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HF radar systems are widely and routinely used for the measurement of ocean surface currents and waves. Analysis methods presently in use are based on the assumption of infinite water depth, and may therefore be inadequate close to shore where the radar echo is strongest. In this paper, we treat the situation when the radar echo is returned from ocean waves that interact with the ocean floor. Simulations are described which demonstrate the effect of shallow water on radar sea-echo. These are used to investigate limits on the existing theory and to define water depths at which shallow-water effects become significant. The second-order spectral energy increases relative to the first-order as the water depth decreases, resulting in spectral saturation when the waveheight exceeds a limit defined by the radar transmit frequency. This effect is particularly marked for lower radar transmit frequencies. The saturation limit on waveheight is less for shallow water. Shallow water affects second-order spectra (which gives wave information) far more than first-order (which gives information on current velocities), the latter being significantly affected only for the lowest radar transmit frequencies for extremely shallow water. We describe analysis of radar echo from shallow water measured by a Rutgers University HF radar system to give ocean wave spectral estimates. Radar-derived wave height, period and direction are compared with simultaneous shallow-water in-situ measurements.

HF radar systems are widely used internationally to provide continuous monitoring of ocean waves and currents for a large range of environmental conditions.

Within the US, coastal ocean current mapping with HF radar has matured to the point where it is now considered an important component of regional ocean observing systems. A mid-Atlantic HF radar network now provides high resolution coverage within five localized networks, which are linked together to cover the full range of the mid-Atlantic coastal ecosystem. Similar regional networks around the US coastline are being formed into a national HF radar network.

While much of the focus of these networks until now has been on offshore current mapping observations, a longer-term objective is to develop and evaluate near-shore measures of waves and currents. These investigations aim to understand the interaction of waves in the shallow coastal waters and how energy is transformed into the creation of dangerous rip currents along the New-Jersey/Long-Island shorelines. Rutgers University radars cover these coastal regions at multiple frequencies from 4.5 to 25 MHz. Their echoes contain information on both currents and waves from deep water up into the shallow coastal zone, providing an excellent archive for such studies. This paper describes the analysis of both simulated and measured radar echo to demonstrate the effect of shallow water on radar observations and their interpretation.

Radar sea-echo spectra consist of dominant first-order peaks surrounded with lower-energy second-order structure. Analysis methods presently in use assume that the waves do not interact with the ocean floor, see [

The assumption of deep water is often invalid close to the coast and for broad continental shelves, and is particularly inadequate to describe the second-order sea-echo used to give information on ocean waves., as second-order echo is often visible above the noise only for close ranges. To interpret this echo correctly, we show that the effects of shallow water must be taken into consideration.

In Section 2, we give the basic equations describing radar echo from shallow water, expanding on the previous description given in [

It follows from the solution of the equations of motion and continuity that long ocean waves are more affected by shallow water. We define the depth at which waves interact with the ocean floor by the approximate relation:

Applying the lowest-order shallow-water dispersion equation to first-order backscatter from the sea gives the following equations for

_{0} is the radar wave vector, of magnitude _{0} , _{B}_{s}_{s}_{s}

The electromagnetic coupling coefficient has the same form as for deep water [

The total radar coupling coefficient Γ^{s}

It can be shown from these equations that at constant wavenumber, the coupling coefficient increases as the water depth decreases, resulting in an increasing ratio of second- to first-order energy as the depth decreases.

In the following analysis, we assume that the deep-water directional wave spectrum is spatially homogeneous and that any inhomogeneity in shallow water arises from wave refraction. When energy dissipation can be neglected, it follows from linear wave theory that since the total energy of the wavefield, is conserved, the shallow-water wave spectrum expressed in the appropriate variables is equal to the deep-water spectrum [

Here _{s}

The shallow- and deep-water rms waveheights are given by:

Substituting _{s}_{s}

The first- and second-order radar cross sections in shallow water at frequency _{s}

_{s}

To compute the second-order integral in _{s}

To calculate the integral in

Frequency contours are defined by:

Frequency contours are hence also discontinuous due to this effect at deep-water wave angles defined by _{0}). Normalized components

The discontinuities in the frequency contours are more pronounced when the contour is drawn in shallow-water wavenumber space, as it follows from

It can be seen from

The effects of shallow-water on measured radar spectra are illustrated in

To gain insight into the effects of shallow water, simulated radar echo spectra were calculated for a narrow-beam radar, using the model directional wave spectrum defined in [

For our model it follows from

This relationship is of course independent of radar frequency and has many angle symmetries.

