# Updates to and Performance of the cBathy Algorithm for Estimating Nearshore Bathymetry from Remote Sensing Imagery

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. cBathy Versions—Version 1.0

_{p}, y

_{p}, t), at a set of discrete pixel locations, x

_{p}, y

_{p}that span the domain of interest and adequately sample the typical ocean wave scales, while not oversampling them (across-shore and alongshore spacing is commonly 5 and 10 m, respectively, see blue dots in Figure 1). The analysis is carried out at a map array of model locations, x

_{m}, y

_{m}, (example red dot in Figure 1) and at each map location is based on the observed wave phases in a tile of observations within some user-specified cross-shore and longshore length scales, L

_{x}and L

_{y}, of the model location (green dots in Figure 1; see Figure 2 for an example phase map). At a set of dominant radial frequencies, σ, the two components of wavenumber, k

_{x}and k

_{y}, are derived from the phase ramp slopes in the x (cross-shore) and y (alongshore) directions and are combined to yield the magnitude of k.

_{p}, y

_{p}), and only the one with the largest eigenvalue is retained. This normalized eigenvector is modelled as a single dominant plane wave form

_{m}, y

_{m}locations from within the tile, weighted by the inverse distance from the current estimation point. In version 1.0, the weighting was taken to depend on the normalized eigenvalue and skill of the model fit, both from phase 1, as well as the inverse distance from the current estimate location in phase 2. The final phase 2 result is the single depth that is the nonlinear best fit to predicted depth using the input suite of frequency–wavenumber pairs and the dispersion relationship. Error estimates, h

_{err}, are produced for each depth (see [1], error estimation is unchanged in recent upgrades).

#### 2.1. Version 1.1 Update

_{0}is the wavenumber in deep water. We can define k

_{0}/k = L/L

_{0}= Γ, where Γ is a non-dimensional wavelength (or wavenumber), going from small in shallow water to a maximum value of 1.0 in deep water. We can define sensitivity to wavenumber error by the equation

_{0}= gT

^{2}/2π is frequency dependent so this weighting puts a strong preference on long-period waves.

#### 2.2. Version 1.2 Update

#### 2.3. Version 2.0

_{0}, at each map point to solving for only k and α, and (c) the introduction of a much better algorithm to find seed values for k and α before the nonlinear search for each tile. The cumulative consequence of these modifications is a major restructuring of the code. Each component will be described in turn.

#### 2.3.1. Automatic Tile Sizes

_{x}and L

_{y}. Suggestions for best values were ad hoc with a belief that the search would work best if the tile was typically about one wavelength long, but an implicit faith that even mismatched tile sizes would solve well somewhere within the nonlinear fitting routine. Thus, the same tile size was used for, e.g., 4 s and 16 s waves despite at least a four-fold difference in their wavelengths. Although sub-optimal, this approach was still successful, since tiles that were poorly designed simply failed to converge and returned nan’s rather than a poor depth.

_{L}times the expected wavelength, where k

_{L}is an empirical scalar taken to be approximately 1.0. However, the expected wavelength depends of the frequency and depth, neither of which is known a priori. Thus, there is strong motivation to develop a seed-finding algorithm (see below) that can provide good initial estimates of k under all wave conditions. Thus, given an initial tile based on generic user input, version 2.0 feeds all of the available pixels to the routine to find the seed k and α, then crops the original tile size to k

_{L}times this wavelength, a size that varies considerably. The number of pixels in this truncated tile is, then, reduced, if necessary, to maxNPix to speed up processing. Because all tiles are roughly one wavelength no matter what the frequency, it is likely that fewer pixels are needed for search convergence, again speeding up processing.

#### 2.3.2. Reduction in the Number of Search Variables

_{0}, is a scalar offset between the measured and modeled phase maps and can jump around a great deal, in ways that are inconsistent with a search for a global cost function minimum. Thus, this variable is not well estimated and likely just confuses the search.

^{2}space (each pixel compared to every other pixel). In addition, visualization of measured and modeled results (e.g., Figure 2) was not as clear in lag space. Thus, we wish to retain the simplicity of working in x-y space maps but using a method for estimating ϕ

_{0}for each search iteration that will allow a sensible search for k and α. The solution is to force the measured and modeled phase to be the same at the tile center (the pixel closest to x

_{m}, y

_{m}). This is done by finding dϕ, the difference in phase at the middle pixel, and multiplying all modeled complex values of the eigenvector, v, by e

^{id}

^{ϕ}. The nonlinear search is, then, reduced to two dimensions.

#### 2.3.3. Improved Seed Algorithm

_{m}, y

_{m}), to be used in finding dϕ.

