# Method to Solve Underwater Laser Weak Waves and Superimposed Waves

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}is 0.995975, which is more stable and accurate. After estimating the parameters, this study used oscillating particle swarm optimization (OPSO) and the Levenberg–Marquardt algorithm (LM) to optimize the estimated parameters; the final results show that the method in this paper is closer to the original waveform. In order to verify the processing effect of the method on complex waveform, this paper decomposes and optimizes the simulated complex waveforms; the final RMSE is 0.0016, R

^{2}is 1, and the Gaussian component after decomposition can fully represent the original waveform. This method is better than other decomposition methods in complex waveform decomposition, especially regarding weak waves and superimposed waves.

## 1. Introduction

## 2. Algorithm Principle

#### 2.1. Principle of the Gaussian Decomposition Method

#### 2.2. Data Preprocessing

#### 2.3. Improved Gaussian Decomposition Method

- (1)
- When Linf = 1 and Rinf > 1 (Figure 2a), the left-hand side of the peak is closer to the real situation. Therefore, taking the half position of the left peak as the full width at half maxima (tgl), and finding the closest point of the half peak to calculate the standard deviation and, finally, comparing these values, allows us to obtain the standard deviation of the peak ($\delta $).
- (2)
- When Rinf = 1 and Linf > 1 (Figure 2b), the right-hand side of the peak is closer to the real situation. Therefore, taking the half position of the right peak as the full width at half maxima (tgr), and finding the closest point of the half peak to calculate the standard deviation and, finally, comparing these values, allows us to obtain the standard deviation of the peak ($\delta $).
- (3)
- When Rinf = 1 and Linf = 1 (Figure 2c) or Rinf > 1 and Linf > 1 (Figure 2d), the peak is a single waveform, or wrapped by multiple waveforms without superposition. Therefore, it is necessary to calculate the distance from the points closet to the half width peak on both sides to the peak point. The point with a smaller distance is taken as the full width at half maxima (dis), and these values are compared, to obtain the standard deviation of the peak ($\delta $).$$Others\{\begin{array}{ccc}\delta =\mathrm{min}(\left|\mu -{P}_{xl}\right|,\frac{\left|\mu -tgl\right|}{\sqrt{2\mathrm{ln}2}})& ,& L\mathrm{inf}=1\&R\mathrm{inf}>1\\ \delta =\mathrm{min}(\left|\mu -{P}_{xl}\right|,\frac{\left|\mu -tgr\right|}{\sqrt{2\mathrm{ln}2}})& ,& R\mathrm{inf}=1\&L\mathrm{inf}>1\\ \delta =\frac{\left|\mu -dis\right|}{\sqrt{2\mathrm{ln}2}}& ,& R\mathrm{inf}>1\&L\mathrm{inf}>1\\ \delta =\frac{\left|\mu -dis\right|}{\sqrt{2\mathrm{ln}2}}& ,& Others\end{array}$$

- (1)
- When Linf = 1 and Rinf > 1 (Figure 3a), the left side of the peak is closer to the real situation; the right side of the peak exhibits a waveform superposition. Therefore, we need to find the corresponding inflection point, and the point of the nearest half peak position on the left side (tgl), and calculate their respective standard deviation ($\delta $). This paper selected a smaller value as the standard deviation, to avoid error due to unrecognized partial inflection points.
- (2)
- When Rinf = 1 and Linf > 1 (Figure 3b), the right side of the peak is closer to the real situation; the left side of the peak exhibits a waveform superposition. Therefore, we need to find the corresponding inflection point, and the point of the nearest half peak position on the right side (tgr), and calculate their respective standard deviation ($\delta $). This paper selected a smaller value as the standard deviation, to avoid error due to unrecognized partial inflection points.
- (3)
- There are three different situations when Rinf > 1 and Linf > 1 (Figure 3c–e); at this time, the distance from the center to the inflection points cannot accurately express the standard deviation, because the inflection points are affected by the superimposed waveform. In this study, the average position of the inflection points on the left and right sides of the peak are denoted as S1 and S2, respectively. The distance from the closest half position of the peak to the peak, on both sides of the peak, is calculated, and the shortest distance is denoted as S3. We sorted S1, S2, and S3, and obtained the middle distance as the half peak width (dis) to calculate the standard deviation ($\delta $).
- (4)
- In this case, because the inflection point is not judged (Figure 3f), this study found the point closest to the half width on both sides of the peak, and calculated the distance between it and the peak, using the shortest distance as dis, to calculate the standard deviation ($\delta $).

