()2𝜋−𝑖𝜔𝜃𝑞²;𝜔²,ℱ_𝑞𝜔)=1Sin𝑞𝑡(𝑞2𝜇_𝑞2³𝑁_𝑞√2𝜋−𝑖𝜔/𝑞𝜃𝑞²;𝑞²⁻¹𝜔²=𝜔²𝑞²ℱ_𝑞Sin(𝑡)(𝜔), (6.1)(6.2)(6.3)(6.4) where the algebraic identity 𝜃(𝑞²;(𝑞²)^𝑛𝜔²)=(𝑞²)^{𝑛(𝑛+1)/2}(𝜔²)^𝑛𝜃(𝑞²;𝜔²), that follows from (2.6), was used to obtain (6.2) and (6.4). Further details are presented in [21].

Proof of Proposition 4.1. We first prove the estimate for the differences involving_𝑞Cos. From (6.1) and (6.2), one has Cos𝑞𝑡−_𝑞𝑞Cos(𝑡)=ℱ⁻¹ℱ_𝑞Cos𝑞𝑡−ℱ⁻¹ℱ_𝑞(𝑡=1Cos)√2𝜋^ℝ𝑒^𝑖𝜔𝑡1𝑞2𝜇_𝑞2³𝑁_𝑞√𝜔2𝜋𝜃𝑞²;²𝜔²−2𝜇_𝑞2³𝑁_𝑞√12𝜋𝜃𝑞²;𝜔²𝑑𝜔≤1√12𝜋𝑞2𝜇_𝑞2³𝑁_𝑞√2𝜋^ℝ𝜔²−𝑞𝜃𝑞²;𝜔²2𝜇𝑑𝜔=^𝑞23𝑁_𝑞22𝜋𝑞^∞₀𝜔²−𝑞𝜃𝑞²;𝜔²𝑑𝜔,=2𝜇𝜋𝑞^𝑞23𝑁_𝑞¹₀𝜔²−𝑞𝜃𝑞²;𝜔²𝑑𝜔+^∞₁𝜔²−𝑞𝜃𝑞²;𝜔²𝑑𝜔, (6.5)(6.6)(6.7) where the triangle inequality gives the inequality in (6.6), and the evenness of |𝜔²−𝑞|/𝜃(𝑞²;𝜔²) allows for reduction to integration over [0,∞) in (6.6).
The change of variables 𝜔=1/𝑢 is made on the first integral in (6.7), and the algebraic identity 𝜃(𝑞²;𝑢⁻²)=𝑢⁻²𝜃(𝑞²;𝑢²) is used to obtain ¹₀𝜔²−𝑞𝜃𝑞²;𝜔²𝑑𝜔=¹_∞𝑢⁻²−𝑞𝜃𝑞²;𝑢⁻²−𝑢⁻²𝑑𝑢=^∞₁𝑢⁻²−𝑞𝜃𝑞²;𝑢²𝑑𝑢. (6.8) Now (6.8) is used to reexpress the bound (6.7) as Cos𝑞𝑡−_𝑞𝑞(𝑡≤2Cos)𝜇𝜋𝑞^𝑞23𝑁_𝑞^∞₁𝜔⁻²−𝑞+𝜔²−𝑞𝜃𝑞²;𝜔²𝑑𝜔. (6.9)
From (2.9), we have that 1𝜃𝑞²;𝜔²≤𝑞^1/4|𝜔|𝑒^{(ln(|𝜔|))2/ln(𝑞)}𝜋ln𝑞−1⁻¹𝑞1=ln𝑞𝑞ln^−1/4|𝜔|⁻¹𝑒^{−(ln(|𝜔|))2/ln(𝑞)}√𝜋−𝑞,ln (6.10) for 1<𝑞<𝑒^𝜋. Deploying the bound (6.10) within the integral in the bound (6.9) gives Cos𝑞𝑡−_𝑞𝑞(𝑡𝑞1Cos)≤𝐹ln𝑞^∞1𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}𝑑𝜔, (6.11) where 𝐹𝑞≡2ln𝑞𝜇_𝑞2³𝑁_𝑞𝜋𝑞^5/41√𝜋−𝑞.ln (6.12) It follows from [21] that lim_𝑞→1+ln𝑞𝜇_𝑞2³𝑁_𝑞=𝜋2⇒lim_𝑞→1+𝐹𝑞=1√𝜋. (6.13)
We now show that |_𝑞Cos(𝑞𝑡)−_𝑞Cos(𝑡)| can be made arbitrarily small, independently of 𝑡, for all 𝑞>1 sufficiently close to 1⁺. In light of (6.13), this is accomplished by first showing the corresponding statement holds for the bracketed expression in (6.11).
