pith. sign in

arxiv: 1907.02714 · v2 · pith:LY246KYRnew · submitted 2019-07-05 · ⚛️ physics.ao-ph

Estimation of spatio-temporal wave grouping properties using Delaunay triangulation and spline techniques

Jos\'e Carlos Nieto Borge , Juan Gerardo Alc\'azar , David Orden , Sara Marazuela Reca , Gerardo Rodr\'iguez This is my paper

Pith reviewed 2026-05-25 01:58 UTC · model grok-4.3

classification ⚛️ physics.ao-ph
keywords wave envelopeDelaunay triangulationtensor-product splineswave groupsnonlinear wavessea surface elevationX-band radar
0
0 comments X

The pith

A new method using Delaunay triangulation and splines computes wave envelopes that can be non-symmetric, unlike the symmetric results from the Riesz transform.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a method to compute the wave envelope of ocean surfaces that avoids the symmetry forced by the Riesz transform. It first identifies local maxima and minima of the sea surface elevation, then uses Delaunay triangulation to connect these points and tensor-product splines to interpolate the upper and lower envelopes. This produces envelopes that reflect the asymmetry present in nonlinear waves. The method is tested on both simulated wave fields and X-band radar measurements, where it reproduces the expected behavior of the waves. The central motivation is that symmetric envelopes limit the detection of features such as groups of high waves.

Core claim

The method computes first the local maxima and minima of the sea surface, and then determines the wave envelope by combining discrete methods, namely the use of the Delaunay triangulation, and tensor-product splines. The proposed method has been applied to simulated wave fields, and also to wave elevations data measured by an X-band radar. The obtained results correctly reproduce the behavior of the simulated waves.

What carries the argument

Delaunay triangulation combined with tensor-product splines applied to identified local maxima and minima of the sea surface elevation to form the wave envelope.

If this is right

  • Groups of high waves can be identified more accurately because the envelope no longer forces symmetry between crests and troughs.
  • Analysis of measured radar data becomes possible without the linear-wave assumption built into the Riesz transform.
  • Envelope-based statistics of wave fields can be extended to cases where nonlinearity produces visibly asymmetric modulation.
  • The discrete triangulation-plus-spline construction supplies a concrete alternative whenever the Riesz transform is known to be inappropriate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local-extrema-plus-triangulation pipeline could be tested on other surface-elevation time series, such as those from laser altimeters or stereo video.
  • If the spline order or triangulation criterion is varied, the resulting envelopes might serve as a tunable family for comparing different degrees of nonlinearity.
  • The method supplies a practical way to generate training labels for machine-learning models that aim to predict envelope asymmetry from radar backscatter.

Load-bearing premise

Locating local maxima and minima followed by Delaunay triangulation and tensor-product splines will produce envelopes that are both non-symmetric and more realistic for nonlinear waves than those from the Riesz transform, without introducing new artifacts.

What would settle it

Apply the method to a known nonlinear wave simulation where the true asymmetric envelope can be computed directly from the governing equations and check whether the new envelopes match the asymmetry while the Riesz envelopes remain symmetric.

Figures

Figures reproduced from arXiv: 1907.02714 by David Orden, Gerardo Rodr\'iguez, Jos\'e Carlos Nieto Borge, Juan Gerardo Alc\'azar, Sara Marazuela Reca.

Figure 1
Figure 1. Figure 1: Simulated linear wave field (left) and the corresponding estimation [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Wave spectrum (left) and envelope spectrum (right) of the ex [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparing elevation with that of neighbors. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Triangulation with long and skinny triangles. Right: Delau [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Construction of an envelope as a Delaunay polyhedral terrain. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Draupner wave record measured at to the Draupner platform on [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Subset of the Draupner wave record (left) and the corresponding [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Refinement with Splines. complexity of computing the matrix C. Since this last complexity dominates the other ones, we get an overall complexity for our method of O(N3/2 ). 3.5 Additional observations on the grid. Although for simplicity in the previous subsections we considered a rect￾angular grid, our method can be used for other grids as well. The only property needed is that, for each point, its set of… view at source ↗
Figure 9
Figure 9. Figure 9: Local heights estimated from the Delaunay polyhedral terrain [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Local height estimated from RT (left), D (center), and S (right). [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Local wave height estimated from RT (top) and its corresponding [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Wave number and frequency spectra from RT (left), D (center), [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Spatio-temporal evolution of the local height along the main wave [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: X-band radar image of the sea surface taken at the FINO 1 [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Estimation of the sea surface elevation derived from the squared [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
read the original abstract