#### 2.4. Algorithm Organization Changes

_{x}, L

_{y}is initially passed into the main analysis routine, csmInvertKAlpha, where the routine prepareTiles is called to (a) find the dominant frequencies, (b) for each frequency, find the dominant eigenvector and the k-α seeds, then c) reduce the tile to an adaptive size and to a maximum number of pixels. These outputs are, then, passed back to csmInvertKAlpha to carry out the nonlinear search for best-fit values and their errors and, then, to build the results structures and find the depths from the σ-k results.

_{m}, y

_{m}points. These include time exposure, brightest and darkest images, all at the coarse resolution of the analysis array but adequate to determine, for example, the locations of wave breaking. Within the phase 1 fDependent sub-field, there are now maps that are included for diagnostic and performance improvement purposes. These include maps of the seed values of k and α used for the nonlinear search as well as maps of the number of pixels used in each tile and of the number of model calls during the nonlinear fitting routine, a key to algorithm speed. In addition, there is a map of which camera is used for each tile. Within the fCombined sub-field, there is now a map of the effective mean frequency, fBar, used in the phase 2 bathymetry calculation, found as the weighted mean of each frequency that contributed to the phase 2 final bathymetry estimate. Finally, the elapsed CPU time for each analysis is saved.

## 3. Bathymetry Test Bed Datasets

## 4. cBathy Phase 2 Version Performance Statistics

_{95}), and the percentage of successful coverage (estimated error <0.5 m) for estimation locations with survey depths greater than 0 m, including tide (called coverage). The results from this example run show a steady improvement in all statistics for versions 1.0 to 1.2 to 2.0 of: bias = [−0.36, −0.05, −0.04] m, rmse = [0.90, 0.59, 0.41] m, Δh

_{95}= [2.58, 1.30, 0.86] m, and coverage = [55%, 77%, 91%].

#### 4.1. Bulk Analysis of cBathy Version Statistics

_{95}, look almost indistinguishable between versions, despite what we had hoped were significant improvements in the algorithm as shown in the example run above. This result is a consequence of the fact that for each version, results with estimated errors, h

_{err}> 0.5 m were rejected from the analysis. Thus, the statistics are computed for different regions of coverage. To make a more direct comparison (apples versus apples), the map of successful coverage for V2.0 was saved for each collection and used as the basis for computing performance statistics for V1.0 and V1.2. This usually larger region, thus, includes poorer performing regions from the earlier algorithm versions. This is confirmed by Figure 15, which shows the same histograms of statistics but is now based on a common area of sampling defined by the V2.0 error estimates. For all three statistics, bias, rmse, and Δh

_{95}, the performance of V2.0 is superior. The fact that statistics are roughly the same when h

_{err}is determined by each individual version is a testament to the robustness of that error estimate. In the following, all statistics will be based on the common area of coverage determined for V2.0.

_{s}is greater than 1.2 m reduces the mean bias to 0.15, 0.11, and 0.12 m, respectively. Removal of estimates for which coverage was less than 50% reduced the original mean bias to 0.15, 0.1, and 0.1, respectively, while removal of both issues (consider only cases with Hs ≤ 1.2 m and coverage ≥ 50%, reducing the dataset from 624 to 563 runs) led to mean biases for the three algorithm versions of 0.15, 0.09, and 0.09 m, respectively.

^{−1}, but with improved wave angle seeds, this was sufficiently close to yield greater success in the nonlinear search.

#### 4.2. Kalman-Filtered Results

_{err}, using a Kalman filter. Thus, regions with bad results due to, for example, wave breaking, can be filled with results from runs in which there was no breaking at that location. Kalman results accumulate over a series of prior data collections, with each new run improving the estimate until the results stabilize after roughly one or two days. In this case, we have included four cBathy bathymetry estimates per day for four days, the three days prior to each survey plus the actual survey day. The Kalman result from approximately 3:00 pm on the day of the survey is used to compare with the survey ground truth. Figure 20 shows the Kalman-filtered results for the example survey used in Figure 8, 21 November 2017. Results from the three versions are similar but with slight differences in coverage and depth estimates, for example, the region around x = 200, y = 830, which appears anomalous in version V1.0 but not in later versions. All statistics are better after Kalman filtering compared to the individual cBathy estimates. Bias for the three versions (in order V1.0, V1.2, V2.0) was 0.006, −0.07, and −0.03 m, while the rmse was 0.41, 0.37, and 0.35 m, and the 95% exceedance was 0.85, 0.83, and 0.77 m, respectively. Coverage of locations that were immersed at the final time were all 100%, a credit to averaging over the varying tide elevations.

