Available online at www.sciencedirect.com Coastal Engineering 55 (2008) 93–111 www.elsevier.com/locate/coastaleng Remote sensing of breaking wave phase speeds with application to non-linear depth inversions ⁎ Patricio A. Catálan ,1, Merrick C. Haller Coastal & Ocean Engineering Program, School of Civil and Construction Engineering, Oregon State University, 220 Owen Hall, Corvallis, OR 97331, USA Received 29 August 2006; received in revised form 7 July 2007; accepted 5 September 2007 Available online 22 October 2007 Abstract A number of existing models for surface wave phase speeds (linear and non-linear, breaking and non-breaking waves) are reviewed and tested against phase speed data from a large-scale laboratory experiment. The results of these tests are utilized in the context of assessing the potential improvement gained by incorporating wave non-linearity in phase speed based depth inversions. The analysis is focused on the surf zone, where depth inversion accuracies are known to degrade significantly. The collected data includes very high-resolution remote sensing video and surface elevation records from fixed, in-situ wave gages. Wave phase speeds are extracted from the remote sensing data using a feature tracking technique, and local wave amplitudes are determined from the wave gage records and used for comparisons to non-linear phase speed models and for non- linear depth inversions. A series of five different regular wave conditions with a range of non-linearity and dispersion characteristics are analyzed and results show that a composite dispersion relation, which includes both non-linearity and dispersion effects, best matches the observed phase speeds across the domain and hence, improves surf zone depth estimation via depth inversions. Incorporating non-linearity into the phase speed model reduces errors to O(10%), which is a level previously found for depth inversions with small amplitude waves in intermediate water depths using linear dispersion. Considering the controlled conditions and extensive ground truth, this appears to be a practical limit for phase speed-based depth inversions. Finally, a phase speed sensitivity analysis is performed that indicates that typical nearshore sand bars should be resolvable using phase speed depth inversions. However, increasing wave steepness degrades the sensitivity of this inversion method. © 2007 Elsevier B.V. All rights reserved. Keywords: Phase speeds; Surf zone; Breaking waves; Depth inversions; Remote sensing 1. Introduction specification of phase speed in this region ambiguous. In addition, most phase speed models are derived explicitly for In the coastal zone, spatial variations in the sea bottom either breaking or non-breaking waves. Thus, for domains that influence both the direction and speed of propagation of surface span the shoaling and breaking zones they are not universally gravity waves. In the modeling of nearshore areas, wave phase applicable and empirical formulations must often be called upon speeds and directions are important parameters because wave- (e.g. Hedges, 1976; Kirby and Dalrymple, 1986). averaged nearshore circulation models require their explicit Surface wave phase speed is also an important remote sensing prediction for the determination of radiation stress forcing. observable. For example, if a functional relationship between water Typical existing models for wave phase speeds are based on a depth and phase speed is specified, then remote sensing data can be prescribed wave form such as sinusoidal, cnoidal, etc.; but, in used to determine bathymetry through depth inversion techniques areas near the onset of wave breaking, wave shape is of non- (e.g. Williams, 1946; Bell, 1999; Stockdon and Holman, 2000). permanent form and can change very quickly, which makes the Due to the large footprint of remote sensing observations and considering the comparative difficulties and expense of collecting data by in-situ means, this is a potentially powerful technique for ⁎ Corresponding author. retrieving bathymetry. However, it is clear that in realistic E-mail addresses: [email protected], [email protected] (P.A. Catálan), [email protected] (M.C. Haller). situations the functional relationship between water depth and 1 Also at: Departmento de Obras Civiles, Universidad Tecnica Federico Santa phase speed is complex and dependent on wave non-linearity, Maria, Valparaiso, Chile. which is a more difficult quantity to observe remotely. In addition, 0378-3839/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2007.09.010 94 P.A. Catálan, M.C. Haller / Coastal Engineering 55 (2008) 93–111 it is disadvantageous to require separate phase speed models for the assumptions made in the description of the underlying wave regions outside and inside the surf zone, since this likely requires motion. additional gymnastics to determine where to apply each model for a given domain and data set. Aarninkhof et al. (2005) have