Wave groups can be detected and studied by using the wave envelope. So far, the method used to compute the wave envelope employs the Riesz transform. However, such a technique always produces symmetric envelopes, which is only realistic in the case of linear waves. In this paper we present a new method to compute the wave envelope providing more realistic results. In particular, the method allows to detect non-symmetry in the wave envelope, something useful, for instance, when detecting groups of high waves. The method computes first the local maxima and minima of the sea surface, and then determines the wave envelope by combining discrete methods, namely the use of the Delaunay triangulation, and tensor-product splines. The proposed method has been applied to simulated wave fields, and also to wave elevations data measured by an X-band radar. The obtained results correctly reproduce the behavior of the simulated waves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that a new method for computing wave envelopes—based on identifying local maxima and minima of the sea surface, followed by Delaunay triangulation and tensor-product splines—yields non-symmetric envelopes that are more realistic for nonlinear waves than those from the always-symmetric Riesz transform. The method is applied to simulated wave fields and X-band radar data, with the assertion that the results correctly reproduce the behavior of the simulated waves and enable detection of high-wave groups.

Significance. If the envelopes are demonstrated to bound the surface elevation without introducing spline-induced artifacts and to recover the crest/trough asymmetry expected from nonlinear theory, the approach would provide a useful computational tool for studying spatio-temporal wave grouping properties where asymmetry matters, such as in rogue-wave or group-detection applications.

major comments (2)
  1. [Abstract] Abstract: the claim that 'the obtained results correctly reproduce the behavior of the simulated waves' is unsupported by any quantitative metrics (e.g., L2 deviation from analytic envelopes, crest-height error, or direct comparison to Riesz-transform results), which is load-bearing for the central assertion of improved realism and faithful reproduction.
  2. [Method description] Method description (paragraph on local-max/min + Delaunay + tensor-product splines): the construction is asserted to produce non-symmetric envelopes that bound the surface and recover nonlinear asymmetry without new artifacts, but no verification against known nonlinear solutions (e.g., Stokes-wave envelopes) is supplied to confirm that the discrete interpolation step does not introduce connectivity or oscillation artifacts.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it stated the number of simulated cases, the specific nonlinear wave parameters tested, and whether any error norms or visual comparisons to Riesz envelopes are shown in the figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which identify opportunities to strengthen the quantitative support for our claims. We respond to each major comment below and will revise the manuscript to incorporate the requested evidence.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'the obtained results correctly reproduce the behavior of the simulated waves' is unsupported by any quantitative metrics (e.g., L2 deviation from analytic envelopes, crest-height error, or direct comparison to Riesz-transform results), which is load-bearing for the central assertion of improved realism and faithful reproduction.

    Authors: We agree that quantitative metrics are needed to support the abstract claim. In the revised manuscript we will add L2 deviations from the known envelopes in the simulated cases, crest-height errors, and side-by-side numerical comparisons with the Riesz-transform envelopes. These additions will directly substantiate the assertions of faithful reproduction and improved realism for nonlinear waves. revision: yes

  2. Referee: [Method description] Method description (paragraph on local-max/min + Delaunay + tensor-product splines): the construction is asserted to produce non-symmetric envelopes that bound the surface and recover nonlinear asymmetry without new artifacts, but no verification against known nonlinear solutions (e.g., Stokes-wave envelopes) is supplied to confirm that the discrete interpolation step does not introduce connectivity or oscillation artifacts.

    Authors: We concur that explicit verification against analytic nonlinear solutions is required. The revised manuscript will include a validation subsection that applies the method to a Stokes wave, for which the exact envelope is known. We will report crest-trough asymmetry recovery, surface-bounding behavior, and quantitative measures of any interpolation artifacts (e.g., maximum overshoot or oscillation amplitude) arising from the Delaunay triangulation and tensor-product spline steps. revision: yes

