## Contours of slopes of a rippled water surface |

Optics Express, Vol. 19, Issue 20, pp. 18789-18794 (2011)

http://dx.doi.org/10.1364/OE.19.018789

Acrobat PDF (1073 KB)

### Abstract

The appearance of a horizontal array of linear lamps below the water surface when viewed from above is approximately in the form of contours of one component of the water surface slope. The degree of approximation is a fraction of one percent when this method is used to describe the slopes of a wind ruffled surface. An extension of the method to image both components of slope requires two arrays of lamps pulsed alternately and in synchronism with a fast camera.

© 2011 OSA

## Introduction

1. P. A. Hwang, D. B. Trizna, and J. Wu, “Spatial measurements of short wind waves using a scanning slope sensor,” Dyn. Atmos. Oceans **20**(1-2), 1–23 (1993). [CrossRef]

3. T. E. Hara, J. Bock, J. B. Edson, and W. R. McGillis, “Observation of short wind waves in coastal waters,” J. Phys. Oceanogr. **28**(7), 1425–1438 (1998). [CrossRef]

4. C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. **19**(5), 055503 (2008), doi:. [CrossRef]

## Contouring one component of slope

*x*and

*y*are horizontal components, lamps parallel to the

*y*axis produce contours of the

*x*component of slope. Viewed from above, when the water surface is flat, the lamps appear as almost straight colored lines (Fig. 1(a) ). When the water surface is ruffled by wind the lamp images are broken into multiple curved and colored stripes (Fig. 1(b)). In this example,

*x*is positive downwind and the lamps are parallel to the

*y*direction. Each color provides, with a slight degree of uncertainty, a contour of

*x*-slope, the value of which varies slowly in the

*x*direction. We shall refer to such stripes as ‘contour stripes’. The uncertainty arises because the true value of

*x*-slope at each point within a contour stripe depends to some extent on the y component of slope at which the refraction of the light ray occurs.

## Uncertainty of contour value within a contour stripe

**be in an upward directed unit vector normal to the water surface at**

*N**z*-axis upwards.

**has components**

*N**θ*is the vertical angle of

**and**

*N**α*its azimuth. Snell’s law for refraction of light rays is contained in the expressionwhere

*n*is the index of refraction of the water relative to air,

**and**

*I***are unit vectors parallel to the incident and refracted rays respectively, and**

*R**k*is positive. Total internal reflection of the incident ray occurs when

*k*sin

*θ*is positive, Eqs. (2a) and (2b) have the signs of cos

*α*and sin

*α*respectively, and

*α*can be found from the four quadrant arc tangent of the ratio of Eq. (2b) to Eq. (2a). The sum of squares of Eq. (2a) and Eq. (2b) divided by the square of Eq. (2c) leads to the square of the tangent of

*θ*. The (

*x*,

*y*) components of slope of the water surface at

*p*are

*h*, must be measured by some means (for example, acoustically) in order to relate pixel coordinates of image points, thus (

*ψ*,

*γ*), to the corresponding (

*x*,

*y*) points on the water surface. In addition the vertical component of distance

*H*to the light source must be known. From these four input data together with the function

**(**

*S**X, Y, -H*) describing the shape of the light source we can compute slope components of the water surface which could possibly contribute to every point within a contour stripe at water surface position, (

*x*,

*y*). In general we find that there is a range of true contour values within each contour stripe. The resulting uncertainties of true values are illustrated in Figs. 3 and 4 .

*h*= 2.5 m above the water surface (thus assuming the waves on the surface are negligibly high), and that the light source is linear, 1.5 m long, parallel to the

*y*axis and 2.5 m below the water surface. In Fig. 3 the center of the lamp is vertically below the camera (

*X*= 0). In Fig. 4 it is displaced 1.0 m (

*X*= 1.0 m). With either configuration the contour stripes are very nearly true contours of

*x*-slope which vary slowly in the

*x*direction but with a degree of ambiguity depending on the range of possible values of

*y*-slope. There would be no ambiguity if the

*y*-slopes were known within the contour stripe. Without such knowledge we show the uncertainty in two ways. The first assumes that rays of light entering the camera may have originated anywhere along the entire length of the lamp. The uncertainties are represented by the standard deviation of

