Building structure-borne noise measurements and estimation due to train operations in tunnel

Editor: Anastasia Paschalidou
编辑：阿纳斯塔西娅·帕斯卡利杜

Structure-borne noise 结构噪声

Vibration and noise correlation
振动和噪声相关性

Acoustics-vibration coupling
声学-振动耦合

Measurement 测量

Noise estimation 噪声估算

A B S T R A C T

The perception of structure-borne noise is particularly salient when train passes through the tunnel under the buildings, which has a negative impact on human health. In the process of constructing buildings along metro lines, it is crucial to estimate indoor structure-borne noise levels in order to enhance design and prevent any negative impact on human comfort. This study conducted measurements of structure-borne noise, reverberation time, and train-induced vibrations in Guangzhou, China to investigate the generation, propagation, and dissipation mechanisms of structure-borne noise. An approach based on Short-Time Fourier Transform and Schroeder integral was proposed for obtaining frequency-dependent reverberation time. Additionally, a deep learningbased approach incorporating indoor vibrations, frequency-dependent reverberation time, and room parameters as inputs was proposed based on Genetic Algorithm-Artificial Neural Network. The estimated structure-borne noise levels demonstrated good agreement with measured values, indicating the feasibility of the approach. The finding of this research facilitates a clear comprehension of the generation, distribution, and dissipation mechanisms of indoor structure-borne noise for engineers while also enabling convenient acquisition of indoor structure-borne noise. The estimated noise levels can be effectively utilized during building design processes along metro lines to mitigate adverse impacts on human comfort.
当火车通过建筑物下方的隧道时，对结构噪声的感知尤为突出，这对人体健康产生了负面影响。在地铁沿线建造建筑物的过程中，估计室内结构噪声水平至关重要，以增强设计并防止对人类舒适度产生任何负面影响。本研究以广州市为研究对象，对结构噪声、混响时间和列车引起的振动进行了测量，研究了结构噪声的产生、传播和耗散机制。提出了一种基于短时傅里叶变换和施罗德积分的求频混响时间方法。此外，提出了一种基于遗传算法-人工神经网络的基于室内振动、频率相关混响时间和房间参数作为输入的深度学习方法。估计的结构噪声水平与测量值吻合良好，表明该方法的可行性。本研究的发现有助于工程师清楚地理解室内结构噪声的产生、分布和耗散机制，同时也能够方便地获取室内结构噪声。在地铁沿线的建筑设计过程中，可以有效地利用估计的噪音水平，以减轻对人类舒适度的不利影响。

Building structure-borne noise is generated by structural vibrations induced by various sources, such as traffic, metro trains, and construction activities (e.g., pile driving impacts and machine vibrations), which can have detrimental effects on both vibration-sensitive equipment and human health (Venkataraman et al., 2022; Li et al., 2024; Qiu et al., 2024). Ground-borne vibration has a predominant impact on human comfort (Liu et al., 2020a) within buildings and vibration-sensitive equipment (Liu et al., 2023; Avci et al., 2020; Li et al., 2023a; Ma et al., 2024). In terms of human perception, noise is considerably more discernible than vibrations (Li et al., 2022; Sadeghi and Vasheghani, 2021). For instance, in the case of train-induced vibrations, the auditory perception of rumbling caused by metro trains is more pronounced compared to vibrations. However, the structure-borne noise generated by metro trains is almost impossible to eliminate, unlike other types of environmental noise due to the long-term transportation demand (Li et al., 2023a; Peplow et al., 2021). Therefore, it is imperative to investigate the propagation characteristics and estimation of structure-borne noise in rooms for the purpose of meeting the requirements of human health and comfort in building constructions along railways (He et al., 2022; Liu et al., 2020b).
建筑结构噪声是由各种来源引起的结构振动产生的，例如交通、地铁列车和建筑活动（例如，打桩冲击和机器振动），这可能对振动敏感设备和人类健康产生不利影响（Venkataraman 等人，2022 年;Li 等人，2024 年;Qiu 等人，2024 年）。地面振动对建筑物和振动敏感设备内的人体舒适度（Liu et al.， 2020a）具有主要影响（Liu et al.， 2023;Avci 等人，2020 年;Li 等人，2023a;马 等人，2024 年）。就人类感知而言，噪音比振动更容易辨别（Li et al.， 2022;Sadeghi 和 Vasheghani，2021 年）。例如，在列车引起的振动的情况下，与振动相比，地铁列车引起的隆隆声的听觉感知更为明显。然而，由于长期的运输需求，与其他类型的环境噪声不同，地铁列车产生的结构噪声几乎不可能消除（Li et al.， 2023a;Peplow 等人，2021 年）。因此，为了满足铁路沿线建筑施工中人类健康和舒适性的要求，必须研究室内结构噪声的传播特性和估计（He et al.， 2022;Liu等人，2020b）。