Circularity Check

0 steps flagged

No circularity: method is an independent algorithmic procedure

full rationale

The paper describes a computational pipeline (identify local maxima/minima of the surface, apply Delaunay triangulation, then tensor-product splines) that constructs an envelope directly from the discrete data points. No equations, fitted parameters, or predictions are shown to reduce by construction to the input data or to prior self-citations. The claim that the resulting envelopes are non-symmetric and reproduce simulated-wave behavior is presented as an empirical outcome of applying this procedure, not as a definitional identity or a result forced by the method's own assumptions. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from the authors' prior work appear in the provided text. The derivation chain is therefore self-contained as a sequence of standard discrete-geometry operations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the method rests on standard assumptions about identifying extrema and the suitability of triangulation plus splines for surface interpolation, with no free parameters or invented entities mentioned.

axioms (2)
  • domain assumption Local maxima and minima of the sea surface elevation can be reliably extracted from the input data (simulated or radar).
    The method description begins with computation of these extrema as the first step.
  • domain assumption Delaunay triangulation combined with tensor-product splines can represent the wave envelope without forcing symmetry.
    This is the core technical premise enabling non-symmetric envelopes.

pith-pipeline@v0.9.0 · 5696 in / 1307 out tokens · 33371 ms · 2026-05-25T01:58:50.323090+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages

  1. [1]

    E., 1977

    Barnhill, R. E., 1977. Representation and approximation of surfaces. In J. R. Rice (editor) Mathematical Software III, pages 69–120. Academic 25 Press, New York, 1977

    work page 1977

  2. [2]

    Benetazzo, A., Fedele, F., Gallego, G., Shih, P. C. and Yezzi, A., 2012. Offshore stereo measurements of gravity waves. Coastal Engineering 64, 127–138

    work page 2012

  3. [3]

    S., Williams, J

    Bell, P. S., Williams, J. J., Clark, S., Morris, B. D. and Vila-Concejo, A., 2005. Nested radar systems for remote coastal observations. Journal of Coastal Research 39, 483–487

    work page 2005

  4. [4]

    Buckley, J. R. and Aler, J., 1997. Estimation of ocean wave height from grazing incidence microwave scatter. IEEE International Symposium on Geoscience and Remote Sensing–IGARSS 1997 2, 1015–1017

    work page 1997

  5. [5]

    Buckley, J. R. and Aler, J., 1998. Enhancements in the determina- tion of ocean surface wave height from grazing incidence microwave backscatter. IEEE International Symposium on Geoscience and Remote Sensing–IGARSS 1998 5, 2487–2489

    work page 1998

  6. [6]

    R., Allingham, M

    Buckley, J. R., Allingham, M. and Michaud, R., 1994. On the use of marine radar imagery for estimation of properties of the directional spectrum of the sea surface. Atmosphere–Ocean 32 (1), 195–213

    work page 1994

  7. [7]

    F., Kosleck, S., Testa, D

    Clauss, G. F., Kosleck, S., Testa, D. and Hessner, K., 2008. Forecast of critical situation in shrot-crested seas. 27th Int. Conf. on Offshore Mechanics and Arctic Engineering. OMAE 2008. ASME, 1–10

    work page 2008

  8. [8]

    Numerical modeling of 3D fully nonlinear potential periodic waves

    Chalikov, D., Babanin and A.V., Sanina, E., 2014. Numerical modeling of 3D fully nonlinear potential periodic waves. Ocean Dyn. 64 (10), 1469–1486

    work page 2014

  9. [9]

    and Rosenthal, W., 2004

    Dankert, H. and Rosenthal, W., 2004. Ocean surface determination from X–band radar–image sequences. J. Geophys. Res. 109 (C4), 1–11

    work page 2004

  10. [10]

    and Overmars, M., 2008

    de Berg, M., Cheong O., van Kreveld, M. and Overmars, M., 2008. Computational Geometry: Algorithms and Applications. Springer- Verlag. ISBN: 978-3-540-77973-5

    work page 2008

  11. [11]

    and Puppo, E., 2000

    de Floriani, L., Magillo, P. and Puppo, E., 2000. Applications of Com- putational Geometry to Geographic Information Systems. Chapter 7 In Sack, J.R., Urrutia, J. (editors), Handbook of Computational Geome- try, pages 333–388. Elsevier. ISBN: 978-0-444-82537-7

    work page 2000

  12. [12]

    A., Drennan, W

    Donelan, M. A., Drennan, W. M. and Magnusson, A. K., 1996. Non- stationary analysis of the directional properties of propagating waves. J. Phys. Oceanogr. 26 (9), 1901–1914

    work page 1996

  13. [13]

