# Micro Radar Surface Velocimetry for Hydrologic Signal Processing Using a Bandpass Filtering Approach

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Hydrologic Signal Processing Approach

^{8}m/s), ${f}_{o}$ is the frequency of the microwave radar antennae (10.525 GHz), ${f}_{D}$ is the Doppler peak frequency caused by the movement of the surface ripples, ${\theta}_{1}$ is the tilt angle, ${\theta}_{2}$ is the yaw angle, and $\theta $ is the angle between the radar waves and the flow direction. The transmission speed of the radar waves, $c$, the frequency of the radar antennae, ${f}_{o}$, and the angle between the radar waves and the flow direction, $\theta $, are known constants; hence, the surface velocity, $v$, can be estimated by measuring the receiving frequency of the radar echo, ${f}_{D}$. Therefore, the precision of the radar echo frequency, ${f}_{D}$, is very important in estimating the surface velocity of a river.

#### 3.1. Fast Fourier Transform

#### 3.2. Wavelet Transform

- 1.
- The energy of the mother wavelets must be finite, as shown in Equation (6):$${E}_{\varphi}={\displaystyle {\int}_{-\infty}^{\infty}{\left|\phi (s)\right|}^{2}}dt<\infty $$

- 2.
- Mother wavelets must meet the admissible conditions, as shown in Equation (7):$${C}_{g}={\displaystyle {\int}_{0}^{\infty}\frac{{\left|\stackrel{\u2322}{\phi}(\omega )\right|}^{2}}{\omega}}d\omega <\infty ,\text{}\stackrel{\u2322}{\phi}(\omega )={\displaystyle {\int}_{-\infty}^{\infty}\phi (s){e}^{-i\omega t}dt}$$

#### 3.3. Bandpass Filter

_{cL}, of a lower frequency band and the cut-off frequency, F

_{cH}, of a higher frequency band; the bandpass width of the filter is the difference between F

_{cL}and F

_{cH}.

_{c}= 0.1) of the signal in the frequency spectrum [31,32]. The signal was only allowed oscillation between the stopband cutoff frequencies (i.e., F

_{cL}–F

_{cH}), and the amplitudes in the other intervals (i.e., 0–F

_{cL}and higher than F

_{cH}) were set to zero to filter out the lower and higher portions of the signal, arising from the measurement errors.

