MIMO radar river flow measurement based on space-velocity-time algorithm and adaptive correction model

2.1. MIMO radar point cloud data model

The radar system utilized in this study is the Huawei ASN850 centralized MIMO radar. Operating at a carrier frequency of 80 GHz, the system enables higher Doppler frequency shifts under identical flow conditions, thereby enhancing the sensitivity to velocity changes. It employs an antenna array configuration comprising eight transmitters and eight receivers, which significantly improves angular resolution and measurement accuracy. Furthermore, the radar operates in the frequency-modulated continuous wave (FMCW) mode, providing high range resolution and real-time capability essential for hydrological monitoring.

In a centralized MIMO radar system, the spacing between array elements – or the effective antenna aperture – is considerably smaller than the distance to the target. As a result, each transmitted and received signal observes essentially identical target parameters, including direction of arrival (DOA), range, Doppler frequency, and radar cross section (RCS) [22]. The MIMO radar estimates the target’s distance, velocity, and angle by transmitting FMCW and analyzing the time delay, frequency shift, and phase differences in the received echoes. The ranging process is based on the time delay of the electromagnetic wave traveling from the transmitter to the target and back, combined with the known frequency sweep rate, to compute the range. Velocity estimation is performed by applying fast Fourier transform (FFT) to a sequence of received FMCW pulses and calculating the target speed from the inter-pulse phase differences. Angular estimation is achieved by exploiting the phase differences between signals received across the antenna array, with angular resolution directly related to the number of antenna elements – greater array density yields higher angular precision.

The radar RF chip generates a Chirp signal, which is transmitted through the transmitter antenna and fed into the mixer; the receiver antenna receives the echo signal reflected from the object and transmits it to the mixer to mix with the transmitter signal to form an Intermediate Frequency (IF) signal. FFT is performed on the IF signal in the distance dimension, and the distance-horizontal angle heat map is obtained by beamforming in the horizontal direction, which is processed by 2D-Constant False-Alarm Rate (2D-CFAR) to extract the distance and horizontal angle information of the target. Finally, the velocity dimension Fast Fourier Transform is done to acquire the velocity information [23]. The distance, horizontal angle, and velocity information of target points are gathered by the preceding stages, and the information of numerous target points is assembled into a point cloud and out-put. The MIMO radar signal processing pathway is presented in Fig. 1.

Fig. 1Radar signal processing flow diagram

The IF signal is received by many antennas and physical information such as relative distance, radial velocity and horizontal angle of the target is extracted to generate multidimensional point cloud data which can be represented as:

where $r$ represents the relative distance between the target and the radar, $v$ represents the radial velocity of the target, $\theta$ satisfies the horizontal angle, and RCS represents the effective radar scattering cross-section area.

When the radar detects the target object, the transmitted signal as well as the received signal after a series of processing will produce multiple point cloud data, which is called point cloud data set and can be expressed as:

2.2. Design of SVT algorithm for 3D point cloud

This study presents a point cloud data processing algorithm based on the three-dimensional SVT model, designed for the acquired point cloud dataset. The algorithm leverages physical parameters such as relative distance, radial velocity, and horizontal angle to perform processing tasks, including point cloud clustering and feature extraction, within the three-dimensional space, velocity, and time domains. This approach facilitates the rapid aggregation and streamlined processing of the point cloud data, effectively mitigating noise interference and reducing data redundancy in the river flow measurement. The detailed workflow of this method is illustrated in Fig. 2.

1. Acquisition and Pre-processing of Point Cloud: The MIMO radar system is initialized to capture 3D point cloud data. Based on the radar installation settings, the point cloud is projected onto the horizontal plane. A global adaptive distance thresholding method is then applied to eliminate outlier noise points, thereby reducing interference in the point cloud data.

2. Spatial Dimension Processing: The pre-processed point cloud data is divided into multiple raster blocks according to spatial dimensions. Noisy points are removed by evaluating the weights of points within each raster, optimizing the point cloud data. The center point of each raster is selected to represent the point cloud position, simplifying subsequent data processing.

