## Abstract

A method is developed to estimate nonlinear internal wave (NLIW) vertical displacement, propagation direction, and propagation speed from single moored acoustic Doppler current profiler (ADCP) velocity observations. The method is applied to three sets of bottom-mounted ADCP measurements taken on the continental slope in the South China Sea in 2006–07. NLIW vertical displacement is computed as the time integration of ADCP vertical velocity observations corrected with the vertical advection of the background flow by the NLIW. NLIW vertical currents displace the background horizontal current and shear by ~150 m. NLIW propagation direction is estimated as the principal direction of the wave-induced horizontal velocity vector, and propagation speed is estimated using the continuity equation in the direction of wave propagation, assuming the wave’s horizontal spatial structure and propagation speed remain constant as the NLIW passes the mooring, typically *O*(10 min). These NLIW properties are estimated simultaneously and iteratively using the ADCP velocity measurements, corrected for their beam-spreading effect. In most cases, estimates converge to within 3% after four iterations. The proposed method of extracting NLIW properties from velocity measurements is confirmed using NLIWs simulated by the fully nonlinear Dubreil–Jacotin–Long model. Estimates of propagation speed using the ADCP velocity measurements are also in good agreement with those calculated from NLIW arrival times at successive moorings. This study concludes that velocity measurements taken from a single moored ADCP can provide useful estimates of vertical displacement, propagation direction, and propagation speed of large-amplitude NLIWs.

## 1. Introduction

Nonlinear internal waves (NLIWs) are often generated as a result of tidal current interaction with topography in stratified water and are ubiquitous in straits and the coastal ocean. These waves may transport energy over hundreds of kilometers, crossing basins and shelf regions, and dissipate far from their generation sites. Vertical isopycnal displacement *η*, propagation direction *θ*, and propagation speed *C* are fundamental NLIW properties that are useful for identifying their generation and dissipation sites and for estimating their intensity, energy, and energy flux.

Shipboard and moored acoustic Doppler current profiler (ADCP) measurements capture velocity fluctuations associated with NLIWs (Chang et al. 2008; Lien et al. 2010, manuscript submitted to *J. Phys. Oceanogr.*; Alford et al. 2010; Ramp et al. 2010). NLIWs may have vertical currents as large as *O*(0.1 m s^{−1}) and are often distinct in ADCP vertical velocity records. ADCP velocity measurements have been used to estimate NLIW vertical displacements (Lien et al. 2005; Moum and Smyth 2006), propagation direction (Scotti et al. 2005; Mirshak and Kelley 2009), and propagation speed (Scotti et al. 2005; Moum and Smyth 2006). Conventionally, vertical displacement is estimated by integrating ADCP vertical velocity in time, and propagation direction is estimated as the principal direction of the wave-induced horizontal velocity (Lien et al. 2005; Moum and Smyth 2006). However, these conventional methods do not consider the advection effect of the background flow by NLIWs.

Moum and Smyth (2006) estimate the wave speed *C* by *C* = −∂* _{t}u*′/∂

*′, where*

_{z}w*u*′ and

*w*′ are the horizontal and vertical velocity components associated with NLIWs and ∂

*and ∂*

_{t}*are the time and vertical derivatives, respectively. Scotti et al. (2005) indicate that the ADCP beam-spreading effect could seriously distort the observed beam-averaged velocity for nonhydrostatic NLIWs: that is, the large aspect ratio. They proposed a phase-lagged beam-to-earth transformation to correct the beam-spreading effect, and this method is applied to velocity measurements taken from an ADCP platform fixed to the ocean floor that is not disturbed by NLIW passage. NLIW propagation direction and speed are required for the transformation and are estimated using the temporal differences of echo intensity signals arriving at four different ADCP beams.*

_{z}Here, we propose a scheme to estimate the vertical displacement, propagation direction, and propagation speed of NLIWs using velocity measurements taken from a single moored ADCP platform. The estimation process is iterative and the beam-spreading correction, depending on the propagation speed and direction, is performed at each iteration step. The method is verified using simulated NLIW data generated by a fully nonlinear Dubreil–Jacotin–Long (DJL) model. Propagation speed estimates are also compared with those computed from in situ NLIW arrival time measurements taken by three successive moorings on the Dongsha Plateau in the South China Sea (SCS; Fig. 1).