Several researchers have conducted studies on the analysis of structure-borne noise (Tao et al., 2022; Smith, 2011; Thompson, 2008; Nelson, 1987; Kurzweil, 1979). The findings support the development of subsequent prediction methods, which can be categorized into empirical approach (Thompson, 2008; Kurzweil, 1979; Vér and Beranek, 2005; Quagliata et al., 2018; Chinese code of Ecology and Environment, 2018), numerical approach (Sadeghi et al., 2023; Vogiatzis, 2012; Colaço et al., 2017a; Nagy et al., 2006; Fiala et al., 2007), and analytical approach (e Sousa and Gibbs, 2011; Kang et al., 2022; Shen and Oldham, 1982), respectively.
一些研究人员对结构噪声的分析进行了研究（Tao 等人，2022 年;史密斯，2011 年;汤普森，2008 年;Nelson，1987 年;Kurzweil，1979年）。这些发现支持了后续预测方法的发展，这些方法可以分为经验方法（Thompson，2008;Kurzweil，1979 年;Vér 和 Beranek，2005 年;Quagliata等人，2018;中国生态环境规范，2018）、数值方法（Sadeghi et al.， 2023;Vogiatzis，2012 年;Colaço等人，2017a;Nagy 等人，2006 年;Fiala 等人，2007 年）和分析方法（e Sousa 和 Gibbs，2011 年;Kang 等人，2022 年;Shen 和 Oldham，1982 年）。

Empirical approach is commonly employed in the preliminary stage of building construction to estimate and assess the sound pressure levels in buildings, which exhibits similar definitions across various standards or research studies. An empirical model was provided by RIVAS D 1.6 to describe the relationship between sound pressure levels and velocity levels (RIVAS Del.1.6, 2012), as expressed:
在建筑施工的初步阶段，通常采用实证方法来估计和评估建筑物的声压级，这在各种标准或研究中都有类似的定义。RIVAS D 1.6提供了一个经验模型来描述声压级和速度级之间的关系（RIVAS Del.1.6,2012），如：

where is sound pressure levels (re: ), is velocity levels (re: ) in -octave band center frequency with a range of . The subscript ' ' and ' ' represent the sound pressure and velocity. The similar formulations have been provided by Chinese code HJ 453-2018 (Chinese code of Ecology and Environment, 2018) and Federal Transit Administration report FTA Guidelines (Quagliata et al., 2018), as expressed in Eq. (2) and Eq. (3), respectively.
其中 是声压级 （Re： ）， 是 -倍频程带中心频率中的速度级 （Re：）， 范围为 。下标 ' ' 和 ' ' 表示声压和声速。中国代码HJ 453-2018（中国生态与环境法规，2018）和联邦运输管理局报告FTA指南（Quagliata et al.， 2018）提供了类似的表述，分别如方程（2）和方程（3）所示。

Where is A-weighting sound pressure levels (re: ), is velocity levels (re: ) in 1/3-octave bands center frequency with a range of is A-weighting adjustment at the octave band center frequency. The above formulas are applicable to the working conditions of small rooms. It is worth noting that the formula of HJ 453-2018 is specifically applicable to the conditions of relatively small rooms. Compared to Eq. (1) and Eq. (3), the structure-borne noise
其中 是 A 加权声压级 （re： ）， 是 1/3 倍频程中心频率中的速度级 （re： ），范围是 倍频程中心频率的 A 加权调整。以上公式适用于小房间的工作条件。值得注意的是，HJ 453-2018的公式特别适用于相对较小的房间的条件。与式（1）和式（3）相比，结构噪声
is overestimated by Chinese code HJ 453-2018 in order to ensure that the building design is not adversely affected by noise disturbances. However, this excessively cautious approach led to unnecessary waste of building materials and escalated labor expenses. Besides, an empirical model proposed by Beranek (Vér and Beranek, 2005) was employed by Tao et al. (2022) in a low-rise over-track building to estimate the structure-borne noise through the following formulation:
被中国规范HJ 453-2018高估，以确保建筑设计不受噪音干扰的不利影响。然而，这种过于谨慎的做法导致了不必要的建筑材料浪费和劳动力成本的增加。此外，Tao et al. （2022） 在一栋低层轨道建筑中采用了 Beranek （Vér and Beranek， 2005） 提出的经验模型，通过以下公式估计结构噪声：