## 4. Experimentation and Analysis

## 5. Conclusions and Suggestions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- LeBoursicaud, R.; Pénard, L.; Hauet, A.; Thollet, F.; LeCoz, J. Gauging extreme floods on youtube: Application of LSPIV to home movies for the post-event determination of stream discharges. Hydrol. Process.
**2015**, 30, 90–105. [Google Scholar] [CrossRef] - Muste, M.; Fujita, I.; Hauet, A. Large-scale particle image velocimetry for measurements in riverine environments. Water Resour. Res.
**2008**, 44, W00D19. [Google Scholar] [CrossRef] - Tauro, F.; Porfiri, M.; Grimaldi, S. Orienting the camera and firing lasers to enhance large scale particle image velocimetry for stream flow monitoring. Water Resour. Res.
**2014**, 50, 7470–7483. [Google Scholar] [CrossRef] - Melcher, N.B.; Costa, J.E.; Haeni, F.P.; Cheng, R.T.; Thurman, E.M.; Buursink, M.; Spicer, K.R.; Hayes, E.; Plant, W.J.; Keller, W.C.; et al. River discharge measurements by using helicopter-mounted radar. Geophys. Res. Lett.
**2002**, 29, 41-1–41-4. [Google Scholar] [CrossRef] - Cheng, R.T.; Gartner, J.W.; Mason, R.R.; Costa, J.E.; Plant, W.J.; Spicer, K.R.; Haeni, F.P.; Melcher, N.B.; Keller, W.C.; Hayes, K. Evaluating a Radar-Based, Non-Contact Streamflow Measurement System in the San Joaquin River at Vernalis; Open File Report of 2004-1015; U.S. Geological Survey: Menlo Park, CA, USA, 2004. Available online: http://pubs.usgs.gov/of/2004/1015/OFR2004_1015_new.pdf (accessed on 1 March 2016).
- Costa, J.E.; Cheng, R.T.; Haeni, F.P.; Melcher, N.; Spicer, K.R.; Hayes, E.; Plant, W.; Hayes, K.; Teague, C.; Barrick, D. Use of radars to monitor stream discharge by noncontact methods. Water Resour. Res.
**2006**, 42, W07422. [Google Scholar] [CrossRef] - Hsu, Y.S.; Tung, T.C.; Chou, H.C.; Chang, K.C.; Lee, J.J.; Yu, B.H.; Huang, P.C. Applications of the radar surface velocimeter. J. Taiwan Water Conserv.
**2006**, 54, 82–91. [Google Scholar] - Lee, J.S.; Julien, P.Y. Electromagnetic wave surface velocimetry. J. Hydraul. Eng.
**2006**, 132, 146–153. [Google Scholar] [CrossRef] - Costa, J.E.; Spicer, K.R.; Cheng, R.T.; Haeni, F.P.; Melcher, N.B.; Thurman, E.M.; Plant, W.J.; Keller, W.C. Measuring stream discharge by non-contact methods: A proof-of-concept experiment. Geophys. Res. Lett.
**2000**, 27, 553–556. [Google Scholar] [CrossRef] - Tamari, S.; García, F.; Arciniega-Ambrocio, J.I.; Porter, A. Testing a handheld radar to measure water velocity at the surface of channels. La Houille Blanche
**2014**, 3, 30–36. [Google Scholar] [CrossRef] - Kim, D.S.; Yang, S.K.; Jung, W.Y. Error analysis for electromagnetic surface velocity and discharge measurement in rapid mountain stream flow. J. Environ. Sci. Int.
**2014**, 23, 543–552. [Google Scholar] [CrossRef] - Kolmogorov, A.N. Stationary sequences in hilbert space. Mosc. Univ. Math. Bull.
**1941**, 2, 1–40. [Google Scholar] - Wiener, N. The Interpolation, Extrapolation and Smoothing of Stationary Time Series; OSRD 370, Report to the Services 19, Research Project DIC-6037; Massachusetts Institute of Technology: Cambridge, MA, USA, 1942. [Google Scholar]
- Kalman, R.E. A new approach to linear filtering and prediction problems. J. Basic Eng.
**1960**, 82, 35–45. [Google Scholar] [CrossRef] - Woody, C.D. Characterization of an adaptive filter for the analysis of variable latency neuroelectric signals. Med. Biol. Eng.
**1967**, 5, 539–554. [Google Scholar] [CrossRef] - Thakor, N.V.; Zhu, Y.S. Applications of adaptive filtering to ECG analysis: Noise cancellation and arrhythmia detection. IEEE Trans. Biomed. Eng.
**1991**, 38, 785–794. [Google Scholar] [CrossRef] [PubMed] - Haykin, S.O. Adaptive Filter Theory, 5th ed.; Prentice-Hall Incorporation: Upper Saddle River, NJ, USA, 2013. [Google Scholar]
- Lin, J.W. Repetitive model refinement for structural health monitoring using efficient Akaike information criterion. Smart Struct. Syst.
**2015**, 15, 1329–1344. [Google Scholar] [CrossRef] - Plant, W.J.; Keller, W.C.; Hayes, K. Measurement of river surface currents with coherent microwave systems. IEEE Trans. Geosci. Remote Sens.
**2005**, 43, 1242–1257. [Google Scholar] [CrossRef] - Cooley, J.W.; Tukey, J.W. An algorithm for the machine calculation of complex Fourier series. Math. Comput.
**1965**, 19, 297–301. [Google Scholar] [CrossRef] - Mallat, S.G. A theory for multi-solution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell.
**1989**, 11, 674–693. [Google Scholar] [CrossRef] - Daubechies, I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory
**1990**, 36, 961–1005. [Google Scholar] [CrossRef] - Daubechies, I. Ten Lectures on Wavelets; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1992. [Google Scholar]
- Chui, C.K. An Introduction to Wavelets; Academic Press Incorporation: Boston, MA, USA, 1992. [Google Scholar]
- Misiti, M.; Misiti, Y.; Oppenheim, G.; Poggi, J.M. Wavelet Toolbox for Use with MATLAB; The Math Works Incorporation: Natick, MA, USA, 1996; Available online: http://web.mit.edu/1.130/WebDocs/wavelet_ug.pdf (accessed on 1 March 2016).
- Gurley, K.; Kareem, A. Applications of wavelet transforms in earthquake, wind and ocean engineering. Eng. Struct.
**1999**, 21, 149–167. [Google Scholar] - Mallat, S. A Wavelet Tour of Signal Processing; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Donoho, D.L. De-noising by soft-thresholding. IEEE Trans. Inf. Theory
**1995**, 41, 613–627. [Google Scholar] [CrossRef] - Taswell, C. The what, how, and why of wavelet shrinkage denoising. Comput. Sci. Eng.
**2000**, 2, 12–19. [Google Scholar] [CrossRef] - Band-Pass Filter. Available online: https://en.wikipedia.org/wiki/Band-pass_filter (accessed on 19 May 2016).
- Etter, D.M. Engineering Problem Solving with MATLAB; Prentice-Hall Incorporation: Upper Saddle River, NJ, USA, 1993. [Google Scholar]
- Lin, J.W. Mode-by-mode evaluation of structural systems using a bandpass-HHT filtering approach. Struct. Eng. Mech.
**2010**, 36, 697–714. [Google Scholar] [CrossRef]