3. Velocity Dimension Processing: Outliers in the flow velocity data within each raster are corrected and removed using a three standard deviation (3σ) scale correction method. The corrected flow velocity mean value is then assigned to the center point of each raster.

4. Time Dimension Processing: The corrected point cloud data is temporally segmented, with real-time radar data divided into distinct frames, where each frame corresponds to a one-minute interval. A sliding T-test is applied to compare current data with historical data, identifying significant changes. The results are integrated and analyzed to produce the final output.

Fig. 2SVT algorithm flowchart

2.2.1. Preprocessing

Based on the radar shore-based lateral mounting height, pitch, and horizontal angles, and the physical information such as relative distance, radial velocity, and horizontal angle in the point cloud data acquired by the radar system, the point cloud is projected onto the $XOY$ horizontal plane and represented in the form of a matrix:

3

$P=\left[\begin{array}{c}{p}_1\\ p_2\\ ⋮\\ p_i\end{array}\right]=\left[\begin{array}{ccc}X1& Y1& V1\\ X2& Y2& V2\\ ⋮& ⋮& ⋮\\ Xi& Yi& Vi\end{array}\right],$

where the point cloud projection into a 2D coordinate system can be expressed in terms of coordinates and river flow rate:

4

$X=r\cdot \cos \alpha \cdot \cos (\beta +\theta ),$

5

$Y=r\cdot \cos \alpha \cdot \sin (\beta +\theta ),$

6

$V=v\cdot \frac{1}{\cos \alpha }\cdot \frac{1}{\cos (\beta +\theta )},$

where $r$ represents the relative distance between the target and the radar, $v$ represents the radial velocity of the target, $\theta$ represents the horizontal angle measured by the radar, $\alpha$ represents the pitch angle of the radar installation, and $\beta$ represents the horizontal angle of the radar installation.

Due to the complex topography of the river, other obstacles such as buildings and vegetation along the banks may interfere with normal signal reception, and floating objects on the surface of the river (e.g., leaves, rubbish, etc.) reflect radar signals and generate echoes that interfere with the water surface echo signals. Therefore, there will be noise points in the initial point cloud produced by radar scanning the river surface [24].

The global adaptive distance thresholding method is applied in the preprocessing to cull out outlier noise points away from the topic point cloud to prevent altering the flow velocity data. Determine the distance threshold based on the global distance average and variance by computing the average of the distance between the target original point cloud and all the points in the neighborhood. When the average of the distance between the point cloud and its nearby points is larger than, the point is declared to be an outlier and rejected. This technique is able to successfully locate and eliminate outlier noise spots, which in turn increases the accuracy of the subsequent processing. As illustrated in Fig. 3, the black dots are the topic point cloud and the red points are the outlier noise points.

Fig. 3Distance thresholding method

2.2.2. Space dimensional point cloud processing

The raster partitioning process is performed on point cloud by traversing each raster and calculating its weights in order to remove redundant points and reduce noise effects. Selecting the center point of the raster to replace the spatial location of the point cloud in the raster further simplifies the number of point clouds, which can more intuitively and accurately show the distribution of flow velocity on the surface of the river.

Dividing the enclosing frame into a number of rasters to improve the accuracy and processing speed of raster clustering. As shown in Fig. 4:

Fig. 4Grid-based partitioning

When the point cloud is denser, the raster edge lengths are smaller to represent the data more finely; when the density is smaller, the raster edge lengths become larger to reduce the computational complexity.

Traverse each raster obtained by the above method in turn, and compute the weights of each point in each raster by means of the neighborhood. The smaller the weight, the denser the distribution of points in the area; conversely, the larger the weight, the more sparsely distributed the points in the area.