where is room reverberant time, is the volume of the room, is the floor vibration radiated sound power, is the average radiation ration for the floor bay, is the velocities, is the air density, is the floor bay area, and is the sound speed in air. The results indicated a significant increase in error at higher frequencies, reaching nearly within the range of . Despite its expedience in determining structure-borne sound pressure levels in rooms, the empirical approach exhibits relatively lower accuracy.
其中 ，是房间的混响时间， 是房间的体积， 是地板振动辐射的声功率， 是地板间隔的平均辐射比， 是速度， 是空气密度， 是地板间隔面积， 是空气中的声速。结果表明，在较高频率下，误差显著增加，几乎 达到 。尽管在确定房间内结构声压级方面是权宜之计，但经验方法的精度相对较低。

Numerical approach is the commonly employed method to establish the room sound field, with the boundary element method (BEM) (Bernhard et al., 1987; Wrobel, 2002; Wu, 2002) being recognized as the most accurate technique for estimating noise from vibrations. BEM can be applied to structures of arbitrary shapes and easily integrated with finite element method (FEM) models of buildings (Nagy et al., 2006). Jean and Guigou-Carter (2021) using the 2.5 dimensions FEM/BEM model to calculate the ground-borne vibration and conduct parametric studies. Fiala et al. (2007) proposed a fast algorithm for acoustic computations based on the spectral finite element method; the results show that the dominant frequency bands of the train-induced structure-borne noise were basically determined by the first acoustic resonance of the room. A FEM model was established by Sadeghi and Vasheghani (2021) to investigate the impact of building structural and acoustic parameters on structure-borne noise.
数值方法是建立房间声场的常用方法，采用边界元法（BEM）（Bernhard等人，1987;Wrobel，2002 年;Wu，2002）被公认为估计振动噪声的最准确技术。BEM可以应用于任意形状的结构，并易于与建筑物的有限元法（FEM）模型集成（Nagy等人，2006）。Jean 和 Guigou-Carter （2021） 使用 2.5 维 FEM/BEM 模型计算地面振动并进行参数化研究。Fiala等人（2007）提出了一种基于频谱有限元方法的声学计算快速算法;结果表明：列车诱发结构噪声的主频带基本由房间的第一次声共振决定;Sadeghi 和 Vasheghani （2021） 建立了一个 FEM 模型，以研究建筑结构和声学参数对结构噪声的影响。

However, despite the simplicity and directness of the traditional numerical method, it lacks attractiveness in terms of practical estimation. The large size of the matrices for numerical calculation makes great computational costs. Improved numerical approaches have been developed by researchers in the field. Nagy et al. (2006) introduced modifications to the numerical calculations using the Rayleigh integralbased method, making it more suitable for practical applications without the need for constructing and inverting large matrices. Colaço et al. (2017b) enhanced calculation efficiency by combining a 2.5dimensional FEM with the method of fundamental solutions (MFS) model for structure-borne noise analysis.
然而，尽管传统数值方法简单直接，但在实际估计方面缺乏吸引力。用于数值计算的矩阵尺寸很大，计算成本很高。该领域的研究人员已经开发了改进的数值方法。Nagy et al. （2006） 使用基于瑞利积分的方法对数值计算进行了修改，使其更适合实际应用，而无需构建和反演大型矩阵。Colaço et al. （2017b） 通过将 2.5 维 FEM 与基本解方法 （MFS） 模型相结合进行结构噪声分析，提高了计算效率。
Analytical approach commonly involves the calculation of acoustic radiation responses for both slabs and beams. Shen and Oldham (1982) computed the directivity patterns of typical building elements based on the Galerkin method. Luo et al. (2021) proposed a computational technique based on the cellular automata (CA) method for analyzing the sound radiation characteristics of arbitrary shape structures. A theoretical model was proposed by Yang et al. (2021), which can be employed for estimating the sound radiation from a semi-infinite unbaffled long enclosure with the ground. The fundamental aspect of acoustic-vibration coupling lies in fluid-structure interaction. Due to the complexity of air fluid propagation in buildings, the analytical method is not commonly employed in practical application engineering.
分析方法通常涉及计算板和梁的声辐射响应。Shen和Oldham（1982）基于Galerkin方法计算了典型建筑构件的方向性模式。Luo等人（2021）提出了一种基于元胞自动机（CA）方法的计算技术，用于分析任意形状结构的声音辐射特性。Yang et al. （2021） 提出了一个理论模型，该模型可用于估计来自与地面的半无限无挡板长外壳的声音辐射。声-振动耦合的基本方面在于流固耦合。由于空气流体在建筑物中传播的复杂性，该分析方法在实际应用工程中并不常用。

