Introduction

The Doppler effect is a well-known phenomenon, which states the frequency of a wave changes according to the relative velocity of the source and the observer. The effect is widely accepted and has been used in radar, sonar and sensors to measure the relative speed of a moving target1,2,3.

Doppler effect is also used to interpret the frequency shift of the HF radio wave’s backscatter from the moving ocean wave. In 1955, Crombie4 observed sea echo with a HF radar at 13.5 MHz. The radar sea-echo spectra contain two dominant peaks shift from the incident radio frequency. Crombie explained that the two dominant peaks were Bragg scatter, coherently backscatter comes from ocean waves half the radar wavelength λ, propagating radially toward, or away from the radar. And the frequency shift 0.38 Hz, is the Doppler shift corresponding to the phase velocity of the Bragg wave. The frequency shift of the Bragg scatter is named as Bragg frequency fB afterwards.

Crombie mentioned there was a subsidiary peak at a frequency of in the sea echo spectrum and Crombie thought it is due to waves having a length twice the length of the Bragg wave with the phase velocity times the Bragg wave. Barrick5 attributed the subsidiary peak with the frequency of to some non-linearity features in the wave formulas and the scattering process. E.D.R.Shearman6 proposed that sea waves of length not only Bragg wave, but also of L = λ, 3λ/2, 2λ, 5λ/2, , etc, which contains harmonics of Bragg wave, may take place Bragg resonant backscatter and these harmonics correspond Doppler-shifted contributions should appear in the echo spectrum at . All of the above and other models7,8,9,10,11,12,13,14,15, were deduced from the scattering measurement data collected from actual sea surface based on Doppler effect and has not been verified under man-made conditions.

It would be very difficult to collect the water wave scatter with a HF radar under man-made conditions due to the tens meters wave length of radio wave. Subscale experiment is helpful to understand the mechanism of the frequency shifts of radio wave backscatter from the Bragg and Non-Bragg water waves. A coherent UHF radar at 340 MHz is designed for the subscale experiment. Backscatter data from monochromatic Bragg water waves and non-Bragg water waves precisely generated by a man-made wave-tank was collected with the UHF radar. The spectrum of the radar data is analyzed and frequency shift can’t be interpreted with the Doppler effect mentioned above. A new mechanism is thus proposed for the frequency shift of backscatter from Bragg and non-Bragg periodic gravity water waves.

Wave tank experiment

The wave tank and UHF coherent radar position are shown in Fig. 1(a).

Figure 1
figure 1

The experiment environment.

(a)The wave tank and radar deployment. (b) The time domain waveforms of the water wave recorded with sensors.

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The coherent UHF radar was a full digital chirp radar16, operating at 340 MHz with bandwidth of 15 MHz. The sweep period was 0.043 s. Spectrum analyzer collected data of 256 sweep periods for coherent processing. It’s easy to see that the radar wave length λ is 0.88 m and the frequency resolution of the spectrum analyzer is 0.091 Hz. The Bragg wave length λB = λ/2 = 0.44 m. The transmitting and receiving antennas were V-V polarized, which were set on the left side of the tank. Radio wave beam was pointing to the water wave maker and the incidence angle to the water surface was about 3°.

The length, width and the depth of the wave tank are 65 m, 201 cm, and 180 cm respectively. The wave-maker is on the right side and the water wave propagates from right to the left as shown in Fig. 1(a). An energy dissipation system is on the left side, to prevent splash and water wave reflections. The wave tank generates periodic gravity sinusoidal waves and the wave’s period ranges between 0.38 s~1.9 s. In order to keep the water wave shape and smooth, the amplitudes of the water wave changed with the wave length. Two pressure sensors, 0.4 m apart, with sample rate 50 Hz, were put in the water to monitor the water wave shape and parameters. Figure 1(b) demonstrates the time domain waveform recorded by the sensors, given the water wave length 3λB, amplitude 5 cm and the water deep 0.8 m. In term of Fig. 1(b), the wave length is 1.316 m and the phase velocity is 1.43 m/s, which is almost exactly the same as the theoretical value .

