Low tide terrace (LTT) beaches are a common type of sandy beach but is one of the least documented [1
]. It can be seen both in low-tide (microtidal) [2
] and high-tide (macro- and meso-tidal) regimes [4
]. While the flat lower terrace area is generally alongshore-uniform, the steeper upper part often presents rhythmic cusps [6
]. The terrace formation is due to a coupling process between the effects of tides and waves. At low tide, waves plunge heavily on the outer terrace, but at high tide, they may pass across the terrace unbroken to surge up the beach face, producing reflection [7
]. LTT are intermediate beaches associated with low incoming wave energy [4
]; theoretically, it can transform into a transverse bar and rip state (TBR) when wave energy increases and into the lowest energy beach state, which is a reflective beach, when energy decreases. LTT beaches commonly features weak waves and coarse sand.
After more than 30 years of development, many coastal studies produced results using video systems, including for long-term morphological evolution [11
]. However, most are limited to the description of shoreline or submerged-bar location because these are possible in all conditions. Beach profiling from bathymetry inversion is another challenge, and is not commonly tackled even though it is in high demand. Image quality has several sources of error: validity of modulation transfer function (link between surface elevation and optical signal, see [13
]), breakpoint discontinuities, wave conditions, weather (lighting), geometrical conditions. Automated extraction techniques are thus known to produce high noise levels [14
] that can make geometrical interpretation difficult. If first-rate data is beyond reach, we need at least to estimate their quality. Recent results from cBathy
—an inversion method using a Kalman filter [15
]—show that the methodological error can be estimated directly from video techniques [16
]. However, other error types are due to physical assumptions used in depth inversion. Inversion relies on the linear wave theory that provides a dispersion relation between wave propagation and bathymetry for shallow and intermediate depths. Even though a few studies produce video-based bathymetry estimates—from drones [17
] or shore-based cameras (e.g., [14
])—little attention is assigned to error estimates, even though the method is often applied from very shallow to deep waters, without much consideration to the theory’s validity.
Quantifying beach morphodynamics requires monitoring beach profiles on a wide range of timescales. Conventional measurement techniques have obvious limitations because they are time consuming, costly and hazardous. These disadvantages do not apply to remote techniques such as shore-based video systems and LiDAR or satellite imagery [20
]. Continuous video monitoring now reveals the beach response to wave forcing at different timescales [16
]. In particular, video-based estimation of shoreline position is now rather straightforward, which makes it possible, for example, to estimate the timescales of beach recovery after a storm—about nine days in Biscarrosse (southwest France) [23
], and three days in Nha Trang, central Vietnam [24
]. But by considering only the shoreline, these studies do not take into account the evolution of the beach profile. For the reasons discussed above, high-frequency (i.e., daily) estimates of beach profiles are subject to high noise levels that are often simply filtered. For example, [26
] produce a dataset covering more than three years but its time resolution is limited to seven days after averaging over the tidal cycle and removing poor-quality data due to lighting issues (e.g., sun glint). This makes it difficult to study short-term events. But even at seasonal scale, the validity of video-based depth inversion during a low-energy season is not demonstrated.
In the present study, we present the three-year morphological evolution of a low tide terrace sandy beach in Nha Trang, Vietnam, based on bathymetry inversion from a shore-based camera system. Special care is given to the validity range of data inversion and removal of faulty data based on a new method for error estimate. Then, an analysis of morphodynamics is given at different scales from seasonal to short-term events. The results show that the width of a low tide terrace is an important parameter of beach evolution. The discussion section finally provides guidelines on the limits of video-based inversion for future applications in the diverse coastlines of the world.
