## 3 Adaptive Time-Frequency Analysis
### Variational Mode Decomposition and Optimization Models
#### 3.2.1 Variational Mode Decomposition
The combination of these three methods allows for more effective noise removal from density logging data, improving data resolution and thin-layer identification accuracy [80]. An optimized VMD algorithm based on tunable diode laser absorption spectroscopy (TDLAS) has been developed for signal denoising [81]. A wavelet threshold function based on a combination of grey wolf optimization (GWO) and VMD has been proposed to address noise interference in ultrasonic echo signals [82]. An adaptive denoising and detrending method based on an improved seagull optimization algorithm (ISOA) and VMD has been developed for processing signals from nuclear circulating water pump impellers [83]. VMD can effectively remove trend components from signals, significantly improving the signal-to-noise ratio of nuclear circulating water pump impeller signals while preserving valuable information. Additionally, a photoacoustic spectroscopy second harmonic signal denoising method that utilizes the sparrow search algorithm (SSA) to optimize VMD combined with wavelet threshold denoising (WTD) technology has been proposed, and it improves the detection accuracy and real-time measurement stability of acetylene sensors [84].To enhance the performance of VMD, researchers have proposed various parameter optimization methods, such as adaptive parameter selection methods based on particle swarm optimization (PSO) and genetic algorithms. These methods automatically adjust VMD parameters, thereby improving decomposition accuracy and efficiency. A fault diagnosis method for electric valves in nuclear power plants based on a random forest model that integrates VMD, MDI, and ISSA has been proposed [85]. Another study introduced an ultrasonic machining vibration signal processing method based on SE, PSO, and VMD that improves signal processing efficiency by optimizing VMD parameters [86]. The pursuit of high anti-interference capability and precision in models is also a direction of model optimization. The complex variational mode decomposition (CVMD) algorithm has been proposed for analyzing complex-valued data [87]. VMD can also be used to decompose projectile signals [88], and by detecting the mathematical model of the skyscreen, the measurement accuracy of projectile muzzle velocity can be improved. Compared to traditional methods, this algorithm has significant advantages in term of anti-interference capability and measurement accuracy. Strength and Weakness
* Susceptible to the influences of noise and spurious components
* Stability and noise resistance
* Adaptability and parameter controllability
* Effective mitigation of mode mixing  
First, VMD demonstrates a robust capacity to mitigate mode mixing. By integrating a regularization term within a variational optimization framework, VMD effectively curtails the occurrence of mode mixing, a pervasive challenge in signal decomposition, particularly when confronting nonlinear and non-stationary signals. Traditional approaches, exemplified by empirical mode decomposition (EMD), are prone to mode mixing due to the perturbations induced by local extrema points. In contrast, variational mode decomposition (VMD) dissects the signal into multiple narrowband mode components through the judicious application of frequency-domain constraints and bandwidth limitations, thereby significantly reducing the likelihood of mode mixing. Second, VMD distinguishes itself through its adaptability and parameter controllability. It possesses the proficiency to adaptively determine the number of modes and frequency centers, obviating the necessity of relying on envelope calculations predicated at signal extrema points. This attribute endows VMD with heightened flexibility and precision in the processing of complex signals. For instance, within the context of structural health monitoring, VMD has the capability to dynamically adjust decomposition parameters in accordance with the idiosyncrasies of vibration signals, thereby facilitating the extraction of fault features with unparalleled accuracy.