For any raster, extract the point $({Xmax}_{max}$ with the largest coordinate value and the point $({Xmin}_{min}$ with the smallest coordinate value among the points it contains, with initial weights:

7

$a_0=\frac{{\left(X_{max}-X_{min}\right)}^2+{\left(Y_{max}-Y_{min}\right)}^2}{n},$

where $n$ denotes the number of points contained in the raster.

Let a certain matrix contain a set of points $Q=\{q_1,q_2,\cdots q_i\}$, where the weight $q_i$ of any point $a_i$ is:

8

$a_i=\frac{1}{k}{\sum }_{j=1}^k{‖q_i-q_j‖}^2,$

where $\{q_j{\}}_{j=1}^k\subset Q$ denotes the $k$ neighborhood of point $q_j$.

If the weight value of a point in this raster is less than the set weight threshold, the point is rejected, and vice versa, it is retained, thus rejecting the noisy points in the point cloud [25].

In the study of river flow velocity, based on the continuity principle of hydrodynamics and actual observation data, in the gentle section of the river channel, the resistance of the water flow is relatively constant due to the small changes in the riverbed morphology, resulting in small changes in the flow velocity between the proximate points. In addition, the viscosity effect of the fluid makes the flow rate tend to be smoothly distributed over a short range. Thus, the center of the raster is selected to re-place the remaining point-space positions within the raster.

2.2.4. Time dimensional point cloud processing

Capturing point cloud features throughout time by separately assessing data collected by the radar in real time at different time periods. Further integrating and analyzing the processing results of each time period can effectively identify and track the dynamic changes of point cloud data.

The point cloud data supplied from the radar in real time is segmented according to the time sequence of one frame per minute, and each frame is processed as a spatial and velocity dimensional point cloud separately. Based on the first ten frames of point cloud data are aggregated and averaged for integrated processing, and each subsequent newly created frame of point cloud data is treated independently. For the current frame, the algorithm takes the data from the previous ten frames as a basis for correcting and evaluating it.

In order to detect whether there is a significant change in the processed single-frame data between different time periods, the algorithm in this research uses a sliding t-test. In this method, the difference between the newly created data and the previous data may be constantly compared every minute, thus improving the accuracy and reliability of data processing.

The window size chosen for the sliding t-test algorithm in this study is 11 frames. The window starts from the start position of the time series and gradually slides back 3-5 frames, moving the window until it covers the entire time series. For each window position, the data within the window is divided into two parts (the first 10 frames and the last 1 frame), and a t-test is performed on the two parts of the data to calculate the t-statistic and p-value. As shown in Fig. 5.

Fig. 5Schematic diagram of sliding t-test

Based on the T-statistic and P-value, it was determined whether there was a significant change in a particular window location. If the P-value is less than a preset level of significance, the window location is considered to have an anomalous flow rate point, which is weighted and corrected according to the actual situation in the field.

2.3. Adaptive correction model

The river surface fluid state is unstable due to geomorphic features, environmental factors, weather, and fluid interactions, and although the SVT algorithm can be used to remove some of the noise during detection, there are still nonlinear disturbances caused by a variety of river features.

To further improve the accuracy of radar-measured flow velocity, an adaptive flow velocity correction model is proposed. Using machine learning and deep learning methods, river features, weather features, signal features, etc. are used as model inputs, and random forest method, BP network and LSTM network are used to predict the river flow rate, calculate the correction factor, and search for the hyper-parameters of the three models by designing an improved slime mould algorithm (ISMA) to accelerate the convergence speed and avoid falling into the local optimal solution [26]. Meanwhile, in order to meet the real-time river detection, the segmented incremental learning method is used to iteratively update the correction factor to improve the model accuracy and robustness. The detailed model principle is shown in Fig. 6.

Fig. 6Optimization model structure diagram

3.1. Radar shore-based side mounting

In order to gain higher precision of speed measurement, there are commonly two techniques of placing classic radar flowmeters: bridge center mounting and shore-based telescopic pole mounting, as shown in Figs. 7(a) and 7(b).