    Space–time extremes in short–crested storm seas

    Fedele, F., 2012. Space–time extremes in short–crested storm seas. Jour- nal of Physical Oceanography. 26

    work page 2012

  14. [14]

    Space–time waves and spectra in the northern adriatic sea via a wave acquisition stereo system

    Fedele, F., Benetazzo, A., Forristall, G., 2011. Space–time waves and spectra in the northern adriatic sea via a wave acquisition stereo system. 30th ASME Int. Conf. Offshore Mechanics and Arctic Engng. Rotter- dam, The Netherlands OMAE2011- 49924 (6), 651–663

    work page 2011

  15. [15]

    Fedele, F., Cherneva, Z., Tayfun, M. A. and Guedes-Soares, C.,

    work page

  16. [16]

    Physics of Fluids 22 (036601)

    NLS invariants and nonlinear wave statistics. Physics of Fluids 22 (036601)

    work page

  17. [17]

    and Cavaleri, L., 2011

    Fedele, F., Gallego, G., Benetazzo, A., Yezzi, A., Sclavo, M., Bastianini, M. and Cavaleri, L., 2011. Euler characteristics and maxima of oceanic sea states. Journal Mathematics and Computers in Simulation 82 (6), 1102–1111

    work page 2011

  18. [18]

    and Teillaud, M., 2014

    Fogel, E. and Teillaud, M., 2014. The computational geometry algo- rithms library CGAL. ACM Communications in Computer Algebra 47 (3/4), 85–87

    work page 2014

  19. [19]

    Z., 2006

    Forristall, G. Z., 2006. Maximum wave heights over an area and the air gap problem. 25th ASME Int. Conf. Offshore Mechanics and Arctic Engng, Hamburg, Germany OMAE2006-92022, 11–15

    work page 2006

  20. [20]

    Voronoi diagrams and Delaunay triangulations

    Fortune, S., 2004. Voronoi diagrams and Delaunay triangulations. Chapter 23 In Goodman, J.E. and O’Rourke, J. (editors), Second Edi- tion, Handbook of Discrete and Computational Geometry, pages 513–

    work page 2004

  21. [21]

    ISBN: 0-8493-8524-5

    CRC Press. ISBN: 0-8493-8524-5

    work page

  22. [22]

    A variational stereo method for the three–dimensional reconstruction of ocean waves

    Gallego, G., Yezzi, A., Fedele and F., Benetazzo, A., 2011. A variational stereo method for the three–dimensional reconstruction of ocean waves. IEEE Trans. Geosci. Remote Sens. 49 (11), 4445–4457

    work page 2011

  23. [23]

    A Course in Ocean Engineering

    Gran, S., 1992. A Course in Ocean Engineering. Elsevier Science Pub- lishers

    work page 1992

  24. [24]

    and Trulsen, K., 2007

    Gramstad, O. and Trulsen, K., 2007. Influence of crest and group length on the occurrence of freak waves. Journal of Fluid Mechanics, 582, 463– 472-

    work page 2007

  25. [25]

    and Trulsen, K., 2011

    Gramstad, O. and Trulsen, K., 2011. Fourth–order coupled nonlinear schr¨ odinger equations for gravity waves on deep water. Phys. of Fluids, 23, 1–9

    work page 2011

  26. [26]

    and Trulsen, K., 2011

    Gramstad, O. and Trulsen, K., 2011. Hamiltonian form of the modified nonlinear Schrdinger equation for gravity waves on arbitrary depth. Journal of Fluid Mechanics, 670, 404–426

    work page 2011

  27. [27]

    Random Seas and Design of Maritime Structures

    Goda, Y., 2010. Random Seas and Design of Maritime Structures. World Scientific. 27

    work page 2010

  28. [28]

    E., Jostad, H

    Hansteen, O. E., Jostad, H. P., Tjelta, T. I. 2003. Observed platform response to a ”monster” wave. In Myrvoll, Frank. Field Measurements in Geomechanics: Proceedings of the Sixth International Symposium on Field Measurements in Geomechanics: 15–18 September, 2003, Oslo, Norway. Taylor & Francis. p. 73

    work page 2003

  29. [29]

    Hamilton, J., Hui, W. H. and Donelan, M. A., 1979. A statistical model for groupiness in wind waves. J. Geophys. Res. 84, 4875–4884

    work page 1979

  30. [30]