**Figure 1.**Bragg scattering and surface velocity measured by a radar wave current meter: (

**a**) lateral view; and (

**b**) top view.

**Figure 3.**Illustration of the design map of a bandpass (A

_{c}= 0.1) filter for radar surface velocimetry.

**Figure 4.**Schematic diagram of the experimental setup in the laboratory for the current meter calibration: (

**a**) lateral view; and (

**b**) top view.

**Figure 6.**Charts of the original signals of radar waves at a tilt angle of 30°: (

**a**) speed = 0.2 m/s; and (

**b**) speed = 1.0 m/s.

**Figure 7.**Frequency spectrum chart of the radar signal analysis at a tilt angle of 30°: (

**a**) speed = 0.2 m/s; and (

**b**) speed = 1.0 m/s.

**Figure 8.**Comparison of the calculated results obtained using different analytical methods at a tilt angle of 30°: (

**a**) speed = 0.2 m/s; and (

**b**) speed = 1.0 m/s.

**Figure 9.**Comparison of the results of the flow velocity experimental analysis using the FFT, bandpass filter, and the wavelet transform. (

**a**) Tilt angle = 20°; (

**b**) Tilt angle = 30°; (

**c**) Tilt angle = 40°; (

**d**) Tilt angle = 50°; (

**e**) Tilt angle = 60°.

**Figure 10.**Comparison of the relative errors in flow velocity experimental analysis using the FFT, bandpass filter, and the wavelet transform. (

**a**) Tilt angle = 20°; (

**b**) Tilt angle = 30°; (

**c**) Tilt angle = 40°; (

**d**) Tilt angle = 50°; (

**e**) Tilt angle = 60°.

**Table 1.**Comparison of the identification results under different tilt angles and movement speeds of the trolley.

Tilt Angle (Degree) | Method | Velocity (m/s) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.2 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 4.0 | 5.0 | 6.0 | ||

20 | FFT | 0.24 | 0.96 | 1.45 | 1.92 | 2.42 | 2.88 | 3.87 | 4.84 | 5.80 |

Wavelet transform | 1.52 | 0.96 | 1.45 | 1.92 | 2.42 | 2.88 | 3.87 | 4.84 | 5.80 | |

Bandpass filter | 0.20 | 0.96 | 1.46 | 1.92 | 2.42 | 2.89 | 3.92 | 4.84 | 5.82 | |

30 | FFT | 0.24 | 0.92 | 1.28 | 2.14 | 2.33 | 2.79 | 3.79 | 4.59 | 5.49 |

Wavelet transform | 1.38 | 0.92 | 1.65 | 2.14 | 2.33 | 2.79 | 3.79 | 4.59 | 5.49 | |

Bandpass filter | 0.20 | 0.92 | 1.50 | 2.14 | 2.32 | 2.79 | 3.79 | 4.63 | 5.49 | |

40 | FFT | 0.19 | 0.90 | 1.35 | 2.28 | 2.09 | 2.60 | 3.48 | 4.34 | 5.25 |

Wavelet transform | 1.56 | 0.90 | 1.35 | 2.28 | 2.09 | 2.60 | 3.48 | 4.34 | 5.25 | |

Bandpass filter | 0.17 | 0.90 | 1.37 | 1.99 | 2.19 | 2.60 | 3.48 | 4.37 | 5.25 | |

50 | FFT | 0.22 | 0.78 | 1.14 | 1.76 | 1.97 | 2.36 | 3.10 | 3.94 | 4.75 |

Wavelet transform | 1.55 | 0.80 | 1.17 | 1.76 | 1.97 | 2.36 | 3.10 | 3.94 | 4.75 | |