Experiment data and analyzing

Wave tank experiments were carried out to study the mechanism of Bragg and non-Bragg backscatter, i.e., the relationship between the frequency shift of the radar echo and the phase velocity of the water wave.

The wave length, frequency, amplitude of the periodic water wave generated by wave tank is shown in Table 1 and so are the phase velocities of the water waves and the calculated Doppler shift according the phase velocities. fB is the Bragg frequency, according to the radar wave length. which is 1.883 Hz.

Table 1 Parameters of the water wave and calculated Doppler shift (λ B = 0.44 m, fB = 1.883).
Full size table

The power density spectra of the backscatter from the wave tank collected by the UHF radar were shown in Fig. 2. These plots represent the received signal power against normalized frequency shift from the carrier (the carrier being located at zero and the Bragg frequency at positions +1). As mentioned above, the frequency resolution of the spectra analyzer is 0.091 Hz and the Bragg frequency is 1.883 Hz, so the normalized frequency resolution is 0.048.

Figure 2
figure 2

Power spectra of the backscatter from different wavelengths (a) λ/2, (b) 2λ/2, (c) 3λ/2, (d) 4λ/2, (e) 5λ/2, (f) 10λ/2.

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Figure 2(a) is the power spectrum according to Bragg wave. Corresponding to the Table 1, the predicted Doppler shift fdpl of the backscatter is the Bragg frequency fB, thus a strong peak should be at position +1 in the power spectrum Figure. In Fig. 2(a), the dominant peak is at the position 1.014. The discrepancy between the measured value and predicted value is less than the frequency resolution of the spectra analyzer. The result seems to confirm the Doppler effect in that the predicted Doppler shift is exactly the same as the measured shift. We should note, however, there is another peak at position −1.014, which can’t be interpreted by Doppler effect because there is no water wave with the wave length of λB propagating away from the radar.

For the water wavelength of 2λB, the predicted Doppler shift of the backscatter, fdpl, should be 1.41 fB. The spectrum measured with the UHF radar is shown in Fig. 2(b). There is a peak at the position 1.401 indeed, but very weak. The dominant peak, in particular, is at the position 0.7245, about half the predicted Doppler shift fdpl. Opposite to the position, at position −0.7245, a secondary peak exists.

For the water wave length of 3λB, the predicted Doppler shift fdpl is 1.73 fB. The measured spectrum was shown in Fig. 2(c). The dominant peak is at the position 0.5796, about fdpl/3. Other peaks appear at positions about −fdpl/3, 2fdpl/3, fdpl, 4fdpl/3, etc.

Figure 2(d) is the measured spectrum of water wavelength 4λB. The predicted Doppler shift fdpl is 1.99 fB. The dominant peak is at the position about fdpl/4. Other peaks appear at positions about −fdpl/4, ±2fdpl/4, ±3fdpl/4, and fdpl etc.

Figure 2(e) is the measured spectrum of water wavelength 5λB. The predicted Doppler shift fdpl is 2.21 fB. There are distinct peaks appear at positions about ±fdpl/5, ±2fdpl/5, ±3fdpl/5, and fdpl etc. In contrast to other spectra mentioned, the strongest peak appears at minus frequency axis of position −fdpl/5.

Many other backscatter data of different water wavelengths were collected by coherent UHF radar. Figure 2(f) is the measured spectrum of water wavelength 10λB. The predicted Doppler shift fdpl of the backscatter should be 2.86 fB. There are distinct peaks appear at positions about ±fdpl/10, ±2fdpl/10, ±3fdpl/10, , etc.