The paper is organized as follows: In section 2, we present the methods used to estimate the vertical displacement *η*, the propagation direction *θ*, and the propagation speed *C*. In section 3, we use simulation experiments to gain insight into the effects of background flow, beam spreading, and random noise of measurements on our estimates of NLIW properties and to validate our iteration scheme. In section 4, we apply our iteration scheme to in situ ADCP measurements taken in the SCS and confirm our estimates of NLIW propagation speed by comparing them with measurements from other independent methods. A summary is presented in section 5.

## 2. Data and methods

### a. Oceanic data

Three bottom-mounted, upward-looking, 75-kHz Long Ranger ADCPs (LR1, LR2, and LR3) were deployed on the continental slope in the northern SCS in 2006 and recovered in 2007. The moorings were aligned in an east–west direction along 21°05′N on the eastern slope of the Dongsha Plateau, roughly along the prevailing westward propagation direction of NLIWs (Fig. 1). These locations were chosen to catch the evolution of NLIWs on the upper flank of the continental slope (Lien et al. 2010, manuscript submitted to *J. Phys. Oceanogr.*). The total measurement period was ~11 months at LR1 and LR2 and ~5 months at LR3 (Table 1 and Fig. 2). From east to west, the distance was ~8.5 nautical miles between LR1 and LR2, and ~17 nautical miles between LR2 and LR3. The water depths at LR1, LR2, and LR3 were ~600, ~430, and ~350 m, respectively. Mooring locations, water depths, and ADCP setup are summarized in Table 1.

The ADCP was mounted inside a syntactic float at a depth of ~10 m above the bottom to take near-full-depth velocity measurements. The blanking distance of the ADCP velocity measurements from the transducer head was about 7 m. ADCP measurements at 10% of the total water depth beneath the sea surface were contaminated by sea surface reflection and were excluded from analysis.

Satellite images frequently capture surface signatures of westward-propagating NLIWs in the northern SCS (Fig. 1). In situ observations (Yang et al. 2004; Ramp et al. 2004; Lien et al. 2005; Alford et al. 2010) suggest that NLIWs appear at tidal periodicity with amplitudes modulated at a fortnightly tidal cycle. Several possible generation mechanisms have been proposed. NLIWs might be generated within the Luzon Strait by currents flowing over rough topography (Bole et al. 1994; Liu et al. 1998; Ramp et al. 2004). Alternatively, NLIWs might evolve from internal tides (Zhao et al. 2004, Lien et al. 2005; Chao et al. 2007). Strong internal tides are generated in Luzon Strait, propagate into the SCS, are amplified by the shoaling continental slope near Dongsha Island, and then evolve into high-frequency NLIWs (Lien et al. 2005). However, further discussion of the generation mechanism of NLIWs is beyond the scope of the present study.

Zonal velocity fluctuations at 100-m depth at the three mooring sites exhibit clear fortnightly modulation (Fig. 2). Packets of large-amplitude NLIWs arrive predominantly during the semidiurnal period (Fig. 2), with amplitudes modulated fortnightly. The leading wave has a typical vertical displacement of >100 m and a maximum horizontal current speed of >1 m s^{−1}, westward in the upper 200 m and eastward below 200 m (Fig. 2d). The maximum speed of the background flow is 0.3–0.5 m s^{−1}, much weaker than that of the NLIW current. NLIWs were weakest between mid-January and mid-February 2007, presumably because of the mesoscale modulation of internal tidal generation and propagation. This topic is currently under investigation.

### b. Separating NLIW velocities from background flow

Both earth- and wave-oriented coordinates are used in the following analysis. In the earth coordinate system, the *x* and *y* components represent the zonal (east–west) and meridional (north–south) components, respectively. In the wave coordinate system, *x*′ and *y*′ represent the along-wave and across-wave components of the NLIW propagation direction, respectively. The *z* component is vertical in both coordinate systems. Velocities in earth and wave coordinates are expressed as (*u*, *υ*, *w*) and (*u*′, *υ*′, *w*), respectively. The background velocities in earth and wave coordinates are (*U*, *V*, *W*) and (*U*′, *V*′, *W*), respectively.