Fig. 7Radar installation method

a) Central bridge installation

b) Shore reach pole installation

Mounting the radar at the center of a bridge is the most effective setup for capturing the maximum Doppler frequency in areas like Central Sichuan. However, not all monitoring sites are equipped with bridges. Alternatively, shore-based pole installations can still provide a wide radar cross-section and strong Doppler signals. That said, these installations are limited by the spacing between poles, and when the river’s water level drops, the radar beam may no longer reach the water surface effectively. The algorithm proposed in this study is not limited to vertical installations; it also supports lateral mounting on riverbank risers, where the radar is tilted to illuminate the water flow direction at an angle, as shown in Fig. 8.

Fig. 8Shore-based lateral installation

3.2. Signal noise processing

The raw data collected by the radar is affected by environmental factors, there will be a large number of noise signals, so the collected data need to be denoised and filtered [34-36]. The SVT algorithm in this paper can quickly filter out and correct the noise to obtain a more stable set of measurement results. Here we take the 753 flow points with grid coordinates of (39, 19) in Xi’an inner river as an example to show the effect of denoising by SVT algorithm, as shown in Fig. 9.

Fig. 9SVT algorithm denoising effect

The red dashed line in the figure represents the original data, the green line corresponds to the data after spatial and velocity-domain filtering, and the blue line reflects the results after incorporating temporal-domain filtering. It is evident that the proposed filtering process effectively eliminates abrupt outliers in flow velocity, resulting in a smoother and more consistent temporal trend. Nevertheless, minor fluctuations remain in the filtered data due to environmental, hydraulic, and meteorological influences. However, it is already possible to achieve a decent accuracy in engineering measurements.

3.3. Medium to high flow rate scenarios

Medium and high flow rate scenarios feature strong water surface ripples and waves, making the RCS huge, and the same hardware parameter settings are prone to echo saturation, which introduces high harmonics and interference and leads to unreliable measurement findings.

1. Jinkou River Hydrological Station, Leshan, Sichuan: The MIMO radar is located 14 meters upstream of the cable radar, 27 meters from the cable radar line, 9 meters downstream from the elevation, with a pitch angle of 77° and a horizontal angle of 65°. The SVT calculation results are shown in Fig. 10.

According to Fig. 10, the flow velocity of the Jinkou river is about 4 m/s, which reaches its maximum at a location about 20 meters from the river bank and diminishes at a position far from the river bank. The cableway survey line connects nicely with the radar FOV coverage of only 40 and 50 meters. The flow velocities at the 40 m and 50 m points of the cable radar were 3.90 m/s and 3.86 m/s, respectively, which are com-pared with the flow velocities at the cable radar points as shown in Figs. 11.

Fig. 10Jinhou river SVT algorithm processing result

Fig. 11Comparison of flow velocity at Jinkou River measurement points

a) Comparison of 40 m measurement points

b) Comparison of 50 m measurement points

2. Qinghe River, Beijing: In the experiment, the MIMO radar was positioned at the head of the bridge, looking upstream, at a height of 8 meters from the water level, with a pitch angle of 78° and a horizontal angle of 30°. The radar installation location is shown in Fig. 12(a), and the results of SVT algorithm processing are shown in Fig. 12(b).

Fig. 12Qing river measurement results

a) Radar installation and deployment at Qing river

b) Qing river SVT algorithm processing result

According to Fig. 12, it can be seen that the flow velocity of the Qing River in Beijing is about 0.8 m/s, and there is a dramatic increase in the flow velocity at a position about 12 meters from the river bank. It was found that the flow rate in the river channel rose at this point, perhaps due to gullies and hydrilla, while at other areas the flow rate stabilized. As shown in Fig. 13(a), we used a rotameter to estimate the actual flow rate at the site due to the shallow water level of the river, and the comparative results are shown in Fig. 13(b).