The following commonness can be reached by viewed the above experiment data: 1). Each of the spectra of the backscatter contains two or more peaks which are equally spaced; 2). The dominant peak is not the predicted Doppler effect peak corresponding to the waves phase velocity except when the wave is Bragg wave; 3). The water wave propagating to the radar results in not only positive frequency shift, but also the minus frequency shift and the intensity of which may be dominant once in a while (e.g. Fig. 2(e)). The Doppler effect according to the water wave phase velocity can’t interpret the above phenomena of the frequency shift and what is the mechanism? The radar’s range resolution of this experiment had been finite and what characters would be present when the radar’s range resolution becomes infinite?

Theoretical modeling

In order to reveal the mechanism of frequency shift of the backscatter, we study the backscatter of the sinusoidal water wave in time domain with finite radar range resolution.

Consider a parallel sinusoidal water wave propagating toward to the radar. Defining xyz rectangular coordinate system, the water surface is expressed as ξ = ξ(x, t), where x = (x, y) is horizontal coordinates and t is the time. The radar is at position x = 0. The vector of radio wave is of x axis positive direction.

Define the water depth d, the parameters of water wave are defined as vector , angle frequency ω1, amplitude a1. If is of x axis positive direction, k1 > 0, it means the water wave propagating along the x axis positive direction. The range resolution of the radar is D. The water surface can be expressed as

Ignoring the boundary effect, the backscattering coefficient σ can be expressed as σ = σ(x, t) and σ(x, t) is periodic both in space domain and time domain17. So, σ(x, t) can be written as σ(x, t) = σ(x + mL1, t + mT1), where L1 = 2π/k1, T1 = 2π/ω1, m is an integer.

Expansion of σ(x, t) in Fourier series,

where cn and ϕn are the amplitude and phase coefficient of the Fourier series. c0 can be neglected on account of it has no influence to the frequency shift.

Assume the incident electric field is

The electric field at the receiver scattering from position x is,

t0 = 2x/c, is the time delay of the radio wave, c is the velocity of light.

The average total electric field of the backscatter from radar range resolution 0~D is

Substitute Eqs 2,3 and 4 into Eq. 5:

Or write Eq. 6 as:

Define the angle frequency shift Δω = ω − ω0. In term of Eq. 7, the power spectrum of the backscatter during the coherent integrate time T can be written as:

n is an integer and .

When T → ∞,

The parameter T is generally set to enough large in order to get high frequency resolution for the real ocean radar systems. So, Eq. 8 can be approximately rewritten as:

Model Discussion

Equation 12 shows, that the power spectrum of the backscatter from periodic water wave is consisted of a series of discrete peaks, which are equally spaced in frequency axis by the fundamental frequency ω1 and the amplitude is decided by factor Cn and Wn.

Consider a periodic sinusoidal water wave with wavelength of L1 = N · λ/2, propagating towards the radar. So, k1 = −2k0/N and . If the wave height a1 is very small comparing to the wave length L1, the backscatter model of the surface is close to a sinusoidal diffraction grating5,12,17,18, that means, c1 ≫ c2, c3, in Eq. 2. So, Cn is symmetric about n = 0 and with maximum value at n = ±1. Wn is a sinc function with symmetric center and maximum value at n = N.

The relationship of Cn, Wn and P(Δω) is illustrated in Fig. 3. Figure 3(a,b) are the simulated data Cn and Wn respectively with k1 = −2k0/3. Figure 3(c) is the power spectrum P(Δω). The relationship expressed in Eq. 12 can be seen as a filter with the input signal spectrum Cn, transfer function Wn and output spectrum P(Δω). When the radar range D is a finite value, Wn acts as a band-pass filter and the passband is highly influenced by the radar range resolution D. The smaller the D, the wider the passband. In Fig. 3(b), the maximum value of Wn occurs at the position 3ω1. After the Cn pass through Wn, the value of Cn at position 3ω1 keeps undamped and which at other frequency positions are depressed corresponding to the shape of |Wn|.