#### 2.4. Two-Order Oscillating Particle Swarm Optimization

- (1)
- According to the estimated parameters used to establish a group of particles, these particles are randomly distributed within the estimated parameter range.
- (2)
- The fitness of each particle (fitness, Equation (10)); if the fitness is better than the individual extremum, we update the individual extremum; if the fitness is better than the global extremum, we update the global extremum.
- (3)
- Optimizing the particles’ position and velocity (${V}_{i}$, ${X}_{i}$, Equation (11)).
- (4)
- We repeat the above steps until the limit error requirement, or the number of iterations, is reached.$$fitness={\displaystyle \sum _{i=1}^{n}{({y}_{i}-f{x}_{i})}^{2}};f{x}_{i}={\displaystyle \sum _{i=0}^{m}{A}_{i}\mathrm{exp}(-\frac{{(x-{u}_{i})}^{2}}{2{\delta}_{i}{}^{2}})}+noise)$$${y}_{i}$ is the objective function; $f{x}_{i}$ is a function composed of multiple Gaussian components after Gaussian decomposition; n is the number of samples.$$\begin{array}{c}{V}_{i}(t+1)=w{V}_{i}(t)+{c}_{1}{r}_{1}({P}_{i}-1+{\alpha}_{1}){X}_{i}(t)-{\alpha}_{1}{X}_{i}(t-1))\\ +{c}_{2}{r}_{2}({P}_{g}-(1+{\alpha}_{2}){X}_{i}(t)+{\alpha}_{2}{X}_{i}(t-1))\\ {X}_{i}(t+1)={X}_{i}+{V}_{i}(t+1)\end{array}$$

#### 2.5. Levenberg–Marquardt Algorithm

## 3. Results

#### 3.1. Parameters Estimation

#### 3.2. Parameter Optimizing and Fitting

#### 3.3. Accuracy Comparison

^{2}, Equation (16)). The results showed that the method in this paper is more accurate than other methods; although the accuracy of some data is not higher than in the iterative method, the accuracy of the iterative method is not much higher than that of this paper, and it can be found that the overall fitting effect of this paper is better than the iterative method. The results showed that the RMSE is 2.544 and the R

^{2}is 0.995975, the average precision parameters are higher than other data, and the decomposition effect is more stable and accurate. In summary, this method is more suitable for the decomposition of multiple waveforms; not only that, but this method can also express waveform data more accurately than the other methods.

^{2}value closer to 1 indicates a better fit.

^{2}was 1, and the optimized waveform was very close to the original waveform.

## 4. Discussion

- (1)
- When solving the superimposed waveform, it is unreasonable to use the highest peak value as the amplitude of the Gaussian component; although this limitation can be corrected in the subsequent parameter optimization process, it undoubtedly does not increase the workload of parameter optimization.
- (2)
- The laser echo signal will be affected by noise in the water. In this paper, the water data under the influence of different noise are not further explored, but this problem will be further studied in follow-up work.
- (3)
- Although this method uses multiple sets of research data, the experimental environment is still limited to ponds with clear water quality. Because the data source problem can only be tested with simulated data, there is no exploration of different types of water body data, and the universality of this method cannot be verified.

## 5. Conclusions

- (1)
- By comparing multiple sets of study data, underwater lasers are mostly complex waveforms of strong wave and weak wave superposition. We found that the improved inflection point selection method proposed in this paper is more suitable for underwater waveform calculation, which can solve underwater waveform data well and has a good effect on the parameter extraction of superimposed waves and weak waves.
- (2)
- To further verify the universality of this method, this study used a variety of methods to decompose the simulated waveform. The results showed that the waveform decomposed by this method is closer to the actual waveform in comparison with other methods, and the accuracy is much higher than that of other decomposition methods.
- (3)
- To verify the accuracy of the subsequent parameters optimization process, this study compared a variety of parameter optimization combinations, and the results showed that although the accuracy of the optimized method was not the highest, the actual results were closer to the original waveform than the other methods.