Fig. 13Qing river comparison experiment

a) Measurements using a rotor current meter

b) Flow velocity comparison at measurement points

3. Xi’an inland river: MIMO radar installed on the bridge, facing downstream, at a height of 15.7 meters above the water surface, with a pitch angle of 78° and a horizontal angle of 30°. The results of the SVT algorithm are shown in Fig. 14(a), and the results of the comparison experiment using the handheld radar are shown in Fig. 14(b).

Fig. 14Xi’an inland river comparison experiment

a) Xi’an river SVT algorithm processing result

b) Flow velocity comparison at measurement points

As can be seen from Table 1, the flow velocity findings generated from the 3D point cloud processing algorithm processing were compared with cable channel flow meter radar, rotor type flow meter, and handheld radar flow meter in the experiment.

1) In the experiment at Jinkou river in Sichuan, the flow velocity at the two measurement points was 3.963 m/s and 3.789 m/s by the three-dimensional point cloud processing algorithm, and the average error was less than 2 % compared with the cable radar flow velocity measurements of 3.900 m/s and 3.860 m/s. Cable-channel radar is often used for hydrological monitoring, where streamflow measurements are made by fixing the radar equipment through a cable. They are expensive to build, have poor real-time performance, require regular maintenance of the cableway, and the length and height of the cableway limit their monitoring range. Compared to cable-based radar, shore-based side-scan MIMO radar offers real-time monitoring, is easy to install while providing larger coverage, is not bound by cable lengths and river conditions, and does not require costly cable maintenance.

2) In the experiment on the Qing river in Beijing, the real flow velocity was measured by a rotor-type tachometer, and the result was proved to be 0.751 m/s. At the same measurement position, the flow velocity detected by the MIMO radar was 0.758 m/s, with an average error of 1.5 percent. The rotor flow meter has good measurement accuracy, but it is greatly impacted by water conditions and other factors, confined to detecting the flow velocity at a single spot, and complex to install and maintain. Shore-based side-scan MIMO radars are straightforward to install, are not influenced by water conditions, and give steady measurements in a variety of severe settings.

3). In the experiment of Xi'an inner river, the flow velocity was measured to be about 0.882 m/s by the three-dimensional point cloud processing method, whereas the flow velocity recorded by the hand-held radar flow meter was 0.9 m/s, with an average error of less than 2 %. Handheld radar flowmeters usually can only measure the flow velocity at specific points on the surface of the water body, and it is difficult to obtain the flow velocity distribution information of the whole river area, which is poor in real time, and the measurement results are easily affected by the operator’s station position, angle, and handheld stability, resulting in unstable data or large errors.

By comparing with different traditional flow rate measurement methods at each site, this approach is easy to install, minimizes the demand of the installation environment, and lowers the installation cost and complexity. Able to achieve continuous real-time monitoring, suitable for dynamic changes in the water flow environment, you do not need to be in direct contact with the water body, to reduce the impact of floating objects in the water body, sediment, or water surface fluctuations, to ensure more stable measurement results. At the same time, it is possible to measure the flow velocity distribution at several sites throughout the river channel or a vast area of water instead of being confined to a single point measurement, which is suited for complex river environments.

And through the three-dimensional point cloud processing algorithm, the MIMO radar has a high measurement accuracy; the algorithm processed flow velocity compared with the true value of each site; the average error is less than 2 %; the maximum error is less than 5 %, in line with the industry's requirements for the relative error of less than 5 %.

3.4. Low flow rate scenario

In low flow river scenarios, the radar IF filter circuit must be set up with a high-pass isolation filter, and at the same time, the RCS is small in low-flow, the echo signal is weak, and the signal-to-noise ratio of the low-frequency signal obtained from the sampling is low, which can easily lead to inaccurate measurement values or no measurement results.