Figure 3
figure 3

The relationship of Cn, Wn and P(Δω).

The amplitude of (a) Cn, (b) Wn, (c) P(Δω).

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For the experiment data, when L1 = λ/2, k1 = −2k0, we have N = 1 and ω1 = ωB. At position Δω = ±ω1, |Cn| has the maximum value and the symmetric center and maximum value of |Wn| is at Nω1 = ω1, So the P(Δω) has a dominant peak at position ω1 and get a secondary peak at position Δω = −ω1 due to the lower sidelobe level of |Wn| at this position. No obvious peaks at other positions can be seen because of and are depressed further by |Wn|. The characters addressed above are clearly shown in Fig. 2(a). The two peaks are at ±1.014ωB and there is a deviation of 0.014ωB which is less than the frequency resolution 0.048ωB of the radar system.

For the water wavelength of L1 = 2 · λ/2, k1 = −2k0/2, . The maximum value of |Wn| is at Δω1 = 2ω1. After the Cn pass through Wn, component with frequency 2ω1 get maximum gain and other components are depressed with different values. The experiment result is shown in Fig. 2(b). The actual frequency position is at ±0.7245ωB, 1.401ωB and the frequency deviation is also less than the frequency resolution 0.048ωB of the radar system.

Other instances include water wavelengths of L1 = 3 · λ/2 with ω1 = 0.576ωB, L1 = 4 · λ/2 with ω1 = 0.498ωB, L1 = 5 · λ/2 with ω1 = 0.442ωB, L1 = 10 · λ/2 with ω1 = 0.287ωB. The experiment data are in good accordance with the Eq.12.

From the experiment data, the main energy of the Wn is concentrated at the positive frequency axis due to the water wave propagating to the radar system. So for most of the power spectrum, P(ω1) is obviously great than P(−ω1) except Fig. 2(e). This is because that Wn is not a monotonic function.

When D → ∞, Eq. 10 changes to Wn → σ(n + 2k0/k1), then, the Eq. 12 can be written as:

Wn is of non-zero value only when the 2k0/k1 is an integer. If the water wave propagating forward to the radar, k1 = −2k0/N,

The backscatter power spectrum of the water wave is

For the gravity water wave with wave number k1, , its phase velocity is , the Doppler shift according to the phase velocity is,

So,

The Eq. 16 means, when the radar range trends to infinite, only those periodic water wave with the wave length integer multiple of the Bragg wave, can generate coherent backscatter and the power spectrum contains only a single peak with a frequency shift Δω = Nω1, which just equals the Doppler shift corresponding to its phase velocity mathematically.

Conclusion

Backscatter data from periodic sinusoidal water wave, both Bragg water wave and non-Bragg water waves precisely generated by a man-made wave tank, was collected with a coherent UHF radar. A few points can be observed from the spectrum of the backscatter data: 1) A single train gravity water wave propagating to the radar system can result in both positive frequency shifts and negative frequency shifts simultaneously; 2) The power spectrum of the backscatter from a gravity water wave with a fixed phase velocity contains a serial frequency shift components.

The above phenomena were obviously in conflict with the interpretation about the Doppler effect to the phase velocity of the gravity water wave.

A new mechanism was proposed to interpret the frequency shift of the backscatter from gravity water wave: The frequency shift is not the Doppler shift corresponding to the wave’s phase velocity, but the water wave’s frequency and its integral multiple with different weighting factor. The weighting factor is related with water wave length, radar wave length and radar range resolution.

Specifically, in the proposed model, when the radar range resolution trends to infinite, the frequency shift of backscatter from a periodic water wave just equals the Doppler shift corresponding to its phase velocity mathematically, but with different mechanism.

Additional Information

How to cite this article: Wen, B. and Li, K. Frequency shift of the Bragg and Non-Bragg backscattering from periodic water wave. Sci. Rep. 6, 31588; doi: 10.1038/srep31588 (2016).