We consider an NLIW with a constant propagation speed *C* and constant propagation direction *θ* in a fluid with the background density and background horizontal velocity [*U*′(*z*), *V*′(*z*)]. NLIW vertical displacement *η*(*x*′, *z*) is defined as

The streamfunction of the wave-induced motion *ψ* is defined following Stastna and Lamb (2002),

where The streamfunction of the total flow field *ψ ^{t}* is the sum of the streamfunction of the wave-induced motion

*ψ*and the streamfunction of the background flow

*ψ*(

*z*)

^{b}: that is,

*ψ*=

^{t}*ψ*+

*ψ*(

^{b}*z*). The along-wave component of total velocity

*u*′ and the vertical velocity

*w*are functions of the streamfunction [i.e., ], yielding

We assume NLIW properties do not vary along the NLIW crest (*y*′ component). Because NLIWs have no across-wave velocity component, the variation of the across-wave velocity *υ*′ is solely due to the background flow vertically advected by the NLIW,

Equations (3)–(5) express the total particle velocity in terms of the vertical displacement and propagation velocity of NLIWs and of the background flow. Our purpose here is to describe the method to extract NLIW vertical displacement, propagation speed, and propagation direction using observed velocity measurements from ADCP mooring and to investigate the importance of the modulation by the background flow.

### c. Calculating NLIW properties from moored ADCP data

We consider NLIWs to be at steady state when passing the moorings, generally at *O*(10 min). Therefore, *η* is a function of . The NLIW vertical displacement can be obtained by substituting into (4) and applying the time integration,

Note that the effect of background flow is included in (6) as *U*′/*C*. The NLIW propagation direction can be derived by transforming (5) to earth coordinates, yielding

Equation (8) is simply the continuity equation in the wave propagation direction. Note that (6)–(8) express the vertical displacement, the propagation direction, and the propagation speed of NLIWs in terms of velocity.

In the absence of the background flow, the expression of *η* is reduced to the generally used form . For a typical SCS NLIW propagating at 2 m s^{−1} in a ~0.3 m s^{−1} background flow, vertical displacement estimated using yields a ~17% error.

Conventionally, the NLIW propagation direction is estimated as , which is the approximation derived from (7). For vertically uniform background flow, the two expressions are identical. However, for large-amplitude NLIWs in a sheared background flow, the approximation may lead to significant errors in estimating the propagation direction. The error in the estimate of *θ* also leads to an error in the estimate of propagation speed (8). Therefore, it is important to use the complete expressions in (6)–(8), including the effect of background velocity and its vertical displacement by NLIWs. Equations (6)–(8) are coupled and nonlinear and are solved via iterations as discussed in section 3d.

## 3. NLIW simulations

Fully nonlinear, steady-state NLIWs can be described by the DJL equation (Stastna and Lamb 2002),

where *N*(*z* − *η*) is the displaced background buoyancy frequency. The boundary conditions are *η*(*x*, 0) = *η*(*x*, −*H*) = 0, where *H* is water depth and . The DJL equation, Eq. (9), can be solved numerically by variational algorithm, minimizing the kinetic energy required to maintain NLIWs with a prescribed available potential energy (Turkington et al. 1991; Stastna and Lamb 2002).

In the following analysis, we present an idealized fully nonlinear, steady-state DJL NLIW that satisfies the typical density profile and background horizontal current at LR1 (Fig. 3). At LR1, the NLIW propagates at a mean direction *θ* = 150° (0° at true east, counterclockwise from east), and the typical background current is at 175°, with a maximum current speed of 0.35–0.4 m s^{−1} in the upper layer (Fig. 3a). The buoyancy frequency squared at LR1 usually has a maximum value of 4 × 10^{−4} s^{−2} at ~50 m (Fig. 3b). The background current is projected onto the along-wave and across-wave components. The along-wave component of the background current and the buoyancy frequency are used in (9) to obtain DJL solutions. The total current velocity along the NLIW propagation direction *u*′ and perpendicular to the NLIW propagation path *υ*′ are obtained through (3) and (5), respectively. The vertical current velocity *w* is obtained through (4). The current velocity field (*u*′, *υ*′, *w*) on the wave coordinate can be transformed to the earth coordinate (*u*, *v*, *w*) using the known propagation direction *θ*.