Shanxi Bai river hydrological station: The MIMO radar is installed in the well tower, facing upstream, at a height of 19 meters from the water surface, with a pitch angle of 74° and a horizontal angle of 55.7°. The radar is installed and deployed as shown in Fig. 15(a), Rainfall occurred on the day of the experiment, due to rainwater striking the surface of the river to form certain ripples, it will have some impact on the surface flow rate measurement. And due to the rainfall that caused the upstream water level to rise, the upstream gate was opened to release water, resulting in the flow rate result of the measuring point at the downstream location being large. The results of the 3D point cloud processing algorithm are shown in Fig. 15(b).

Fig. 15Bai river measurement results

a) Radar installation and deployment at Bai river

b) Bai river SVT algorithm processing result

According to Fig. 15, the flow velocity at the near-shore point of the Bai river is about 0.4 m/s. The flow velocity at the far-shore point is high due to the reflux of the river water caused by the collision with the bridge abutment. The measurements were compared with the on-site shipboard ADCP, and the ADCP measurements are shown in Fig. 16:

Fig. 16Measurement results from vessel-mounted ADCP

a) Trajectories and flow vectors

b) Earth coordinate flow velocity magnitude

There are three measurement points, 39 m, 43 m, and 47 m, within the measuring ship course and radar FOV coverage, which are compared with the flow velocity of ADCP, and the results are shown in Fig. 17.

According to Table 2, it can be shown that the Shanxi Bai river site employs ship-board ADCP to measure the flow velocity, and the shipboard ADCP detects the water flow information using acoustic wave, and the measurement accuracy is high. But ADCP measurements need the vessel to travel along the survey line, which does not give real-time, full-time monitoring data and is expensive to install and operate. Through the three-dimensional point cloud processing algorithm, the three measurement points measured the flow velocity of 0.314 m/s, 0.261 m/s, and 0.475 m/s, while the flow velocity measured by the ADCP was 0.296 m/s, 0.241 m/s, and 0.463 m/s. The flow velocity processed by the algorithm in the low flow velocity scenario is less than 5 cm/s in comparison with the true value of the site under the absolute error, which is in line with the industry's criteria. Tests have shown that at low flow rates, the approach can give real-time accurate data faster and at cheaper installation and maintenance costs.

Fig. 17Comparison of streamflow at Bai river gauging sites

a) 39 m measurement point of Bai River

b) 43 m measurement point of Bai River

c) 47 m measurement point of Bai River

3.5. Adaptive correction flow rate experiment

In the above experiments, the data from Xi'an station met the requirements of the algorithm, so adaptive correction experiments were conducted to detect the flow velocity there. A total of 145 887 data were obtained from the experimental data, and three models were used for prediction training, and the data set was divided according to the ratio of 8:1:1. The original features of the river and the enhanced features after feature engineering are shown in Table 3. It contains 7 original features and 9 enhanced features totaling 16 features.

A combination of mutual information, Pearson and Spearman correlation analyses, along with feature–target scatter plots, was used to evaluate feature relevance. After cross-validation, the following features were retained: range, frame, RCS_Range2, Angle Range, angle, Frame Diff. Where $x$, $y$ as a core feature of the SVT algorithm is also preserved. Mutual information is analyzed as shown in Fig. 18.

3.5.1. Comparison of optimization algorithms

Fast search configuration of hyperparameters of benchmark models RF, BP, LSTM using optimization algorithms. Among them, RF needs to configure the number of decision trees, the maximum depth of the tree, the maximum number of separated samples, the number of leaf node samples and the maximum number of separated features, and BP needs to configure the activation function, the learning rate, the number of neurons and the number of layers, etc., and the two belong to the low-dimensional search problem. While LSTM needs to configure nearly 30 hyperparameters such as the number of layers, batch size, learning rate, etc., which belongs to the high-dimensional search problem. Therefore, CEC test functions are used to compare multiple optimization models, some test functions are shown in Table 4, and the test results are shown in Fig. 19.