_{95}are all very good with V2.0 results outperforming earlier versions. Note that the coverage (of submerged depths) is now always greater than 95% due to the Kalman filtering, with V2.0 performing the best (the lowest V2.0 coverage was 99.3%). Again, performance is best for lower waves, although the Kalman filtering averages over multiple prior runs. Mean statistics for the three algorithms averaged over the 39 surveys are shown in Table 2. Again, version 2.0 outperforms earlier versions.

_{err}) of the measured absolute error to both the Kalman and non-Kalman (phase 2) predicted error was taken, averaged over the domain for which Kalman and non-Kalman predicted errors were less than or equal to 1.0 and 0.5 m, respectively (this removes the influence of extreme predicted errors). In HPH13, this value was only tested for Kalman results and was found to be approximately 7. For this more extensive dataset and the version 2.0 algorithm, the values are smaller, as shown in Figure 23. The mean of the Kalman-filtered results is 4.47, smaller than the value of 7.0 noted by HPH13. The mean of the non-Kalman-filtered results was 2.0. During this comparison, it was also noted that the process of Kalman filtering reduced the actual error of the Kalman estimates by an average factor of 4.6 compared to the non-Kalman single run estimates. The same analysis, carried out for version 1.0 results, showed that the mean of the predicted to measured errors for Kalman- and non-Kalman-filtered results were 5.22 and 2.29, with the Kalman ratio being a bit smaller than the factor of 7.0 found by HPH13 for their dataset. The reduction in the algorithm error for Kalman- and non-Kalman-filtered predictions was 5.3, slightly larger than with V2.0, likely due to the improvements in the non-Kalman estimates.

## 5. Discussion

_{95}all showed improvements with V2.0. In some ways, it is reassuring that the quality measurements (predicted h

_{err}) are performing sensibly.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**cBathy pixel array used for the analyses in this paper, overlain on a merged snapshot from 17 May 2015. For clarity, only ¼ of the pixels are shown (decimated by two in both x and y directions). The wavenumber for each analysis point, for example, x

_{m}= 250; y

_{m}= 750 shown above by the red asterisk, is found using phase map data from a surrounding region, shown by green dots above. Imagery is derived from six oblique-viewing cameras and merged into rectified images such as this [44,45].

**Figure 2.**Phase map for the example tile shown in Figure 1 for the dominant frequency, f = 0.0956 Hz. Observed phase is on the left, best-fit modeled (Equation (2)) phase on the right. The x and y components of wavenumber are derived from the components of slope of the phase ramps (colors from seaward to landward going from blue through green, yellow then red with 360° phase jumps as blue–red transitions).

**Figure 3.**Sensitivity of the dispersion relationship showing the increasing sensitivity of the inversion process as deep water is approached.

**Figure 4.**Radon transform results for the example tile and model point shown in Figure 1 and Figure 2.

**Left**panel shows the actual Radon transform versus candidate wave angle (x-axis) and projected distance (y-axis). The

**right**panel shows the variance of the Radon transform as a function of wave angle. The angle with maximum variance is chosen as the wave angle seed, in this case 10.8°.

**Figure 5.**Organization of cBathy version 1.2 (and 1.0) showing the three phases of the algorithm by color.

**Figure 6.**New organization of cBathy version 2.0 with the partitioning into tiles happening later. “PrepImages” is added to create timex and other image types from the time stack and is not integral to the cBathy calculations.

**Figure 7.**Time series of wave height, H

_{s}, wave period, T

_{m}, peak wave direction, and tide elevation for each of the 624 cBathy estimates collected over almost four years. Blue dots for the upper three panels correspond to estimates from the 26 m depth waverider buoy. These are overplotted with red dots from the 8 m array. Shore normal wave direction is 72°, indicated by the blue horizontal line in the wave direction panel.

**Figure 8.**Comparison of bathymetry from the three versions of cBathy, V1.0, V1.2, and V2.0 (left three panels) for 19 November 2017, 1959 GMT, with the ground-truth survey from two days following (right panel). All cBathy results have been filtered to remove any values with an estimated error greater than 0.5 m (dark red color).

**Figure 9.**cBathy performance statistics for 19 November 2017, partitioned by depth. Values are plotted at the mean depth for each bin. No survey depths greater than 6 m were observed for this run.

**Figure 10.**Map of fBar, the weighted mean frequency that contributed to phase 2 depth estimates in version 2.0.

**Figure 11.**Comparison of seed wave angle and wavenumber to the final values from the nonlinear fitting routine. Example shows only values for the dominant frequency.

**Figure 13.**Percentage of successful coverage for sub-aqueous pixels versus significant wave height at the 8 m array for each of the three versions. Versions were plotted in order from versions 1.0 through 2.0 so the results from version 2.0 (blue dots) overplot earlier results.