The apparent contradiction between the accuracy and efficiency of structure-borne noise estimation poses a challenge in achieving an optimal balance. Alternatively, employing deep-learning based approaches may be considered as a potential method for reconciling the conflict (Li et al., 2023b). Redonnet et al. (2024) employed deep learning to predict airfoil self-noise, showcasing its superiority over a widely-used semi-empirical prediction tool. Zhang et al. (2021) predicted the traffic noise using the recurrent neural network, which can help the regulation and policy makers to make early decisions. Mostafavi and Cha (2023) proposed a novel high-performance deep learningbased feedforward active noise controller to attenuate constructionrelated noise. Liang et al. (2022) proposed a deep learning-based approach for the identification of traffic noise sources and their occurrence at different time intervals. The measurement and prediction results demonstrated that lower floors of buildings were more susceptible to structure-borne noise induced by underground railways, while higher floors experienced a greater level of airborne noise caused by road traffic. These practical applications demonstrate the efficacy of noise prediction using neural networks, thereby providing valuable insights for predicting structure-borne noise caused by train-induced vibrations. Consequently, this study utilizes on-site measurements and employs a deep-learning based approach to accurately estimate the structure-borne noise generated by building vibrations.
结构噪声估计的准确性和效率之间存在明显的矛盾，这给实现最佳平衡带来了挑战。或者，采用基于深度学习的方法可以被认为是调和冲突的潜在方法（Li et al.， 2023b）。Redonnet et al. （2024） 采用深度学习来预测翼型自噪声，展示了其优于广泛使用的半经验预测工具。Zhang et al. （2021） 使用递归神经网络预测了交通噪声，这可以帮助监管和政策制定者及早做出决策。Mostafavi 和 Cha （2023） 提出了一种新型的基于深度学习的高性能前馈有源噪声控制器，用于衰减与建筑相关的噪声。Liang et al. （2022） 提出了一种基于深度学习的方法，用于识别交通噪声源及其在不同时间间隔的发生。测量和预测结果表明，建筑物的较低楼层更容易受到地下铁路引起的结构噪声的影响，而较高楼层的建筑物则更容易受到道路交通引起的空气噪声的影响。这些实际应用证明了使用神经网络进行噪声预测的有效性，从而为预测由列车引起的振动引起的结构噪声提供了宝贵的见解。因此，本研究利用现场测量并采用基于深度学习的方法来准确估计建筑物振动产生的结构噪声。

Objectives of this research aim: (1) to investigate the characteristics and interrelationships of building vibrations and structure-borne noise induced by train-induced vibrations; (2) to estimate the structure-borne noise within building rooms while acquiring building vibrations, using the approach that is more accurate than empirical formulations. The approach can be applied when the spatial parameters of the room (including area, height, and location in the room), acoustical parameter of the room (frequency-dependent reverberation time), and excitation parameter (velocity levels) are available. Accurate estimation of noise response enables a more comprehensive assessment of effective measures for noise control.
本研究目的：（1）研究列车振动引起的建筑振动和结构噪声的特征和相互关系;（2）使用比经验公式更准确的方法，在获取建筑振动的同时估计建筑室内的结构噪声。当房间的空间参数（包括房间的面积、高度和位置）、房间的声学参数（与频率相关的混响时间）和激励参数（速度水平）可用时，可以应用该方法。准确估计噪声响应，可以更全面地评估噪声控制的有效措施。