### a. Background flow effect

We chose the DJL solution with *C* = −2.21 m s^{−1} as our test wave. The horizontal and vertical velocities of this test wave are shown in Figs. 4a,d. The along-wave component of velocity in (3) includes the strain term *Cη _{z}* associated with the NLIW (Fig. 4b) and the effect of the background flow (Fig. 4c). Prior to the arrival of the NLIW (i.e., the initial condition), the background current represents the total flow field. As the NLIW approaches, the background isopycnal surface, as well as the background current, is displaced by the NLIW. The vertical velocity

*w*expressed in (4) includes the NLIW term

*Cη*(Fig. 4e) and the effect of the background flow (Fig. 4f). The latter represents the error of the estimate of the NLIW vertical displacement using . The error could be as large as ~15 m for the test wave (Fig. 3a): that is, 10% of the maximum vertical displacement. This error increases with the magnitude of the background current, shear, and NLIW amplitude.

_{z}This test wave is used to validate the method and the iteration scheme for extracting the NLIW propagation speed and direction and vertical displacements using velocity observations (section 2c). The propagation direction and speed of the wave are computed using (7) and (8) (Figs. 5a,b, respectively). The estimates of *C* = −2.21 m s^{−1} and *θ* = 150° are identical to those of the test wave, by construction. In estimating the propagation direction, ignoring the effect of the vertical displacement of the background current, i.e. , results in the estimate being biased by 3°. Ignoring the background current entirely, i.e. , leads to a bias of 5° on the propagation direction. The former is better but inaccurate.

### b. ADCP beam-spreading effect and correction

The 75-kHz Long Ranger ADCP used on moorings LR1, LR2, and LR3 has four beams of angle *φ* that are 20° from the vertical and separated 90° horizontally. The horizontal separation between beams (i.e., the beam spreading) increases with the distance from the transducer heads. Vertical profiles of horizontal and vertical velocity components are computed using the along-beam velocities observed by the four beams. Therefore, the average horizontal spatial scale of ADCP velocity measurements increases with the distance from the transducer head, referred to as the “beam-spreading effect.”

Here, we use the test wave to examine the beam-spreading effect on the velocity measurements of NLIWs and on estimates of NLIW properties. We project the current velocity (*u*, *υ*, *w*) of the DJL simulated test wave onto four ADCP beams at a sampling rate identical to that on mooring LR1, thus introducing the beam-spreading effect. The simulated ADCP velocity in earth coordinates is computed as

Here, *b*_{1}, *b*_{2}, *b*_{3}, and *b*_{4} are along-beam components of velocity on four ADCP beams. For the sake of simplicity, we assume that *b*_{1} and *b*_{2} are aligned east–west and *b*_{3} and *b*_{4} are aligned north–south.

Comparison between Figs. 6a,d shows three distorted features due to the beam-spreading effect: 1) the nodal point depth is raised from ~260 to ~220 m, 2) a maximum eastward velocity core exists at 300–400 m, and 3) an M-shaped westward velocity feature is present at depths between 200 and 300 m in the distorted velocity field (Fig. 6a). The vertical displacement, estimated as , yields an error up to ~30 m (Fig. 6c), 20% of the true NLIW isopycnal displacement.

Scotti et al. (2005) proposed a modified phase-lagged transformation scheme to correct the beam-spreading effect,

where , , , , , , and *d* is the vertical distance from the transducer head. This method requires knowledge of the NLIW propagation speed and direction. The phase-lagged beam-to-earth transformation corrects the beam-spreading effect and recovers the test wave (Figs. 6d–f).

We compute the propagation speed of the test wave using beam-spreading distorted velocities according to (8). The beam spreading introduces noise into the estimate (cf. Figs. 5a, 7a) and results in an estimate of propagation speed *C* = 3.22 m s^{−1}, which is 46% faster than that of the test wave. The scatterplot between ∂* _{z}w* and ∂

*′ shows significant deviations from the regression line in the upper 250 m, where the beam-spreading effect is strong (Fig. 7a). The beam-spreading effect does not introduce significant error into the estimate of propagation direction for the test wave (cf. Figs. 5b–d with Figs. 7b–d).*

_{t}u### c. Random noise effect in ADCP measurements

The random noise intrinsic to the ADCP velocity measurements may affect the estimate of propagation direction [i.e., the regression slope between *u* − *U*(*z* − *η*) and *υ* − *V*(*z* − *η*)] and the estimate of the propagation speed [i.e., the regression slope between ∂* _{t}u*′ and ∂

*′]. To examine the effects, we added a random noise value of 0.05 m s*

_{z}w^{−1}to the test wave’s horizontal velocity and 0.025 m s

^{−1}its vertical velocity. Using the standard linear regression method, our estimates of the NLIW’s propagation direction and wave speed have errors of ~3° and ~0.5 m s

^{−1}, respectively.