Table 3Characteristic table

Characteristic classification	Feature name	Calculation method	Explanation
Original features	frame(f)	–	The frame in which the radar collects data
	range	–	Target distance measured by radar
	v	–	Flow rate detected by radar
	angle	–	Angle of radar to target
	RCS	–	Radar Cross section
	$x$	–	Coordinates of the target in space
	$y$	–
Enhanced features	Distance	$\mathrm{s}\mathrm{q}\mathrm{r}\mathrm{t}\left(x^2+y^2\right)$	Euclidean distance from target to radar origin
	Cos Angle	$\mathrm{c}\mathrm{o}\mathrm{s}\left(Angle\right)$	Flow velocity versus radar direction
	RCS_Range2	RCS / range²	Reflective intensity per unit distance
	Angle range	angle*range	Joint characterization of angle and distance
	Frame diff	$f\left(i+1\right)-f\left(i\right)$	Sequence of back minus front frames
	$XY$ Product	$x*y$	Revealing coordinate distribution
	$X$ plus $Y$	$x+y$	Goal-oriented
	$X$ minus $Y$	$x-y$	Target-directed
	Polar angle	$\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}2(Y,X)$	Target polar angle direction

Fig. 18Characteristic mutual information analysis

The three test functions listed above correspond to the effect of each algorithm in high-dimensional search under the condition of different variable range sizes, which is to simulate the situation that there are different value ranges in the hyperparameter configurations, e.g., the configuration of the learning rate only needs a single-digit range, while the number of neurons needs a tens or even hundreds of ranges, and in low-dimensional search, the individual algorithms have similar results. The global optimum of all three algorithms is 0, so the closer the fitness of the curve is to 0, the better the search is indicated. The fewer the number of iterations, the more efficient the search is indicated. Therefore, one of the actual search processes in the experiment is extracted and compared, and the comparison effect is shown in Fig. 20.

Table 4Test function

Function name	Function expression	Dimension	Variable range values	Global optimum
F9	${\sum }_{\mathrm{i}=1}^{\mathrm{n}}(x_1^2-10\cos (2\pi x_i)+10)$	30	[–5.12, 5.12]	0
F10	$-20\exp \left(-0.2\sqrt{\frac{1}{n}{\sum }_{i=1}^n{x}_i^2}\right)\mathrm{}-\exp \left(\frac{1}{n}{\sum }_{i=1}^n\cos (2\pi x_i)\right)+20+e$	30	[–32, 32]	0
F11	$\frac{1}{4000}{\sum }_{i=1}^n{x}_i^2-\prod _{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)+1$	30	[–600, 600]	0

Fig. 19Comparison results of optimization algorithms

a) F9 test function

b) F10 test function

c) F11 test function

d) F9 test results

e) F10 test results

f) F11 test results

Fig. 20Comparison of low and high dimensional real search

a) Comparison of low-dimensional BP search

b) Comparison of high-dimensional LSTM search

As can be seen from the figure, the ISMA method effect has some advantages for both high and low dimensional searches. Therefore, ISMA is finally chosen as the optimization algorithm for the adaptive correction model.

3.5.2. Practical application

30 sets of parameter search experiments were conducted for the three models using ISMA, with the upper limit of 200 iterations for each set and the convergence threshold of 1e-5. The average values of the optimal parameters were calculated to generate the correction factors. The calibration results are shown in Table 5.

Table 5Comparison of three methods of training and results

Methods	Training MAE / (m·s-1)	Training MSE / (m2·s-2)	Calibration factor / (m·s-1)	Target error / %
ISMA-RF	0.041316	0.0025412	0.90093	0.93
ISMA-BP	0.039810	0.0021796	0.89243	1.41
ISMA-LSTM	0.040490	0.0022191	0.90366	0.79

The same scene of Fig. 14 was selected as the target test set, and the flow rate was fitted using the above correction factors, and the results are shown in Fig. 21.

As can be seen from the figure, the flow rate accuracy is improved after correction by all three algorithms, with the most obvious effect being the LSTM network, thanks to its strong correlation with time, followed by Random Forest, and finally the BP algorithm.

Fig. 21Calibration results of the three algorithms are plotted