- (1)
- The underwater laser echo signal is susceptible to various noise sources. In follow-up work, multiple experiments will be carried out under the influence of various noise sources, to further verify the accuracy and reliability of the proposed method.
- (2)
- In follow-up work, a variety of different sensors will be used to test the algorithm in this paper, to further understand whether the limitations and deviations of the sensors will affect the results.
- (3)
- The data background selected in this paper is a pond in a park in Guilin, Guangxi, China; the water quality has little effect on the laser echo signal. The planned follow-up research will study different types of water bodies, such as oceans and rivers, to further verify and evaluate the universality of this method in different environments.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PSO | Particle swarm optimization |

OPSO | Oscillating particle swarm optimization |

LM | Levenberg–Marquardt |

Interpretation | |

Savitzky–Golay smoothing filter | The Savitzky–Golay filter is a digital filter that can be applied to a set of data to smooth the data, which can improve the accuracy of the data without changing the signal trend and width. It is mainly realized via the process of convolution; that is, the continuous subset of adjacent data points is fitted with a low-order polynomial by a linear least-squares method. |

five-point three-time smoothing method | After removing the first point from the known number of equidistant points, a curve-smoothing method using a cubic polynomial to approximate the two points before and after each data point is proposed. |

Jacobian matrix | In vector analysis, a Jacobian matrix is a matrix whose first-order partial derivatives of a function are arranged in a certain way, and its determinant is called the Jacobian determinant. |

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**Figure 2.**Different images of$\text{}L{\mathrm{inf}}_{\mathrm{l}}=1$ and $R{\mathrm{inf}}_{\mathrm{r}}=1\text{}$ ((

**a**–

**d**) represent different cases of $L{\mathrm{inf}}_{\mathrm{l}}=1$ and $R{\mathrm{inf}}_{\mathrm{r}}=1$, respectively).

**Figure 4.**Waveform multiple decomposition process ((

**a**–

**e**) represent the diagrams of each stage in the process of waveform processing).

**Figure 5.**Comparison of Gaussian component elimination ((

**a**,

**b**) represent the image before and after the removal of the pseudo component, and the red-box-highlighted area is the difference between the two images).

**Figure 6.**Negative oscillation occurs after decomposition (the red box highlights the negative oscillation area).

**Figure 7.**Comparison of different parameter optimization methods ((

**a**,

**b**) represent the waveform solution using PSO and OPSO, respectively; the red box highlights the area with a large gap between the different methods).

**Figure 8.**Comparison of fixed damping coefficient. ((

**a**,

**b**), respectively, represent whether to use fixed damping coefficient to solve the waveform, the red box highlights the area with the large difference between the different methods).

**Figure 9.**Data processed by inflection point (the image representation uses the inflection point to solve different data; the red box highlights the area that is poorly fitted).

**Figure 10.**Data processed by iterative decomposition (the image representation uses iterative decomposition to solve different data; the red box highlights the area that is poorly fitted).

**Figure 11.**Data processed by inflection point selection decomposition (the image representation uses inflection point selection decomposition to solve different data; the red box highlights the area that is poorly fitted).

**Figure 12.**Data processed by this paper (the image representation uses the method of this paper to solve different data; the red box highlights the area that is poorly fitted).

**Figure 14.**Different methods decompose the simulation waveform. ((

**a**) is the result of using inflection point to solve the simulated waveform data; (

**b**) is the result of using iterative to solve the simulated waveform data; (

**c**) is the result of using inflection point selection decomposition to solve the simulated waveform data; (

**d**) is the result of using the method in this paper to solve the simulated waveform data.)

**Figure 16.**Fitting results of different optimization methods. ((

**a**) is the result of using PSO to fit waveform data; (

**b**) is the result of using OPSO to fit waveform data; (

**c**) is the result of using fixed damping LM to fit waveform data; (

**d**) is the result of using update damping LM to fit waveform data; (

**e**) is the result of using PSO and fixed damping LM to fit waveform data; (

**f**) is the result of using PSO and update damping LM to fit waveform data; (

**g**) is the result of using OPSO and fixed damping LM to fit waveform data; (

**h**) is the result of using OPSO and update damping LM to fit waveform data.)

Data | Iterative | Inflection Point | Inflection Point Selection Decomposition | This Paper | |
---|---|---|---|---|---|