The standard linear regression analysis is inappropriate when both fitted (*y* axis) and reference (*x* axis) variables are subject to random errors. On the other hand, Deming regression analysis, also called the errors-in-variables regression, accounts for measurement errors in both fitted and reference variables (Deming 1943). Using this method, we recovered the true propagation speed and direction of the test wave.

### d. An iterative method

An iterative method is used to estimate *η*, *θ*, and *C* and to correct the ADCP beam-spreading effect from the velocity measurements. The iteration begins from the estimate of the vertical displacement (6) with the initial estimate . The propagation direction *θ* is estimated using (7). The wave speed *C* is estimated using (8) after rotating the coordinate system to (*x*′, *y*′), using the estimate of *θ*. The estimated *θ* and *C* are used to perform the phase-lagged beam-to-earth transformation (11) to correct for the beam-spreading effect. These estimates of *η*, *θ*, *C*, and beam-spreading-effect corrected velocities are used to repeat the above procedures. The iteration analysis continues until the estimates of *η*, *θ*, and *C* converge, typically after the fourth iteration (Fig. 8). The estimate of the propagation speed is 2.26 m s^{−1}, an error of ~2%. The estimate of the propagation direction is almost identical to that of the test wave. Estimates of vertical displacement and horizontal and vertical velocities from our iteration analysis are in good agreement with those of the test wave. The root-mean-square of the difference between estimates of vertical displacement and that of the test wave, averaged over the wave, is 6 m. Averaged over the wave event, the root-mean-square difference between estimates of the zonal, meridional, and vertical components of NLIW velocity and those of test wave are 0.055, 0.03, and 0.025 m s^{−1}, respectively.

This test wave analysis shows that 1) the background flow may lead to errors in the estimates of *η* and *θ* by ~15 m and ~5° for the test wave; 2) the beam-spreading effect leads to the distortion of the measured velocities of the test wave and produces serious errors in estimates of *η* and *C* (i.e., ~30 m and ~2 m s^{−1}); and 3) the iteration scheme to estimate *η*, *C*, and *θ* using (6), (7), (8), and (11) is effective.

### e. Temporal sampling effect in ADCP measurements

As the NLIW passes the mooring location, the temporal sampling number *N _{s}* could have an impact on the estimate of properties because the sparse measurements are expected to distort the flow field. The sampling interval of ADCP is fixed (Table 1), so

*N*is primarily determined by the propagation speeds and horizontal scales of NLIW. The speed of NLIW propagation correlates negatively to

_{s}*N*. To assess the effect, the iterative method mentioned above was applied to the temporally resampled test wave to estimate

_{s}*C*and

*θ*. We found that the scarcity of

*N*results in the propagation speed of NLIW being underestimated, possibly because

_{s}*u*′(

*t*) is oversmoothing because of sparse sampling. The estimate error decreases when

*N*increases, and the significant estimate errors of

_{s}*C*(e.g., >25% of the true values) occur when

*N*≤ 3. The estimate errors of

_{s}*C*are <8% of the true values when

*N*≥ 6 and then decrease below 4% of the true values when

_{s}*N*≥ 7. However, the effect of temporal sampling does not have a significant impact on the estimate of

_{s}*θ*. The estimate errors of

*θ*are <3% of the true values if

*N*≥ 3. We will further discuss the effect of

_{s}*N*in the next section.