*x*-slope for an even distribution of light rays originating along the lamp such that |

*Y*| < 0.75 m. This would imply a very wide and unlikely range of

*y*-slopes. The other method weighs the deviations in

*x*-slope according to the probability of those

*y*-slopes required to produce rays from any position along the lamp. The

*y*-slope probability is taken as the nearly Gaussian form found in the open sea [5

5. F. M. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. **111**(C6), C06005 (2006), doi:. [CrossRef]

*x*-slope contour stripes is a maximum when the wind is parallel to the

*y*direction and increases with wind speed. The slope probability is known to be skewed: highest slopes are directed upwind and the most probable slope is a small value directed downwind. As a result, when the wind is coming from the positive

*y*direction the uncertainties in

*x*-slope contour stripes are larger at positive

*y*locations than at negative

*y*locations. This is illustrated in Figs. 3(c) and 4(c) where it is assumed the wind is blowing from the positive

*y*direction at 15 m/s (as measured 10 m above the sea). The uncertainties would be reduced at lower windspeeds and when the wind direction is along the

*x*axis.

## Lamp spacing and reconstruction from contours

## Exposure time and evolution of slopes in time

## Contouring both slope components

*y*parallel and one

*x*parallel. A single high speed camera, with exposures synchronized with the lamps can then image

*x*-slope and

*y*-slope contour stripes in alternate images.

## Spectral range

*x*and

*y*directions (wavenumbers 0.036 and 0.048 per mm). Multiple cameras or cameras with higher resolution can extend these long wavelength limits.

## Possible use at sea

## Acknowledgment

## References and links

1. | P. A. Hwang, D. B. Trizna, and J. Wu, “Spatial measurements of short wind waves using a scanning slope sensor,” Dyn. Atmos. Oceans |

2. | C. Cox and X. Zhang, “Optical methods for study of sea surface roughness and microscale turbulence,” in Proc. SPIE, Optical Technology in Fluid, Thermal, and Combustion Flow, III, Vol. 3172, S. S. Cha, J. D. Trolinger, and M. Kawahashi, ed. (1997). |

3. | T. E. Hara, J. Bock, J. B. Edson, and W. R. McGillis, “Observation of short wind waves in coastal waters,” J. Phys. Oceanogr. |

4. | C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol. |

5. | F. M. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res. |

**OCIS Codes**

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(120.6660) Instrumentation, measurement, and metrology : Surface measurements, roughness

(080.4295) Geometric optics : Nonimaging optical systems

**ToC Category:**

Ocean Optics

**History**

Original Manuscript: July 14, 2011

Revised Manuscript: August 25, 2011

Manuscript Accepted: August 27, 2011

Published: September 12, 2011

**Virtual Issues**

Vol. 6, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Charles Cox and Xin Zhang, "Contours of slopes of a rippled water surface," Opt. Express **19**, 18789-18794 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-18789

Sort: Year | Journal | Reset

### References

- P. A. Hwang, D. B. Trizna, and J. Wu, “Spatial measurements of short wind waves using a scanning slope sensor,” Dyn. Atmos. Oceans20(1-2), 1–23 (1993). [CrossRef]
- C. Cox and X. Zhang, “Optical methods for study of sea surface roughness and microscale turbulence,” in Proc. SPIE, Optical Technology in Fluid, Thermal, and Combustion Flow, III, Vol. 3172, S. S. Cha, J. D. Trolinger, and M. Kawahashi, ed. (1997).
- T. E. Hara, J. Bock, J. B. Edson, and W. R. McGillis, “Observation of short wind waves in coastal waters,” J. Phys. Oceanogr.28(7), 1425–1438 (1998). [CrossRef]
- C. J. Zappa, M. L. Banner, H. Schultz, A. Corrada-Emmanuel, L. B. Wolff, and J. Yalcin, “Retrieval of short ocean wave slope using polarimetric imaging,” Meas. Sci. Technol.19(5), 055503 (2008), doi:. [CrossRef]
- F. M. Bréon and N. Henriot, “Spaceborne observations of ocean glint reflectance and modeling of wave slope distributions,” J. Geophys. Res.111(C6), C06005 (2006), doi:. [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.