Let 𝛼>0 be arbitrary, with 𝛼 to be specified later. The integral in 𝜔 over the interval [1,∞) in (6.11) is now subdivided into two integrals, the first over [1,𝑒^𝛼ln(𝑞)] and the second over [𝑒^𝛼ln(𝑞),∞). First, considering 𝜔 restricted to the interval [1,𝑒^𝛼ln(𝑞)], one has that 1ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}≤1𝑑𝜔ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻²−𝑞𝜔²+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}=1𝑑𝜔ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻¹−𝑞𝜔+𝜔−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}𝑑𝜔. (6.14) Now the function |𝜔⁻¹−𝑞𝜔| assumes its maximum value on [1,𝑒^𝛼ln(𝑞)] at the right endpoint 𝑒^𝛼ln(𝑞). In addition, for all 𝑞 sufficiently close to 1, the function |𝜔−𝑞𝜔⁻¹| assumes its maximum value on [1,𝑒^𝛼ln(𝑞)] at the right endpoint 𝑒^𝛼ln(𝑞). The latter statement follows since 𝑔(𝜔)=𝜔−𝑞𝜔⁻¹ is increasing, and one needs only to compare endpoints |𝑔(1)|=𝑞−1 and |𝑔(𝑒^𝛼ln(𝑞))|=𝑒^𝛼ln(𝑞)−𝑞𝑒^{−𝛼ln(𝑞)} to determine the larger value. An application of L'Hopital's rule gives that lim_𝑞→1⁺|𝑔(𝑒^𝛼ln(𝑞))|/|𝑔(1)|=∞. Thus, for 𝑞 near 1, on the interval [1,𝑒^𝛼ln(𝑞)], one has 𝜔⁻¹−𝑞𝜔+𝜔−𝑞𝜔⁻¹≤𝑒^{−𝛼ln(𝑞)}−𝑞𝑒^𝛼ln(𝑞)+𝑒^𝛼ln(𝑞)−𝑞𝑒^{−𝛼ln(𝑞)}=𝑞𝑒^𝛼ln(𝑞)−𝑒^{−𝛼ln(𝑞)}+𝑒^𝛼ln(𝑞)−𝑞𝑒^{−𝛼ln(𝑞)}=𝑞+1𝑒^𝛼ln(𝑞)−𝑒^{−𝛼ln(𝑞)}. (6.15) Thus, we bound the integral in (6.14) by the length of the interval times the bound (6.15) on the numerator. This gives 1ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻¹−𝑞𝜔+𝜔−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}≤1𝑑𝜔ln𝑞^{𝑒𝛼ln(𝑞)1}𝑞+1𝑒^𝛼ln(𝑞)−𝑒^{−𝛼ln(𝑞)}≤1𝑑𝜔𝑞𝑒ln𝑞+1^𝛼ln(𝑞)−𝑒^{−𝛼ln(𝑞)}𝑒^𝛼ln(𝑞)−1. (6.16) An application of L'Hopital's rule gives that lim_𝑞→1+𝑒^𝛼ln(𝑞)−𝑒^{−𝛼ln(𝑞)}ln𝑞=2𝛼, (6.17) which implies that in (6.16) we have lim_𝑞→1+𝑒𝑞+1^𝛼ln(𝑞)−𝑒^{−𝛼ln(𝑞)}𝑞𝑒ln^𝛼ln(𝑞)−1=2⋅2𝛼⋅0=0. (6.18) Combining (6.18) with (6.16) and (6.14) gives that lim_𝑞→1+1ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}𝑑𝜔=0. (6.19)
We next estimate the portion of the integral in (6.11) over the interval [𝑒^𝛼ln(𝑞),∞). 1ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}=1𝑑𝜔ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔⁻³−𝑞𝜔⁻¹+𝜔−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}≤1𝑑𝜔ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔⁻³+2𝑞𝜔⁻¹+𝜔𝑒^{(ln𝜔)2/ln(𝑞)}𝑑𝜔, (6.20) by bounding the integrals of the summands in (6.20). For each exponent 𝑘∈ℤ one has 1ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔^𝑘𝑒^{(ln𝜔)2/ln(𝑞)}1𝑑𝜔=ln𝑞^{∞𝑒𝛼ln(𝑞)}𝑒^𝑘ln𝜔𝑒^{−(ln𝜔)2/ln(𝑞)}=𝑞𝑑𝜔_𝑘2/4ln𝑞^{∞𝑒𝛼ln(𝑞)}𝑒^{−(1/ln(𝑞))[ln𝜔−𝑘ln(𝑞)/2]2}𝑑𝜔, (6.21) where a completion of squares gives (6.21). We use the following estimate from [21] ^∞_𝐶𝑒^{−𝐴[ln𝜔+𝐵]2}√𝑑𝜔≤𝜋2𝑒^{1/(4𝐴)−𝐵}√𝐴𝑒^{−𝐴[ln(𝐶)+𝐵−1/(2𝐴)]2}, (6.22) which is applied to (6.21) with 𝐴=1/ln(𝑞),𝐵=−𝑘ln(𝑞)/2, and 𝐶=𝑒^𝛼ln(𝑞) to obtain 𝑞^𝑘2/4ln𝑞^{∞𝑒𝛼ln(𝑞)}𝑒^{−(1/ln(𝑞))[ln𝜔−𝑘ln(𝑞)/2]2}≤𝑞𝑑𝜔_𝑘2/4𝑞√ln𝜋2𝑒^{ln(𝑞)/4+𝑘ln(𝑞)/2}𝑞𝑒1/ln^{−(1/ln(𝑞))[ln(𝑒𝛼ln(𝑞))−𝑘ln(𝑞)/2−ln(𝑞)/2]2}=𝑞^(𝑘+1)2/4√𝜋2𝑒^{−(1/ln(𝑞))[𝛼ln(𝑞))−(𝑘+1)ln(𝑞)/2]2}=𝑞^(𝑘+1)2/4√𝜋2𝑒^{−𝛼2+𝛼(𝑘+1)ln(𝑞)−(𝑘+1)2ln(𝑞)/4}=√𝜋2𝑒^−𝛼2𝑒^{𝛼(𝑘+1)ln(𝑞)}. (6.23) Combining (6.21) and (6.23) gives that, for each 𝑘∈ℤ, 1ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔^𝑘𝑒^{(ln𝜔)2/ln(𝑞)}√𝑑𝜔≤𝜋2𝑒^−𝛼2𝑒^{𝛼(𝑘+1)ln(𝑞)}. (6.24) Revisiting (6.20) and applying (6.24) now gives 1ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}𝑑𝜔≤1ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔⁻³+2𝑞𝜔⁻¹+𝜔𝑒^{(ln𝜔)2/ln(𝑞)}≤√𝑑𝜔𝜋2𝑒^−𝛼2𝑒^{𝛼(−2)ln(𝑞)}+2𝑞𝑒^{𝛼(0)ln(𝑞)}+𝑒^{𝛼(2)ln(𝑞)}. (6.25)(6.26) Since the expression in (6.26) approaches 2√𝜋𝑒^−𝛼2 as 𝑞→1⁺, by choosing 𝛼 sufficiently large one can make (6.26), and hence (6.25), arbitrarily small for 𝑞 sufficiently close to 1.
Now, let 𝜖>0 be given. By (6.11) we have for all 𝛼>0Cos𝑞𝑡−_𝑞𝑞(𝑡𝑞1Cos)≤𝐹ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}𝑞1𝑑𝜔+𝐹ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}𝑑𝜔. (6.27) One has from (6.13) lim_𝑞→1+𝐹𝑞√𝜋2𝑒^−𝛼2𝑒^{𝛼(−2)ln(𝑞)}+2𝑞𝑒^{𝛼(0)ln(𝑞)}+𝑒^{𝛼(2)ln(𝑞)}=2𝑒^−𝛼2. (6.28) Choose 𝐴>2 and then fix 𝛼>0 such that 𝐴𝑒^−𝛼2<𝜖/2. By (6.26) and (6.28), there exists 𝑞₁=𝑞₁(𝐴,𝛼,𝜖) such that for all 1<𝑞<𝑞₁ one has 𝑞10<𝐹ln𝑞^{∞𝑒𝛼ln(𝑞)}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}𝑞√𝑑𝜔≤𝐹𝜋2𝑒^−𝛼2𝑒^{𝛼(−2)ln(𝑞)}+2𝑞𝑒^{𝛼(0)ln(𝑞)}+𝑒^{𝛼(2)ln(𝑞)}<𝐴𝑒^−𝛼2<𝜖2. (6.29) For this value of 𝛼, by virtue of (6.19), one has lim_𝑞→1+𝐹𝑞1ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}=1𝑑𝜔√𝜋⋅0=0, (6.30) which in turn says that there is a 𝑞₂=𝑞₂(𝛼,𝜖) such that for all 1<𝑞<𝑞₂ one has 𝑞10<𝐹ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻²−𝑞+𝜔²−𝑞𝜔⁻¹𝑒^{(ln𝜔)2/ln(𝑞)}<𝜖𝑑𝜔2. (6.31) Thus, for 𝑄^𝜖_𝐶=𝑄(𝐴,𝛼,𝜖)=min{𝑞₁,𝑞₂}, and for all 1<𝑞<𝑄^𝜖_𝐶, applying (6.31) and (6.29) to (6.27) yields Cos𝑞𝑡−_𝑞𝑞(𝑡<𝜖Cos)2+𝜖2<𝜖, (6.32) independently of 𝑡∈ℝ. The proposition is now proven for_𝑞Cos.