    Hasselmann, K. 1962. On the non-linear energy transfer in a gravity- wave spectrum. Journal of Fluid Mechanics. 12, 481–500

    work page 1962

  31. [31]

    G., Reichert, K., 2006,Sea surface elevation maps obtained with a nautical X-Band radar

    Hessner, K. G., Reichert, K., 2006,Sea surface elevation maps obtained with a nautical X-Band radar. 11th International Workshop on Wave Hindcasting and Forecasting. November 11-16th, 2007at Turtle Bay Resort, North Shore, Oahu, Hawaii, USA

    work page 2006

  32. [32]

    G., Nieto-Borge, J.C., Bell, P

    Hessner, K. G., Nieto-Borge, J.C., Bell, P. S., 2008, Nautical Radar Measurements in Europe: Applications of WaMos II as a Sensor for Sea State, Current and Bathymetry. In: Barale V., Gade M. (eds) Remote Sensing of the European Seas. Springer, Dordrecht

    work page 2008

  33. [33]

    Jha, A. K. and Winterstein, S. R., 2000. Nonlinear Random Ocean Waves: Prediction and Comparison with Data. ETCE/OMAE 2000 Proc., ASME. February 14-17. New Orleans. USA

    work page 2000

  34. [34]

    Statistical properties of random wave groups

    Kimura, A., 1980. Statistical properties of random wave groups. 17th International Conference on Coastal Engineering. ASCE, 2955–2973

    work page 1980

  35. [35]

    W., 2009

    King, F. W., 2009. Hilbert Transforms: Volume 1 and 2. Cambridge University Press

    work page 2009

  36. [36]

    Krogstad, H. E. and Trulsen, K., 2010. Interpretations and observations of ocean wave spectra. Ocean Dynamics 60, 973–991

    work page 2010

  37. [37]

    C., Nieto-Borge, J

    Liu, P. C., Nieto-Borge, J. C., Rodr´ ıguez, G., MacHutchon, K. R. and Chen, H. S., 2014. From Single Point Gauge to Spatio-Temporal Mea- surement of Ocean Waves: Prospects and Perspectives. ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineer- ing. Volume 8B: Ocean Engineering San Francisco, California, USA, June 8-13, 2014

    work page 2014

  38. [38]

    S., 1984

    Longuet-Higgins, M. S., 1984. Statistical properties of wave groups in a random sea state. Philos. Trans. R. Soc. London, Ser. A 312, 219–250

    work page 1984

  39. [39]

    S., 1986

    Longuet-Higgins, M. S., 1986. Wave groups statistic. In: Monahan, E. C., Niocaill, G. M. (Eds.), Oceanic whitecaps and their role in air– sea exchange processes. Springer, pp. 15–35. 28

    work page 1986

  40. [40]

    and Mørken, K., 2011

    Lyche, T. and Mørken, K., 2011. Spline Methods Draft, freely down- loadable from http: //www.uio.no/studier/emner/matnat/ifi/INF- MAT5340/v10/undervisningsmateriale/book.pdf

    work page 2011

  41. [41]

    Medina, J. R. and Hudspeth, R. T., 1990. A review of the analyses of ocean wave groups. Coastal Eng. 14, 515–542

    work page 1990

  42. [42]

    and Rikiishi, K., 1975

    Mitsuyasu, H., Tasai, F., Suhara, T., Mizuno, S., Ohkusu, M., Honda, T. and Rikiishi, K., 1975. Observations of the Directional Spectrum of Ocean WavesUsing a Cloverleaf Buoy. J. Phys. Oceanogr. 5, 750–760

    work page 1975

  43. [43]

    and Ramamonjiariosa, A., 1980

    Mollo-Christensen, E. and Ramamonjiariosa, A., 1980. Modelling the presence of wave groups in a random wave field. J. Phys. Oceanogr. 83, 4117–4122

    work page 1980

  44. [44]

    and Izquierdo- Gonz´ alez, P., 2004

    Nieto-Borge, J.C., Rodr´ ıguez-Rodr´ ıguez, G., Hessner, K. and Izquierdo- Gonz´ alez, P., 2004. Inversion of Marine Radar Images for Surface Wave Analysis. J. of Atmospheric and Oceanic Technology. 21(8), 1291–1300

    work page 2004

  45. [45]

    and Lehner, S., 2005

    Nieto-Borge, J.C., Schulz-Stellenfleth, J., Niedermeier, A. and Lehner, S., 2005. Analysis of statistical wave properties of linear and non linear two-dimensional wave fields derived from stochastic simulations. ASCE Waves 2005 Proceedings. Madrid

    work page 2005

  46. [46]

    and Hessner, K., 2013

    Nieto-Borge, J.C., Reichert, K. and Hessner, K., 2013. Detection of spatio-temporal wave grouping properties by using temporal sequences of X-band radar images of the sea surface. Ocean Modelling. 61:21-37

    work page 2013

  47. [47]