_{s}## 4. NLIW observations in the South China Sea

### a. Iterative method applied to ADCP velocity measurements

NLIWs are active in the northern SCS, with horizontal velocities as strong as ~2 m s^{−1} and amplitudes > 100 m. The background flow is primarily associated with internal tides with a maximum current speed of 0.3–0.5 m s^{−1}. At LR1, LR2, and LR3, the separation between ADCP beams is ~1 m near the transducer head and reaches to ~400, ~300, and ~200 m near the surface, respectively. The typical NLIW horizontal scale is ~1 km (Chang et al. 2008). The assumption of a spatially homogenous velocity field, as in the standard ADCP data processing algorithm, is clearly inappropriate for these small-scale NLIWs. The beam-spreading effect is significant for our velocity measurements.

We compute spectra of *u*, *υ*, and *w* velocities and determine the spectral noise floor. The random noise for *u* and *υ* is 0.05–0.06 m s^{−1} and for *w* is 0.01–0.02 m s^{−1}. These random noise estimates are used for the Deming regression analysis.

Scotti et al. (2005) proposed a method to correct the beam-spreading effect in ADCP measurements. This method is straightforward if the ADCP platform is fixed. During NLIW events in the SCS, the passing waves jolted our ADCP platform, mounted in a buoyant float. The ADCP pitch and roll remained within 4° and may be ignored. However, the ADCP platform often experienced strong rotational forces with heading changes of nearly 90° during the NLIW passage.

We identified 358 NLIW packets passing the LR1 and LR2 moorings. The LR3 mooring was deployed for a shorter period and only captured 151 NLIW packets. For the following analysis, we identified leading waves with ADCP heading fluctuations within 20° and maximum horizontal speeds > 0.5 m s^{−1}. A total of 205 leading waves passed LR1 and LR2 sequentially and satisfied the criteria, of which 70 also sequentially passed LR2 and LR3 and were also selected.

An example wave event is shown in Fig. 9, with a maximum NLIW horizontal velocity of ~1.7 m s^{−1} and a maximum vertical displacement of ~130 m. The background flow velocity is ~0.3 m s^{−1} (Fig. 9a). The structure of the observed zonal velocity field shows features typical of beam-spreading distortion (cf. Fig. 6a). Given estimates of the propagation speed and the direction of the NLIW, the beam-spreading effect is corrected using (11) and the true velocity field is recovered (Fig. 9b).

The iterative method applied to ADCP velocity observations at LR1, LR2, and LR3 yields estimates of *η*, *θ*, and *C* (Fig. 10). The analysis converges after the fourth iteration for estimates of NLIW *θ*, *C*, and velocity. The vertical displacements converge after the ninth iteration, and the estimated vertical displacement agrees well with the water scatter fluctuation induced by the NLIW as shown by ADCP echo intensity (Fig. 9c). Propagation speed is estimated at −1.83 m s^{−1} (Fig. 11c), close to the maximum westward current speed. Without the beam-spreading correction, the propagation speed estimate is −3.91 m s^{−1} (Fig. 11a) and is much noisier. The propagation direction is 154° using the beam-spreading corrected velocities (Fig. 11d) and is 151° without the correction (Fig. 11b). Again, as seen in the simulated test wave, the beam-spreading has a minimal effect on the estimate of propagation direction.

Moored ADCP velocity observations show large variations in the slope between ∂* _{z}w* and ∂

*′, revealing significant scattering (Figs. 11a,c), most of which appears in the upper 100–200 m, where the beam-spreading effect is stronger. The iteration analysis yields convergent results (Figs. 11b,d); the regression slope between ∂*

_{t}u*and ∂*

_{z}w*′ for propagation speed and between*

_{t}u*u*−

*U*(

*z*−

*η*) and

*υ*−

*V*(

*z*−

*η*) for propagation direction are much better defined. Through the iteration procedure, the correlation between ∂

*and ∂*

_{z}w*′ increases from 0.45 to 0.86 and the correlation between*

_{t}u*u*−

*U*(

*z*−

*η*) and

*υ*−

*V*(

*z*−

*η*) increases from 0.84 to 0.92 (cf. Figs. 11a,c and cf. Figs. 11b,d). The iteration procedure reduces the error of the propagation speed estimate by ~76% and for the of the propagation direction by ~40%.