It can be seen from

It follows from

It can be seen from

It can be seen from

It is shown in [_{s}_{s}

Since the coupling coefficient increases as the depth decreases, it follows from

It can be seen from

This subsection demonstrates an important point. Since we have shown that the waveheight itself actually decreases slightly upon moving into shallow water, while the second-order echo increases significantly due to the rapid growth of the coupling coefficient, wrongly using deep-water inversion theory to estimate waveheight will overestimate this important quantity. We note that all previous treatments and demonstrations of wave extraction have been based on deep-water theory, even when in fact many of the radar observations have been made in shallow water.

When the magnitude of the second-order energy approaches that of the first-order, it is apparent that the perturbation expansions on which _{Sat}

For shallow-water, the saturation of the radar spectrum is exacerbated by the increase of the coupling coefficient and the radar spectrum saturates for waveheights less than that defined by

In practice the theory may fail before this limit is reached. _{Sat}

For waveheights above the saturation limit, the waveheight predicted by the theory will be too high. However the theory cannot be applied at all when the second-order spectrum merges with the first, as then separation is not possible.

We estimate depths for which shallow-water effects become significant as follows: For first-order echo, the depth limit is defined by equality in ^{S}^{S}

^{S}^{S}

The results presented here are based on analysis of 10-minute radar spectra measured by a 25MHz SeaSonde located at Breezy Point, NJ. The time period from December 29 to 30, 2005, was chosen because simultaneous coverage provided by the SeaSonde and a bottom-mounted ADCP allowed a direct comparison to be made between results from the two sensors. The ADCP was located in the second radar range cell in water of depth 8m. The bathymetry in the area and the locations of the two sensors are shown in

In our analysis, depth contours near the radar are assumed to be parallel to shore and the depth profile is obtained from

Lipa and Barrick [_{n}

As described in [

The first- and second-order regions are separated.

The first order region is analyzed to give the ocean wave spectrum at the Bragg wavenumber. It is assumed that deep-water theory is adequate for this step, as Bragg waves are short and hence insensitive to the effects f shallow water, see

Second-order radar spectral data is collected from the four second-order sidebands of 10-minute averaged cross spectra and fit to a model of the deep-water ocean wave spectrum. Least-squares fitting to the radar Fourier coefficients is used to derive estimates of the significant wave height, centroid period and direction. During this step, the second-order spectrum is effectively normalized by the first-order, eliminating unknown multiplicative factors produced by antenna gains, path losses etc.

Shallow-water analysis requires a further step:

The shallow-water wave spectrum is calculated from the deep-water spectrum using

For our analysis, we define a model for the deep-water ocean wave spectrum as the product of directional and nondirectional factors:

The directional factor in _{c}

This model has proven satisfactory for use in deep-water wave extraction software that produces waveheight, period, and direction. It has been used for real-time SeaSonde systems for many years, providing good agreement with in-situ measurements e.g. as shown in [

It can be seen from

To emphasize the necessity of taking shallow water into account for this location, we estimated the waveheight assuming infinitely deep water.