(a) Floor （a） 地板

(c) Parking garage at basement
（c） 地下室停车库

(b) Southeastern view （b） 东南视图

(d) Northeastern view （d） 东北面景观

Fig. 1. Schematic representation of measurement site scene.
图 1.测量现场场景示意图。

Fig. 2. Measurement setup.
图 2.测量设置。

The technical route of this research involves conducting on-site measurements of the building vibrations and structure-borne noise at various locations within the building. The application of the transfer functions between building vibrations and structure-borne noise was utilized to investigate vibration-acoustic transfer issues related to traininduced vibrations. Additionally, the room reverberation time in the frequency domain was obtained using a method based on the Short-Time Fourier Transform (STFT) (Portnoff, 1980) combined with Schroeder integral (Schroeder, 1965), which is employed to investigate the indoor sound field mechanism of structure-borne noise and serves as one of the inputs for the deep-learning based model. Subsequently, the Genetic Algorithm (Holland, 1992) - Artificial Neural Network (Yegnanarayana, 2009) (GA-ANN) approach was employed to estimate structure-borne noise based on the acquired building vibrations.
这项研究的技术路线包括对建筑物内不同位置的建筑物振动和结构噪声进行现场测量。利用建筑振动与结构噪声之间的传递函数来研究与列车致振动相关的振动-声学传递问题。此外，使用基于短时傅里叶变换（STFT）（Portnoff，1980）和Schroeder积分（Schroeder，1965）的方法获得了频域中的房间混响时间，该方法用于研究结构噪声的室内声场机制，并作为基于深度学习的模型的输入之一。随后，采用遗传算法（Holland，1992）-人工神经网络（Yegnanarayana，2009）（GA-ANN）方法，根据获得的建筑物振动来估计结构噪声。

The contributions of this study are the utilization of STFT and Schroeder integral for determining reverberation time in the frequency domain, followed by employing a deep learning-based approach to estimate structure-borne noise. This methodology yields more precise estimations within a shorter timeframe compared to conventional empirical formulas.
本研究的贡献是利用STFT和Schroeder积分来确定频域中的混响时间，然后采用基于深度学习的方法来估计结构噪声。与传统的经验公式相比，这种方法可以在更短的时间内产生更精确的估计。

A teaching building in Guangzhou, China, was selected as the measurement site for assessing building vibrations and structure-borne noise, where the discernible rumble of the metro trains could be perceived. The building is located directly above two tunnels through which the metro train operates at a constant speed of . The photos and schematic representation of the measurement site scene are depicted in Fig. 1, where the main test room is highlighted by a yellow dotted line frame.
中国广州的一栋教学楼被选为评估建筑振动和结构噪声的测量地点，在那里可以感知地铁列车的隆隆声。该建筑位于两条隧道的正上方，地铁列车通过这些隧道以恒定的速度 运行。测量现场场景的照片和示意图如图 1 所示，其中主测试室由黄色虚线框突出显示。

The measurements were conducted during clear midnights and early morning to mitigate the impact of adverse weather conditions and human-induced noise, ensuring minimal interference from rain, strong wind, and human activities. During the measurements, all doors and windows were kept closed to eliminate potential external factors that could introduce interference.
测量是在晴朗的午夜和清晨进行的，以减轻恶劣天气条件和人为噪音的影响，确保将雨水、强风和人类活动的干扰降至最低。在测量过程中，所有门窗都保持关闭状态，以消除可能引入干扰的潜在外部因素。

As shown in Fig. 2, to investigate the structure-borne noise characteristics induced by train operations, microphones were mounted at the floor center (indicated by a yellow triangle) from the 1 st to the 5 th
如图2所示，为了研究列车运行引起的结构噪声特性，从1号到5号，在地板中心（用黄色三角形表示）安装麦克风

(a) Microphone （a） 麦克风

(b) Instruments setting （b） 仪器设置

(c) Accelerometer （c） 加速度计

Fig. 3. Instrumentations.
图 3.仪器。

Time (s) 时间

Frequency ) 频率 ）

Frequency ( ) 频率 （ ）

Frequency  频率

Power/Frequency (
功率/频率 （

Time  时间

Frequency  频率

Ftequency ( ) 英尺 （ ）

Frequency  频率

Time ( ) 时间 （ ）

Frequency (  频率 （

Fig. 4. Time history and spectrogram of a typical train pass-by event.
图 4.典型火车通过事件的时间历史和频谱图。