    K., 2005

    Ochi, M. K., 2005. Ocean Waves: The Stochastic Approach. Cambridge University Press

    work page 2005

  48. [48]

    Patrikalakis N. M. and Maekawa T., 2002. Shape Interrogation for Com- puter Aided Design and Manufacturing. Springer. ISBN: 3-540-42454-7

    work page 2002

  49. [49]

    Piotrowski, C. C. and Dugan, J. P., 2002. Accuracy of bathymetry and current retrievals from airborne optical time–series imaging of shoaling waves. IEEE Trans. Geosci. Remote Sens. 40, 2606–2618

    work page 2002

  50. [50]

    and Guedes Soares, C

    Rodr´ ıguez, G., Pacheco, M. and Guedes Soares, C. (2005). Maximum Wave Height Distribution in a Sea State: Effects of Record Length and Spectral Peakedness. J. Offshore Mech. Arct. Eng 127(4), 340-344

    work page 2005

  51. [51]

    Rutten, J., de Jong, S

    J. Rutten, J., de Jong, S. M. and Ruessink, G., 2017. Accuracy of Nearshore Bathymetry Inverted From X–Band Radar and Optical Video Data. IEEE Transactions on Geoscience and Remote Sensing. 3 (2), 282–228. 55 (2), 1106–1116. 29

    work page 2017

  52. [52]

    and Babanin, A.V., 2015

    Sanina, E.V., Suslov, S.A., Chalikov, D. and Babanin, A.V., 2015. Detection and analysis of coherent groups in three-dimensional fully- nonlinear potential wave fields. Ocean Modelling. 0:1-14

    work page 2015

  53. [53]

    Støle-Hentschel, S., Seemann, J., Nieto Borge, J. C. and Trulsen, K.,

    work page

  54. [54]

    Consistency between Sea Surface Reconstructions from Nauti- cal X-Band Radar Doppler and Amplitude Measurements. J. of Atmo- spheric and Oceanic Technology. 35(6), 1201–1220

    work page

  55. [55]

    Narrow-band nonlinear sea waves

    Tayfun, M.A., 1980. Narrow-band nonlinear sea waves. Jour. Geophys. Res., 85, C3, 1548-52

    work page 1980

  56. [56]

    and Fedele, F., 2007

    Tayfun, A. and Fedele, F., 2007. Wave–height distributions and nonlin- ear effects. Ocean Engineering. 34, 1631–1649

    work page 2007

  57. [57]

    B., 2001

    Trizna, D. B., 2001. Errors in bathymetric retrievals using linear dis- persion in 3D FFT analysis of marine radar ocean wave imagery. J. Geophys. Res. 39 (C7), 12529–12537

    work page 2001

  58. [58]

    Walker, D. A. G. Taylor, P. H. Eatock Taylor, R., 2004, The shape of large surface waves on the open sea and the Draupner New Year wave. Applied Ocean Researc. 26(3–4), 73–83

    work page 2004

  59. [59]

    C., Chuang, L

    Wu, L. C., Chuang, L. Z. H., D. J. Doong, D. and Kao, C. C.,

    work page

  60. [60]

    International Journal of Remote Sensing 32 (23), 8779–8798

    Ocean remotely sensed image analysis using two–dimensional con- tinuous wavelet transforms. International Journal of Remote Sensing 32 (23), 8779–8798

    work page

  61. [61]

    and Dittmer, J., 1994

    Ziemer, F. and Dittmer, J., 1994. A system to monitor ocean wave fields. IEEE Oceans’94 Conf. Proc. 2, 28–31

    work page 1994

  62. [62]

    and Senet, C., 2004

    Ziemer, F., Brockmann, C., Vaughan, R., Seemann, J. and Senet, C., 2004. Radar survey of near shore bathymetry within the OROMA project. EARSeL eProceedings. 3 (2), 282–228. 30

    work page 2004