Improving the time resolution of surfzone bathymetry using in situ altimeters

Abstract

Surfzone bathymetry often is resolved poorly in time because watercraft surveys cannot be performed when waves are large, and remote sensing techniques have limited vertical accuracy. However, accurate high-frequency bathymetric information at fixed locations can be obtained from altimeters that sample nearly continuously, even during storms. A method is developed to generate temporally and spatially dense maps of evolving surfzone bathymetry by updating infrequent spatially dense watercraft surveys with the bathymetric change measured by a spatially sparse array of nearly continuously sampling altimeters. The update method is applied to observations of the evolution of shore-perpendicular rip current channels (dredged in Duck, NC, 2012) and shore-parallel sandbars (observed in Duck, NC, 1994). The updated maps are compared with maps made by temporally interpolating the watercraft surveys, and with maps made by spatially interpolating the altimeter measurements at any given time. Updated maps of the surfzone rip channels and sandbars are more accurate than maps obtained by using either only watercraft surveys or only the altimeter measurements. Hourly altimeter-updated bathymetric estimates of five rip channels show rapid migration and infill events not resolved by watercraft surveys alone. For a 2-month observational record of sandbars, altimeter-updated maps every 6 h between nearly daily surveys improve the time resolution of rapid bar-migration events.

Appendix: Interpolation and mapping of irregularly sampled observations

Often, a set of bed-level observations z(x _j , y _j , t _j ), where x _j and y _j are the cross- and alongshore coordinates of the jth observation made at time t _j , are mapped using linear interpolation to a regular spatial (x _i , y _i ) and temporal (t _i ) grid:

$$ Z\left({x}_i,{y}_i,{t}_i\right)={\displaystyle \sum_j}{W}_{ij}z\left({x}_j,{y}_j,{t}_j\right) $$

(6)

where Z is the linearly interpolated bed-level elevation estimate at a set of “mapping coordinates” (x _i , y _i , t _i ) and W _ij is the weight of the jth observation at the ith mapping coordinate.

One common choice of interpolation weights is inverse separation weighting, e.g., in time:

$$ {W}_{ij}=A{\left|{t}_i-{t}_j\right|}^{-1} $$

(7)

The factor A (which may be a function of the observation and mapping coordinates) is sometimes set such that the only observations with nonzero weights are those immediately preceding and following the mapping coordinate, and may be normalized by the sum of the weights such that weights at each mapping coordinate sum to one.

Other mapping methods take advantage of knowledge of the signal covariance to seek an estimate of the bathymetry that minimizes the root-mean-square (rms) difference between the true and the mapped bathymetry (Bretherton et al. 1976). Optimal weights are:

$$ {W}_{ij}={\displaystyle \sum_{j\hbox{'}}}{\left[{P}_{j\hbox{'}j}\right]}^{-1}{R}_{j\hbox{'}i} $$

(8)

where P _j ' j is the covariance between observed elevations at locations with indices j' and j, R _j ' i is the covariance between observed and mapped elevations, and [ ]^− 1 is the matrix inverse. This method is referred to as objective mapping or optimal interpolation. Often a Gaussian model for the covariance is used for mapping either in space or in time, e.g., in one dimension:

$$ {R}_{mn}=V \exp \left(-\frac{{\left({p}_m-{p}_n\right)}^2}{2{L}^2}\right) $$

(9)

where V is the estimated signal variance, p is the spatial or time coordinate, m and n are arbitrary indices, and L is a decorrelation length or time scale. The covariance between all observed elevations is:

$$ {P}_{j\hbox{'}j}={R}_{j\hbox{'}j}+{\varepsilon}_O^2\left({x}_j,{y}_j,{t}_j\right){\delta}_{j\hbox{'}j} $$

(10)

where ε _O (x _j , y _j , t _j ) is the rms observational error associated with the jth observation. It is assumed that observation errors are uncorrelated with errors at other locations and times (the delta function δ _j ' j  = 0 if j ' ≠ j,and δ _j ' j  = 1 if j ' = j).

Often a mean or trend M is removed before mapping and then added back in after mapping (this can be considered a scale separation):

$$ Z\left({x}_i,{y}_i,{t}_i\right)={\displaystyle \sum_j}{W}_{ij}\left[z\left({x}_j,{y}_j,{t}_j\right)-M\right]+M $$

(11)

The function M may be an estimate of the true signal mean, a linear trend, a higher-order trend, or an ensemble-averaged estimate of a mean state. The choice becomes particularly important for data that are under-sampled because far from observations the interpolation weights tend to approach zero, and thus the bathymetric estimate approaches M (Rybicki and Press 1992).

The estimated interpolation error is:

$$ \varepsilon \left({x}_i,{y}_i,{t}_i\right)=V-{\displaystyle \sum_j}{W}_{ij}{R}_{ij} $$

(12)

If there are small-scale features (e.g., ripples, megaripples, cusps) that are not resolved by the surveys (e.g., there is aliasing owing to undersampling) or are not desired in the estimate of the bathymetry (e.g., considered noise), weights may be derived to minimize the rms difference between the mapped bathymetry and a filtered (e.g., smoothed) true bathymetry (Ooyama 1987; Plant et al. 1999). When seeking the optimal estimate of smoothed bathymetry, smoothed covariance functions of the true bathymetry (Ooyama 1987) are used. Here, the covariance function is assumed to be a Gaussian (Eq. 10) with the scale L set to the smoothing scale (a resolvable scale of interest) and the signal variance V set to the estimated variance of the smoothed bathymetry. In the presence of unresolved scales, ε _O should include both the rms measurement error and an rms estimate of the error associated with unresolved scales (e.g., the rms amplitude of bedforms). The results are optimal only if the covariance function is chosen correctly (e.g., a spatially variable covariance function could be used), but more detailed information about the true bathymetry would be needed to improve the covariance function estimate, and it is expected that the interpolation errors are not highly sensitive to errors in the choice of covariance function (Rybicki and Press 1992).