Bandpass filter | 0.16 | 0.80 | 1.31 | 1.75 | 1.97 | 2.36 | 2.90 | 4.00 | 4.75 | |

60 | FFT | 0.66 | 0.69 | 0.96 | 1.31 | 1.13 | 2.12 | 2.26 | 3.44 | 3.65 |

Wavelet transform | 2.15 | 0.85 | 1.05 | 1.31 | 1.57 | 2.12 | 2.26 | 3.44 | 3.65 | |

Bandpass filter | 1.26 | 0.69 | 1.73 | 1.31 | 1.50 | 2.11 | 4.15 | 3.44 | 3.65 |

**Table 2.**Comparison of the relative identification errors under different tilt angles and movement speeds of the trolley (Note that the values in bold indicated the lowest relative error when the tilt angle ranged between 20 and 40 degrees).

Tilt Angle (Degree) | Method | Velocity (m/s) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.2 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 4.0 | 5.0 | 6.0 | ||

20 | FFT | 19.01 | 4.27 | 3.19 | 4.01 | 3.13 | 3.86 | 3.30 | 3.13 | 3.30 |

Wavelet transform | 662.20 | 4.27 | 3.19 | 4.01 | 3.13 | 3.86 | 3.30 | 3.13 | 3.30 | |

Bandpass filter | 0.13 | 3.76 | 2.92 | 4.01 | 3.13 | 3.58 | 1.90 | 3.13 | 3.02 | |

30 | FFT | 21.83 | 8.02 | 14.72 | 6.77 | 6.86 | 7.06 | 5.31 | 8.26 | 8.52 |

Wavelet transform | 591.44 | 8.02 | 9.96 | 6.77 | 6.86 | 7.06 | 5.31 | 8.26 | 8.52 | |

Bandpass filter | 1.39 | 7.75 | 0.09 | 6.77 | 7.14 | 7.06 | 5.31 | 7.42 | 8.52 | |

40 | FFT | 3.39 | 9.92 | 9.89 | 14.23 | 16.37 | 13.50 | 12.88 | 13.11 | 12.57 |

Wavelet transform | 678.32 | 9.92 | 9.89 | 14.23 | 16.37 | 13.50 | 12.88 | 13.11 | 12.57 | |

Bandpass filter | 13.05 | 9.64 | 8.73 | 0.54 | 12.52 | 13.50 | 12.88 | 12.52 | 12.57 | |

50 | FFT | 11.20 | 21.80 | 23.69 | 12.02 | 21.21 | 21.24 | 22.56 | 21.21 | 20.91 |

Wavelet transform | 672.56 | 19.66 | 21.82 | 12.02 | 21.21 | 21.24 | 22.56 | 21.21 | 20.91 | |

Bandpass filter | 17.83 | 19.66 | 12.48 | 12.33 | 21.21 | 21.24 | 27.52 | 19.94 | 20.91 | |

60 | FFT | 227.87 | 31.41 | 36.21 | 34.43 | 54.77 | 29.32 | 43.46 | 31.11 | 39.11 |

Wavelet transform | 975.61 | 15.02 | 29.73 | 34.43 | 37.37 | 29.32 | 43.46 | 31.11 | 39.11 | |

Bandpass filter | 527.97 | 30.75 | 15.64 | 34.43 | 40.15 | 29.69 | 3.66 | 31.11 | 39.11 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lin, J.-W.; Huang, C.-W.; Hsu, Y.-S.
Micro Radar Surface Velocimetry for Hydrologic Signal Processing Using a Bandpass Filtering Approach. *Water* **2016**, *8*, 262.
https://doi.org/10.3390/w8060262

**AMA Style**

Lin J-W, Huang C-W, Hsu Y-S.
Micro Radar Surface Velocimetry for Hydrologic Signal Processing Using a Bandpass Filtering Approach. *Water*. 2016; 8(6):262.
https://doi.org/10.3390/w8060262

**Chicago/Turabian Style**

Lin, Jeng-Wen, Chih-Wei Huang, and Yin-Sung Hsu.
2016. "Micro Radar Surface Velocimetry for Hydrologic Signal Processing Using a Bandpass Filtering Approach" *Water* 8, no. 6: 262.
https://doi.org/10.3390/w8060262