RMSE | Test 1 | 2.4600 | 4.1989 | 2.4647 | 2.6383 |

Test 2 | 3.8405 | 4.1085 | 3.6739 | 3.7071 | |

Test 3 | 2.4133 | 2.4133 | 8.1702 | 2.3909 | |

Test 4 | 1.9612 | 1.9612 | 2.8015 | 1.4397 | |

average | 2.66875 | 3.170475 | 4.277575 | 2.544 | |

R^{2} | Test 1 | 0.9964 | 0.9894 | 0.9963 | 0.9958 |

Test 2 | 0.9929 | 0.9919 | 0.9935 | 0.9934 | |

Test 3 | 0.9964 | 0.9964 | 0.9591 | 0.9965 | |

Test 4 | 0.9966 | 0.9966 | 0.9931 | 0.9982 | |

average | 0.995575 | 0.993575 | 0.9855 | 0.995975 |

i | 1 | 2 | 3 | 4 | 5 | 6 |

A | 45 | 40 | 60 | 50 | 50 | 8 |

μ | 98 | 110 | 123 | 135 | 153 | 166 |

δ | 5.5 | 5 | 6.5 | 3.5 | 4 | 3.5 |

i | 1 | 2 | 3 | 4 | 5 | 6 | |

A | Actual value | 45 | 40 | 60 | 50 | 50 | 8 |

Optimized parameters | 44.2589 | 40.3164 | 59.2484 | 49.1012 | 49.3643 | 7.6628 | |

μ | Actual value | 98 | 110 | 123 | 135 | 153 | 166 |

Optimized parameters | 97.8985 | 109.9738 | 123.0346 | 134.9677 | 153.0104 | 166.0331 | |

δ | Actual value | 5.5 | 5 | 6.5 | 3.5 | 4 | 3.5 |

Optimized parameters | 5.4893 | 5.1777 | 6.4799 | 3.6108 | 4.0575 | 3.6352 | |

RMSE | 0.0016 | ||||||

R^{2} | 1 |

Method | RMSE | R^{2} |
---|---|---|

Original data | 3.7071 | 0.9934 |

PSO (Figure 16a) | 1.9853 | 0.9976 |

OPSO (Figure 16b) | 2.2267 | 0.9970 |

Fixed damping LM (Figure 16c) | 1.5947 | 0.9988 |

Update damping LM (Figure 16d) | 1.9386 | 0.9982 |

PSO and fixed damping LM (Figure 16e) | 1.6405 | 0.9987 |

PSO and update damping LM (Figrue 16f) | 1.9485 | 0.9988 |

OPSO and fixed damping LM (Figure 16g) | 1.5947 | 0.9988 |

OPSO and update damping LM (Figure 16h) | 2.0387 | 0.9980 |

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## Share and Cite

**MDPI and ACS Style**

Kang, C.; Lin, Z.; Wu, S.; Yang, J.; Zhang, S.; Zhang, S.; Li, X.
Method to Solve Underwater Laser Weak Waves and Superimposed Waves. *Sensors* **2023**, *23*, 6058.
https://doi.org/10.3390/s23136058

**AMA Style**

Kang C, Lin Z, Wu S, Yang J, Zhang S, Zhang S, Li X.
Method to Solve Underwater Laser Weak Waves and Superimposed Waves. *Sensors*. 2023; 23(13):6058.
https://doi.org/10.3390/s23136058

**Chicago/Turabian Style**

Kang, Chuanli, Zitao Lin, Siyi Wu, Jiale Yang, Siyao Zhang, Sai Zhang, and Xuanhao Li.
2023. "Method to Solve Underwater Laser Weak Waves and Superimposed Waves" *Sensors* 23, no. 13: 6058.
https://doi.org/10.3390/s23136058