3. Analysis of Beach Profile Evolution
Here, we present a beach profile analysis for Nha Trang during the three year measurement period. We will focus on the terrace width and its evolution at seasonal scale and as a response to intense winter monsoon events. As explained in the previous section, data quality control is essentially provided by the error estimate based on tides. But some comparison with bathymetry surveys were also done for validation. The example of Figure 4
during the NT4 survey of fall 2016 shows that error seems to increase with distance to shore but the main aspect of the profile, i.e., the terrace, is captured by the inversion method (more discussion on the error in Section 4
3.1. Bathymetry Evolution
a shows off-shore wave forcing and beach response over the 3.5 years of the study period for the profiling location presented in Figure 1
Wave forcing exhibits variability at multiple scales. Wave height is provided by ERA-Interim while the average period (zero-crossing) is derived from the video images and thus better accounts for local waves. The seasonal scale clearly stands out with high-energy northeast moderate waves during winter monsoon (October to April; m, s) and small local wind waves during summer monsoon (May to September; m, s). Interannual variability is also present but largely as a result of intrinsic variability of strong winter monsoon events (three days to three weeks; m) that largely imprints the signal in winter. Short-term events (<3 days) are associated with typhoon landing but their imprint is more sporadic (from October to December essentially) and not considerably stronger than monsoon events.
Daily mean cross-shore time stack video images are shown in Figure 5
b. This allows us to visualize and understand the evolution of shoreline position. This has been analyzed in detail in [25
] and the results will be summarized in the following for interpretation of beach profile evolution. Figure 5
c shows the daily cross-shore video-derived beach profile, where the low tide terrace area is indicated. This low tide terrace shows a significant variability at short event (during the winter season) and seasonal scales. In the following, we concentrate on the response of the terrace to seasonal and monsoon events.
3.2. Seasonal Pattern
shows wave forcing and beach response over a mean annual period over the 3.5 years of the study and a low-pass filter reveals seasonal patterns (moving average with a window of 30 days). Unfortunately, during summer, the amount of “good” bathymetry data is too small to allow any reliable observation (see Discussion). Nevertheless, the end and beginning of winter monsoon (with decreasing and increasing wave energy—see Figure 5
a) provides useful information on the transition to summer period, so that a sense of seasonal evolution can be determined. Shoreline evolution—which is available all year long—can help completing the picture on beach evolution.
From February to the end of the summer monsoon in October, the shoreline position is stable at first, then accretes more rapidly. In September, it begins an erosive phase due to winter monsoon conditions until February when it reaches its most landward location, 10 m from its summer position [25
The terrace follows an opposite behavior, with a terrace width becoming narrower with decreasing wave energy and wider with increasing energy. There is thus good correlation at the seasonal scale between the evolution of wave energy, shoreline and terrace width. From September to December, when wave energy increases, the shoreline erodes at the same time that the terrace expands. Then, from January to April, wave energy decreases while the shoreline begins its recovery and the terrace width decreases from 18 m (January) to 9 m (April). Interestingly, the speed of terrace evolution is much faster in this period than that of shoreline recovery, which is maximum in summer. This can be explained by different sediment fluxes during the periods of increasing and decreasing wave energy. During increasing energy, the terrace expands at the expense of the shore face, while during the decreasing energy, the terrace appears to loose sand to the offshore slope as much as to the shore face.
3.3. Impact of Winter Monsoon Events
Besides a strong seasonal cycle, the coast of Vietnam is largely impacted by intense winter monsoon events that last from three days to three weeks [25
]. Typhoon events are also present but they are less numerous and of shorter duration (see next section). To identify the events, the Peak Over Threshold method (POT) [1
] was applied on the offshore significant wave height (
) provided by ERA-Interim. The 2% exceedence threshold was used to identify strong events. Then, to characterize their general impact, we construct a composite beach profile as an ensemble mean of all events, thereby providing statistical reliability. Note that the composite hides some of the variability associated with winter monsoon events. For example, shoreline variations can reach 10 m during the strongest winter events but only half of that remains in the composite. Figure 7
presents the composite event for the periods before and after occurrence (day 0): wave forcing (wave height and period), shoreline position, and terrace width.
Day 0 for each event is defined as the time when wave height crosses a given threshold that must be high enough to be considered an event. For that reason, the composite event appears to start a bit earlier around day −2. The wave height rises during three days from the onset of the event where it reaches a peak value of a bit less than 2 m. At the same time, the period rises as well, indicating the presence of longer waves that has developed across the South China Sea. The wave height and period then slowly reduces over about a week.