pursued 2.1.1. Linear theory an alternative track of relating breaking-induced dissipation to the The simplest case is from linear wave theory, under the bathymetry. However, the approach still relies on a characterization assumption of small wave amplitude and locally horizontal of the energy flux, which is related to the phase speed. Hence, since bottom. The resulting free surface can be expressed as η=A exp the accuracy of depth inversions is directly tied to the accuracy of {i(kx−σt)}, where k=2π/L is the wavenumber, σ=2π/T is the phase speed model, there is clearly a need for a general surface the radian frequency, A is the wave amplitude and the quantity wave phase speed model that is applicable from the shoaling zone i(kx−σt) is the phase. Choosing the wavelength, L, as the through the surf zone to the shoreline. Yet, it is not clear that such a characteristic distance (l) the wave period T is then the charac- model exists. teristic time (τ). For linear waves, c, σ, and k are related by The accuracy of depth inversions based on remote measure- means of the linear dispersion relation ments is dependent on two criteria: 1) the ability of the remote r2 L2 g sensor to measure phase speed and wave amplitude, and 2) the c2 ¼ ¼ ¼ tanhðÞkh ; ð2Þ applicability of the chosen phase speed (and depth inversion) k2 T 2 k model to the given wave conditions. The purpose of this paper is where h is the water depth, g is the gravitational acceleration, to directly compare phase speed models that include non-linear and k is the magnitude of the wavenumber vector k. Here, c is and dispersive effects with remotely sensed phase speed defined relative to a fixed frame of reference. Despite its measurements. It is also to study the potential improvement in simplicity, Eq. (2) has proven useful in a number of cases. For depth inversion when measured finite amplitude effects are example, in intermediate depths and moderate wave heights, the included in the relationship between the measured phase speed full linear dispersion relation (including currents) has been used and the local water depth. The analysis utilizes a combination of to estimate water depths and mean currents with errors as low as remotely sensed video intensity data and in-situ measurements of O(5%) (e.g. Dugan et al., 2001; Piotrowski and Dugan, 2002). free surface elevations. With respect to previous depth inversion However, we are mostly concerned here with waves in the studies, the present data set provides remote sensing data at much surf zone, and in shallow water (khbπ/10) Eq. (2) reduces to higher resolution than previously considered, which reduces the the simple form amount of inherent smoothing that often occurs in field situations. pffiffiffiffiffiffiffi In addition, the laboratory setting allows us to control the wave c ¼ gh; ð3Þ conditions and to measure them with high accuracy throughout the surf zone. Finally, we concentrate on the surf zone (though not which shows a direct relation between the local water depth and necessarily shallow water) because the accuracy of depth the speed. This makes it very simple to use for depth inversion; inversions has been found to significantly degrade in this region. yet, comparisons with measured data have shown that this linear This paper is organized as follows: in Section 2 we perform a shallow water approximation underpredicts the observed phase comprehensive review of existing phase speed models. Section 3 speeds, both in the field (e.g. Inman et al., 1971; Thornton and describes the measurement techniques and experimental condi- Guza, 1982; Stockdon and Holman, 2000; Holland, 2001) and tions used in this study. Phase speed results and the application in the laboratory (e.g. Svendsen and Buhr Hansen, 1976; of selected phase speed models to depth inversions are given in Svendsen et al., 1978; Stive, 1980; Stansell and MacFarlane, Section 4. Section 5 provides a discussion of possible error 2002). In some of these laboratory cases a limited degree of sources and an analysis of the sensitivity of phase speed-based non-linearity was incorporated by including the measured wave depth inversions to the amplitude of a given bottom perturbation, setup in the water depth used in Eq. (3) (Svendsen et al., 1978; such as a sand bar. Conclusions are given in Section 6. Stive, 1980), but the overall behavior was still an under- prediction of the phase speeds. In addition, using on a large 2. Phase speeds amount of field data from a cross-shore array of pressure sensors, Holland (2001) showed that, in shallow water, errors in 2.1. Models for non-breaking waves estimated depths using the linear dispersion relation commonly exceeded 50% and were correlated with the offshore wave The speed of propagation (phase speed or celerity) of surface height. This was a clear indication of the importance of finite gravity waves, c, can be defined for a surface of permanent form amplitude effects for depth inversions.