floors, as well as in the basement. The microphones were also mounted adjacent to the column (green triangle) and wall (blue triangle) for additional measurements conducted on the 1st floor and basement. Correspondingly, the accelerometers were placed beneath the microphones at their respective locations on the floors to obtain the vibrations. The basement covers a floor space equal to that of the entire building, which is around . The total area of the rooms on the 1st through 4th floors amounts to , whereas those on the 5 th floor occupy an area of .
地板，以及地下室。麦克风还安装在柱子（绿色三角形）和墙壁（蓝色三角形）附近，以便在一楼和地下室进行额外测量。相应地，加速度计被放置在麦克风下方的地板上的相应位置，以获得振动。地下室占地面积相当于整栋建筑的建筑面积，大约 是。1 楼至 4 楼的房间总面积为 ，而 5 楼的房间面积为 。

The instrumentation employed for measurement is illustrated in Fig. 3, including the Rion UC-59 microphone (Fig. 3(a)) and the JM3873
用于测量的仪器如图 3 所示，包括 Rion UC-59 麦克风（图 3（a））和 JM3873

(a) Comparison of different floors
（a） 不同楼层的比较

(c) Comparison of different locations
（c） 不同地点的比较

(b) Comparison of different room areas
（b） 不同房间面积的比较

(d) Comparison of different locations
（d） 不同地点的比较

floor
地板

(Basement) （地下室）

Fig. 5. Measured velocity levels at different locations.
图 5.测量不同位置的速度水平。

wireless data acquisition system (Fig. 3(b)). As shown in Fig. 3(b), the microphone (blue rectangular) is connected with a tripod adapter to ensure that the instrument is at a height of above the floor during measurements. The accelerometer (yellow rectangular) was mounted on the floor under the microphone at the corresponding location. Both the microphones and accelerometers have time synchronizers and storage cards. All instruments were calibrated based on the laptop's time prior to measurement initiation, ensuring simultaneous data recording. The sampling frequency was set to , which provided spectral information up to the Nyquist frequency of . The dominant frequency range requirement mentioned in HJ 453-2018 (Chinese code of Ecology and Environment, 2018) for structure-borne noise analysis is . Thus, the setting of sampling frequency was enough for the measurements and analysis.
无线数据采集系统（图3（b））。如图 3（b） 所示，麦克风（蓝色矩形）与三脚架适配器连接，以确保仪器在测量过程中处于 高于地面的高度。加速度计（黄色矩形）安装在相应位置麦克风下方的地板上。麦克风和加速度计都有时间同步器和存储卡。所有仪器在测量开始前都根据笔记本电脑的时间进行校准，确保同时记录数据。采样频率设置为 ，这提供了高达奈奎斯特频率的 频谱信息。HJ 453-2018（中国生态环境规范，2018）中提到的结构噪声分析的主要频率范围要求是 。因此，采样频率的设置足以进行测量和分析。

Sound pressure and acceleration time signals were downloaded from the wireless unit, which was read by a MATLAB script to compute spectrogram and frequency spectra. The signals of a typical train pass-by are shown in Fig. 4, different colors in the figure correspond to different locations. The spectrograms reveal that the dominant frequencies of both velocity and sound pressure fall within the range of . The frequency spectra are analyzed in terms of the envelope, where the figure displays the minimum and maximum values obtained for each frequency. The solid lines represent the average values. In subsequent analysis, the average value is utilized to facilitate a clear comparison.
从无线单元下载声压和加速度时间信号，由MATLAB脚本读取以计算频谱图和频谱。典型列车经过的信号如图4所示，图中不同的颜色对应不同的位置。频谱图显示，速度和声压的主频率都在 的范围内。根据包络分析频谱，其中图显示了每个频率获得的最小值和最大值。实线表示平均值。在随后的分析中，使用平均值来促进清晰的比较。