The average half-amplitude full width of the NLIWs is ~1 km (Chang et al. 2008), and the full width of NLIWs is ~1.5 km. At LR1, the sampling intervals are 90 s (Table 1). Accordingly, *N _{s}* is ~7 and ~8 for NLIWs propagating with typical speeds of 2 and 2.5 m s

^{−1}, respectively. The simulation from the previous section suggests that the estimate errors for

*C*can be expected to be <4% of the true value, and the maximum estimate errors of

*C*are ~0.1 m s

^{−1}. At LR2 and LR3, the sampling intervals are 60 or 30 s (Table 1). The estimate errors of

*C*are

*O*(0.01 m s

^{−1}). For this reason, we did not correct the temporal sampling effect here. Otherwise, the sampling interval effect is also related to the wave shape and the observation points where the wave is sampled. The estimate errors calculated from test wave simulated from the DJL model may not accurately represent individual waves as measured by ADCP.

### b. Validation of propagation speed estimates

Arrival time data for NLIWs at the three mooring sites provide another independent estimate of the zonal component of NLIW propagation speed. Using the estimates of propagation direction from the iteration analysis, we compute the meridional component and the average propagation speeds (including both zonal and meridional components) of NLIWs between LR1 and LR2, *C*_{LR12}, and between LR2 and LR3, *C*_{LR23}. Using the iteration results, we compare estimates of the propagation speed *C*_{LR12}, from the arrival time of NLIWs at LR1 and LR2, with (*C*_{LR1} + *C*_{LR2})/2, which is the average of the propagation speed estimates at LR1 and LR2, and also compare *C*_{LR23} with (*C*_{LR2} + *C*_{LR3})/2. These two independent estimates are in general agreement, albeit with large scatterings (Fig. 12). The linear correlation coefficient is 0.7, and the 95% significance level is 0.21.

Several factors may explain the discrepancies between the two independent propagation speed estimates. Our estimate of *C* using the iteration analysis might contain errors because 1) the motion of ADCP platform motion during the NLIW event could result in an inaccurate beam-spreading correction, 2) some oceanic processes are excluded in the iteration analysis (6)–(8) (e.g., unsteady internal waves, turbulence mixing, and trapped core), and 3) the NLIW propagation speed has a strong nonlinear dependency on the local depth. The second and third reasons suggest that NLIW propagation speed may not be linear between mooring sites, so the average speed between the two moorings (e.g., *C*_{LR12}) is different from the average speed at two mooring sites [e.g., (*C*_{LR1} + *C*_{LR2})/2].

## 5. Conclusions

Moored ADCPs are used to capture velocity measurements during NLIW events. However, important NLIW parameters, including propagation speed, propagation direction, and vertical displacement, are not directly measured by the moored ADCPs. We present an iterative method to estimate these parameters using velocity measurements from a single moored ADCP. This method also corrects the beam-spreading effect of ADCP measurements.

A simulated NLIW that satisfies the DJL solution and three sets of moored ADCP measurements taken in the SCS validate our approach. NLIW vertical displacement is computed as the integration time of the direct vertical velocity observations, with the effect of the background flow corrected. The propagation direction is estimated as the principal direction of the wave-induced horizontal velocity, with the effect of background current removed. The propagation speed is estimated as ∂* _{t}u*′/∂

*′ in the NLIW’s propagation direction, assuming that the wave structure and the propagation speed remain constant. We use Deming regression analysis to compute the slope between ∂*

_{z}w*′ and ∂*

_{t}u*′. Our analysis concludes that NLIW vertical displacement, propagation direction, and propagation speed can be estimated from velocity measurements from a single moored ADCP. The background flow and its vertical advection by NLIWs significantly affect the estimates of NLIW properties significantly and cannot be ignored, especially for large-amplitude NLIWs and in strong shear flows.*

_{z}w## Acknowledgments

The authors thank the crew of the Taiwanese R/V *Ocean Researcher 1*, Mr. Her, and graduate students at the Institute of Oceanography of National Taiwan University for the mooring preparation and operation. Discussions with Frank Henyey at the Applied Physics Laboratory, University of Washington, and Stastna Marek at the Department of Applied Mathematics, University of Waterloo, greatly assisted in our analysis and presentation. We also thank Kevin Lamb for providing his DJL model program codes. This work was supported by National Science Council (Grants NSC98-2745-M-019-00, NSC95-2611-M-002-016-MY3, NSC 94-2611-M-012-001, and NSC 95-2611-M-012-003-MY3) of Taiwan and by the U.S. Office of Naval Research (Grant N00014-04-1-0237).