As waves pick up the shoreline quickly retreats about 4 m in 3–4 days and then slowly recovers in about nine days. The terrace width follows a very similar pattern, but with a negative relationship. It first rapidly increases about 4 m following the rise of wave height and then slowly recovers as waves decline. The average post-event beach recovery duration is thus about 9 days both for the shoreline and terrace (i.e., whole beach profile accretion). Clearly, shoreline erosion is a process that provides sand to the terrace and inversely, shoreline accretion feeds on the terrace during recovery. The low tide terrace is thus the sand reservoir of the beach and is easily accessible for wave-induced onshore transport to nourish the upper beach.
3.4. LTT Beach State Dynamics and Recovery
Interestingly, while the beach shoreface adapts to changing wave conditions, it never changes its state as a low tide terrace. For example, transition to a barred beach could be expected during high energy conditions, but does not seem to occur. This suggests that LTT beaches are a stable type and can handle a large variety of waves and tides. One of the most striking properties of LTT is its rapid response time (days typically). This can be attributed to the fast exchange of sediment between the upper beach part and the terrace. It can be compared to a barred beach, which typically exhibits a response time larger than 10 days [23
], due to slow exchange of sediment between the bar and upper beach.
The relation between event duration and recovery is presented in Figure 8
. It shows that long-lasting monsoon events have a more persistent impact on both shoreline position and beach profile, with longer recovery (i.e., up to 10 days), than short-term typhoons (1–3 days), which takes a few days to recover at most (sometimes less than a day). This result is striking at Nha Trang beach, compared with more documented barred beaches in mid-latitudes [23
], where a larger range of high-energy wave event durations are observed. Interestingly at Nha Trang, recovery duration R
is almost linearly related to event duration D
(see details in legend of Figure 8
), albeit with substantial dispersion illustrated by error bars in the figure. This result illustrates again the plasticity (resiliancy) of the terrace due to rapid sediment exchange with the upper beach. It should have important consequences for shoreline modeling. A single beach decay value is often used in models [44
] whereas this parameter should account for variable storm durations.
4. Discussion on Error Estimate
Video-based bathymetry inversion has been developed for more than 20 years [16
], with a range of errors between 0.2–2.0 m (RMSE) and −0.01 to −1.0 m (bias). The method was calibrated and validated in various environments and often showed acceptable results compared with traditional in situ measurements. However, little attention was given to the general characterization of errors. A proper criterion was probably missing. Here, the comparison of tides between in situ and video-based inversion (i.e., daily difference, method described in Section 2.4
, see also Figure 3
) provides a new estimator that seems to cover most of the reported types of errors in a single result.
shows the cross-shore time stack composite of error estimate for the whole time series and profiling section. Three sources of error are clearly standing out in the figure (middle panel) and time series of each error type are extracted in the bottom panel. The three types of errors were decomposed based on visual inspection of their spatial patterns that show contrasting signatures. A more generic approach is left for future work. By order of importance, they are: deep water limitation (DW), breakpoint optical effect (BP) and surfzone nonlinear effect (NL). They have contrasted spatial patterns (see time-mean cross-shore patterns in the outset attached to the second panel): NL is localized at the shore and largest in the shallowest area; BP is distributed along the breakpoint, which has large temporal variability (top panel); and DW is in deep water with an error that seems to increase linearly with distance to shore. A diagram is given in Figure 10
to synthesize the location and processes behind the different error types, which are discussed below, along with their peculiar seasonal patterns.
4.1. Nonlinear Effects during Shoaling and in Shallow Water
The source of error with least intensity and cross-shore extent (but largest persistence all year long) is physical and concerns the use of linear wave dispersion in a nonlinear nearshore environment. Ref. [16
] show that nonlinear processes become important in the shoaling zone and in shallow water (depth to wave length ratio
), producing an overestimation of inverted depth based on linear theory. In theory, for wave height to depth ratios
greater than 1, a nonlinear dispersion relation should be used. However,
is not currently available from video systems, as opposed to LiDAR systems, which thus provide improved accuracy for bathymetry inversion. In addition, wave setup in the surf zone and very shallow water, which increases with wave height, can have substantial contribution to the total depth and is not accounted for in current studies. This also tends to overestimate water depth.