**Figure 14.**Histograms of the four basic performance statistics for each of the three algorithm versions (colors).

**Figure 15.**Histograms of the same statistics as Figure 14, but now based on common regions of sampling determined by the regions of successful estimates from the V2.0 version of cBathy.

**Figure 16.**Scatter plots of the four performance statistics for each of the three algorithm versions (see legend at top right) for the full dataset of 624 runs. Points are plotted in order, so blue points (V2.0) overplot earlier versions.

**Figure 17.**Comparison scatter plots of the four performance statistics for each of the three algorithm versions (see legend at top right) for the reduced dataset of 563 runs. Points are plotted in order, so blue points (V2.0) overplot earlier versions.

**Figure 18.**Error bulk statistics binned by depth. Bin depths are plotted at the mean bin depth. Algorithm versions are indicated in the legend. Coverage is omitted, since all versions have been forced to use the V2.0 coverage.

**Figure 19.**Histograms of the bias (

**left**) and rmse (

**right**) statistics for each of the three version of the cBathy algorithm (see legend). The data have been partitioned into non-breaking regions (solid lines) and surf zone regions (dashed lines; legend has “SZ” appended).

**Figure 20.**Comparison of Kalman-filtered bathymetry from the three version of cBathy, V1.0, V1.2, and V2.0 (

**left**three panels) for 21 November 2017, with the ground-truth survey from that day (

**right**panel). This is the same survey used in Figure 8. All cBathy results have been filtered to remove any values with an estimated error greater than 0.5 m.

**Figure 21.**The four basic performance statistics plotted versus surveyed depth for the example survey date of 19 November 2017. Note that each plot includes three lines, one for each algorithm, with the third (blue) often overplotting the others.

**Figure 22.**The four performance statistics describing the results from the 39 Kalman results. Version colors are shown in the legend.

**Figure 23.**Histograms of the mean ratio of observed to estimated version 2.0 errors, averaged over each of the 39 surveys. Red and blue lines indicate Kalman and non-Kalman results, respectively.

**Table 1.**Means of the four basic performance statistics (rows) comparing across the three versions of the cBathy algorithm (columns). Statistics are computed for the full dataset of 624 runs (upper section) as well as for the reduced dataset of 563 runs for which successful coverage was greater than 50% (based on V2.0 coverage) and significant wave height was less than 1.2 m (lower section).

Full Dataset | |||

Statistic | V1.0 | V1.2 | V2.0 |

Bias (m) | 0.19 | 0.14 | 0.16 |

rmse (m) | 0.64 | 0.82 | 0.56 |

Δh_{95} (m) | 1.27 | 1.25 | 1.19 |

Coverage (%) | 78.7 | 78.0 | 84.7 |

Reduced Dataset | |||

Bias (m) | 0.15 | 0.09 | 0.09 |

rmse (m) | 0.58 | 0.66 | 0.41 |

Δh_{95} (m) | 1.16 | 1.01 | 0.85 |

Coverage (%) | 82.8 | 82.3 | 90.0 |

**Table 2.**Means for Kalman-filtered bathymetries of the four basic performance statistics (rows) comparing across the three versions of the cBathy algorithm (columns). Statistics are computed for the full dataset of 39 surveys, comparing the mid-afternoon Kalman bathymetry for each survey date to ground truth.

Full Dataset | |||
---|---|---|---|

Statistic | V1.0 | V1.2 | V2.0 |

Bias (m) | 0.15 | 0.08 | 0.08 |

rmse (m) | 0.47 | 0.66 | 0.38 |

Δh_{95} (m) | 0.96 | 0.86 | 0.78 |

Coverage (%) | 99.1 | 99.2 | 99.9 |

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**MDPI and ACS Style**

Holman, R.; Bergsma, E.W.J.
Updates to and Performance of the cBathy Algorithm for Estimating Nearshore Bathymetry from Remote Sensing Imagery. *Remote Sens.* **2021**, *13*, 3996.
https://doi.org/10.3390/rs13193996

**AMA Style**

Holman R, Bergsma EWJ.
Updates to and Performance of the cBathy Algorithm for Estimating Nearshore Bathymetry from Remote Sensing Imagery. *Remote Sensing*. 2021; 13(19):3996.
https://doi.org/10.3390/rs13193996

**Chicago/Turabian Style**

Holman, Rob, and Erwin W. J. Bergsma.
2021. "Updates to and Performance of the cBathy Algorithm for Estimating Nearshore Bathymetry from Remote Sensing Imagery" *Remote Sensing* 13, no. 19: 3996.
https://doi.org/10.3390/rs13193996