The proof for _𝑞Sin parallels the above argument for _𝑞Cos, so we only outline it here, indicating points of slight differences. From (6.3) and (6.4), the analogue of (6.6) becomes (6.33) below Sin𝑞𝑡−_𝑞𝑞Sin(𝑡)=ℱ⁻¹ℱ_𝑞Sin𝑞𝑡−ℱ⁻¹ℱ_𝑞(𝑡≤1Sin)√12𝜋𝑞²2𝜇_𝑞2³𝑁_𝑞√2𝜋^ℝ𝜔³−𝑞²𝜔𝜃𝑞²;𝜔²=2𝜇𝑑𝜔_𝑞23𝑁_𝑞𝜋𝑞^2∞₀𝜔³−𝑞²𝜔𝜃𝑞²;𝜔²𝑑𝜔,=2𝜋𝑞²𝜇_𝑞2³𝑁_𝑞^∞₁𝜔⁻³−𝑞²𝜔⁻¹+𝜔³−𝑞²𝜔𝜃𝑞²;𝜔²𝑑𝜔, (6.33)(6.34) where a reciprocation change of variables on the integral on [0,1] converts to an integral on [1,∞), yielding (6.34) as the analogue of (6.9). Deploying the bound (6.10) to (6.34) now gives Sin𝑞𝑡−_𝑞𝑞(𝑡≤𝐹𝑞Sin)𝑞1ln𝑞^∞1𝜔⁻⁴−𝑞²𝜔⁻²+𝜔²−𝑞²𝑒^{(ln𝜔)2/ln(𝑞)}𝑑𝜔, (6.35) with 𝐹(𝑞) defined in (6.12). For the integral in 𝜔 in (6.35) over the interval [𝑒^𝛼ln(𝑞),∞), an application of (6.24) in conjunction with the triangle inequality lets us proceed directly to an analogue of (6.26), with key factor 𝑒^−𝛼2 still intact and the remaining factor approaching a constant as 𝑞→1⁺. On the other hand, the integral over the interval [1,𝑒^𝛼ln(𝑞)] in (6.35) proceeds as follows. One has 1ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻⁴−𝑞²𝜔⁻²+𝜔²−𝑞²𝑒^{(ln𝜔)2/ln(𝑞)}≤1𝑑𝜔ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻⁴−𝑞²𝜔⁻²𝜔²+𝜔²−𝑞²=1𝑑𝜔ln𝑞^{𝑒𝛼ln(𝑞)1}𝜔⁻²−𝑞²+𝜔²−𝑞²≤1𝑑𝜔𝑞𝑒ln^{−2𝛼ln(𝑞)}−𝑞²+𝑒^{2𝛼ln(𝑞)}−𝑞²𝑒^𝛼ln(𝑞)−1=1ln𝑞−𝑒^{−2𝛼ln(𝑞)}+𝑒^{2𝛼ln(𝑞)}𝑒^𝛼ln(𝑞)−1, (6.36)(6.37) where (6.36) and (6.37) hold for 𝑞 sufficiently near 1⁺. Since lim_𝑥→0[(−𝑒^−2𝛼𝑥+𝑒^2𝛼𝑥)/𝑥]=4𝛼, we have the limit as 𝑞→1⁺ of expression (6.37) is 0. The remainder of the proof is now entirely parallel to that of the _𝑞Cos case.

Acknowledgments

The rogue data from the Draupner platform was graciously supplied by Dr. Sverre Haver from Statoil, Norway. The tsunami data came from DART buoys and was graciously provided by Dr. George Mongov at NOAA. The DART data is kept at the National Geophysical Data Center/World Data Center (NGDC/WDC) Historical Tsunami Database, Boulder, CO, USA. Figure 1 profile was permitted for our use here by Dr. Benny Grosen. The authors are grateful for their assistance. Finally, they thank the referee for suggestions for improving the paper.