4.2. Breakpoint Optical Effect
The sudden change of modulation transfer function between optical signal and actual wave phase from non-breaking to breaking waves (breakpoint) is a common problem of shore-based video camera systems. According to [16
] in a recent study, a dominant uncertainty in video-based depth inversion stems from wave crest celerity detection. They suggest that the order of error in celerity detection may be twice larger than that from linear assumption of the dispersion relation. This is confirmed here with our new error estimate (at least in winter with bigger waves). The largest error in wave crest celerity detection occurs at breakpoint in both spectral and temporal methods, which means that it is intrinsic to the intensity images. This comes from a sudden change of the so-called modulation transfer function [13
] at breakpoint. Prior to breakpoint, waves show a dark face and bright backface, but in the breaking zone, they have a bright roller and dark backface. The associated change in transfer function generates an apparent (spurious) increase of celerity. Usual methods ignore this issue or discard this specific region, but corrections were recently proposed relying on a different filter in the breaking and non-breaking zones [14
4.3. Deep Water Asymptote
The third limitation is the largest, if not in magnitude, at least in space-time coverage. It stems from wave-bottom interactions that vanishes in deep waters, i.e., according to the linear theory, when . Our study is particularly challenging in this respect because for a substantial part of the year (mostly summer), the sea state is composed of short wind waves for which video data is rarely accurate. Fortunately, we processed enough data in the inter-seasons to reveal the seasonal evolution of the low tide terrace. However, our conclusion is that video-based depth inversion is generally not appropriate for wind wave conditions, at least for their shortest periods.
A synthesis of water wavelength as a function of depth is given in Figure 11
for winter and summer conditions. In summer, typically for
s, the deep water limit is 3 m, beyond which the wavelength reaches a deep water asymptote
m. Beyond this depth limit, the linear wave theory cannot be used for depth inversion. A second potential limitation associated with short waves is linked to the image resolution that increases with distance to the camera. The minimum wavelength that can be well resolved is
is the image resolution), which should limit the use of video cameras in the offshore zone. In Figure 11
, we estimate the offshore resolution limit for summer wind waves to a maximum distance of ∼250 m. Clearly, the resolution error is not dominant because even for a period of 3 s in summer, the physical deep water limit occurs much closer to shore than the resolution error (compare red and black lines in Figure 11
). Depth inversion is invalid in this case. In winter (blue curve), with coastal wave periods closer to 5 s, the deep water asymptote is pushed much further offshore, although a domain of invalidity remains.
A video-based bathymetry inversion technique is applied to long-term data with varying fetch-limited wind-wave conditions. A main result is the overall stability but high resiliency demonstrated for the LTT beach, with rapid exchange of sediment between the terrace and the upper beach during typhoons, monsoon events or seasonal cycles. The high resiliency of these tropical environments may provide faster beach recovery compared with mid-latitude configurations.
At the same time, the study details the treatment techniques needed for a long-term dataset, and focuses on the limitations of the method in the case of wind waves. For the first time, a tide gauge is used to quantify the error produced by video estimates and produce a quality criterion. Three sources of error are exhibited in a single analysis. They are clearly identified (see diagram in Figure 10
): (1) nonlinearities and wave setup localized in very shallow water; (2) change in the optical wave signature at breakpoint; (3) deep water limit for applying the linear dispersion relation, which is very restricting for wind waves (
s). The error types have contrasted spatial and temporal patterns, but the main error component in Nha Trang is deep water limitation. It has strong seasonality and is greatest in summer, when short wind waves are dominant, making depth inversion invalid. On the contrary, the second and third dominant error contributions at breakpoint and in the surf zone are maximum for large waves (during winter monsoon in Nha Trang). Corrections will probably be proposed for these errors in the future.
More generally, this study provides guidelines for users of video-based depth inversion methods and a base fo r standalone error assessment, which is essential for operational and data assimilation systems. For data assimilation, the estimation of error is required but generally not available [50
]. A comprehensive error estimate, such as the one proposed here, is thus an excellent step for improving the assimilation approach.