We have presented the theory of narrow-beam HF radar sea-echo from shallow water and illustrated the effect of decreasing water depth using simulations for a simple swell model of the ocean wave spectrum. The second-order spectral energy increases relative to the first-order as the water depth decreases, resulting in spectral saturation when the waveheight exceeds a limit defined by the radar transmit frequency. This effect is particularly marked for lower radar transmit frequencies. For waveheights above the saturation limit, the perturbation expansions on which Barrick's

The shallow-water theory was then extended to apply to broad-beam systems such as the SeaSonde and applied to the interpretation of two days of radar data measured by a 25Mhz SeaSonde located on the New Jersey shore. During the measurement period, a storm passed over the area. An ADCP was operated in the second radar range cell in water 8m deep. Radar results were compared with simultaneous ADCP measurements. The comparison confirms aspects of the theory presented in Section 3. For the longer period waves occurring after the passage of the storm front, the standard deviation between SeaSonde and ADCP waveheight measurements decreased by a factor of three when the effects of shallow water were included in the analysis, and the bias decreased by a factor of five. Possible explanations for the remaining discrepancies are (a) the assumption of parallel depth contours (b) the assumption that the wave spectrum is homogeneous in the circular radar range cell (c) saturation in the radar spectrum around the peak of the storm, which, as discussed in Section 3, leads to the over-prediction of the waveheight.

Measured radar and buoy wave data from New Jersey as well as its analysis were funded through the New Jersey NOAA Sea Grant program.

Schematic geometry of the radar beam and an ocean wave train at a depth contour, denoted by the dashed line. Wave angles are measured counter-clockwise from the radar beam to the direction the wave is moving. Increasing _{s}

Examples of frequency contours for water of depth 10m (continuous lines) compared with the corresponding contours for deep water (dashed lines).

Normalized frequency: _{B}

Radar frequency: (a) 5Mhz, (b) 25Mhz

Spectra from a 5MHz SeaSonde monopole antenna. Range/ Water depth: (a) 18km/ 5 -20m (b) 30km/10-50m (c) 42km/ 20-70m (d) 48km/ 35-80m (e) 54km/ 40-100m

The ratio of shallow- to deepwater waveheight plotted vs. depth for a 12 s wave. Wave direction in deep water relative to the radar beam: Red 180°, Blue 135°

Bragg frequency plotted as a function of depth. Radar transmit frequency: Red 5Mhz, Blue 25Mhz

The frequency shift of the second-order peak from the Bragg frequency for an 11s wave. (a) 5Mhz (b) 25Mhz. Angle between wave and radar beam: Yellow 0°, Blue 45°, Green 135°, Red 180°

The absolute value of the coupling coefficient Γ_{s}

The absolute value of the coupling coefficient vs. depth for waves of different period. Radar transmit frequency: 5Mhz. Wave period: Red 15s, Blue 12s. Green 9s.

Ratio of second - to first-order energy for an 11s wave. Significant waveheight: 2.4m. Radar transmit frequency: (a) 5 Mhz, (b) 25 Mhz. Angle between wave and radar beam: Yellow 0°, Blue 45°, Green 135°, Red 180°

Significant waveheight saturation limits for an 11-second wave coming straight down the radar beam. Radar transmit frequency: (a) 5 Mhz , (b) 25 Mhz Red: deep-water saturation limit _{Sat}

Depths at which shallow-water effects become significant vs. radar transmit frequency. Red: second-order echo. Blue: First-order echo.

The coastline and bathymetry (contours in meters) around Breezy Point, New Jersey, showing the positions of the SeaSonde and the bottom-mounted ADCP.

(a) Spectra measured by the 25MHz SeaSonde at Breezy Point. at 1:00pm 12/30/2005. Range: (a) 3 km (b) 6 km (c) 9 km.

SeaSonde (red) and ADCP (blue) results for (a) Significant waveheight (b) Wave period (c) Wave direction (d) Wind direction.

Significant waveheight:

Red: SeaSonde calculated assuming infinite water depth. Blue: ADCP.

Comparison statistics, radar vs. ADCP.

Before storm | After storm | |
---|---|---|

Standard deviation | 0.25m | 0.35m |

Bias | -0.23m | 0.17m |

Standard deviation | 0.76s | 0.60s |

Bias | 0.70s | -0.27s |

Standard deviation | 13.8° | 19.7° |

Bias | -9.5° | 17.0° |

Comparison statistics, radar vs. ADCP assuming deep water.

Before storm | After storm | |
---|---|---|

Standard deviation | 0.25m | 0.95m |

Bias | 0.19m | 0.90m |