The building vibration spectra of the average of all recorded train pass-by events are depicted in Fig. 5, with a reference velocity of , which is obtained from the Chinese code HJ 453-2018 (Chinese code of Ecology and Environment, 2018). The ambient vibration predominantly contributes to the vibration energy within the frequency range of , possibly due to operational activities in the pumping house and electricity distribution room located within the testing building during measurements. However, for train-induced vibrations, which are of interest, frequencies ranging from 30 to exhibit relatively lower magnitudes of ambient vibration that can be disregarded during analysis. As shown in Fig. 5 (a), the trend of vibration amplitude reduction of building floor centers with increasing floor levels is observed as the frequency increases. The basement exhibits significantly lower vibrations compared to the superstructure at low frequencies, while the low-frequency vibrations from the 1st to 4th floors exhibit a tendency towards consistency. The attenuation of building propagation is less pronounced for low-frequency vibrations compared to high-frequency vibrations. The high-frequency vibration experiences a more significant degree of attenuation as a result of its shorter wavelength. The vibrations of load-bearing structures are consistent on relatively higher floors, which is concluded from the authors' previous experimental and analytical studies (Zou et al., 2018; Zou et al., 2022; Zou et al., 2017). Therefore, the vibration energy received on the test rooms on the fourth and fifth floors is deemed to exhibit consistency, and the comparison of velocity levels of different room areas is shown in Fig. 5 (b). However, the smaller size floors have a smaller response due to that the floor of smaller size has greater dynamic stiffness with the same thickness. The vibrations of different building components are illustrated in Fig. 5 (c) and (d), with the floor exhibiting the largest vibration while the structural column has the smallest vibration. The structural column serves as the axial propagating subject of vertical
所有记录的列车通过事件的平均值的建筑振动谱如图5所示，参考速度为 ，从中国代码HJ 453-2018（中国生态与环境规范，2018）中获得。环境振动主要影响 频率范围内的振动能量， 可能是由于测量期间位于测试大楼内的泵房和配电室的操作活动。然而，对于感兴趣的列车引起的振动，频率范围从 30 到 表现出相对较低的环境振动幅度，在分析过程中可以忽略不计。如图5（a）所示，随着频率的增加，建筑物楼层中心的振动幅度随楼层的增加而降低。与上层建筑相比，地下室在低频下的振动明显较低，而1至4楼的低频振动则表现出一致性的趋势。与高频振动相比，低频振动的建筑物传播衰减不那么明显。由于波长较短，高频振动的衰减程度更高。承重结构的振动在相对较高的楼层上是一致的，这是从作者之前的实验和分析研究中得出的结论（Zou et al.， 2018;Zou 等人，2022 年;Zou等人，2017）。因此，在四楼和五楼的测试室接收到的振动能量被认为具有一致性，不同房间区域的速度水平比较如图5（b）所示。 然而，较小尺寸的地板具有较小的响应，因为较小尺寸的地板在相同厚度下具有更大的动态刚度。图5（c）和（d）显示了不同建筑构件的振动，其中地板的振动最大，而结构柱的振动最小。结构柱作为垂直的轴向传播主体

(a) Comparison of different floors
（a） 不同楼层的比较

(c) Comparison of different locations
（c） 不同地点的比较

(b) Comparison of different room areas
（b） 不同房间面积的比较

(d) Comparison of different locations
（d） 不同地点的比较

floor)  地板）

(Basement) （地下室）

Fig. 6. Measured sound pressures at different locations.
图 6.测量不同位置的声压。

vibration, while the structural beam functions as the transverse propagating subject of vertical vibration (Zou et al., 2022). Additionally, the floor acts as the response subject of vertical vibrations.
振动，而结构梁则充当垂直振动的横向传播主体（Zou et al.， 2022）。此外，地板还充当垂直振动的响应对象。

Fig. 6 shows the structure-borne noise of the average of all the recorded train pass-by events, with a reference sound pressure of . The gray dashed lines represent ambient noise, which is considerably smaller in magnitude compared to the structure-borne noise in the frequency of interest range of train-induced vibration from 30 to . Consequently, the analysis can disregard the influence of ambient noise. As shown in Fig. 6 (a), the trend of structureborne noise reduction of the building floor center with increasing floor levels is observed above . The basement exhibits significantly higher structure-borne noise compared to the superstructure at low frequencies, while structure-borne noise from the 1st to 4th floors exhibits a tendency towards consistency at the range of . The sound pressure level magnitude in rooms with the same area is determined by the corresponding vibrations. The significant attenuation of vibration in the high-frequency range results in the reduction of structural noise at high frequencies. The noise in the basement is significantly higher than that in the upper structure within the frequency range of . The absence of interior decorations in the basement results in minimal absorption and significant reflection of acoustic waves upon reaching the boundary surface, with dissipation primarily reliant by air damping.
图 6 显示了所有记录的列车通过事件的平均结构噪声，参考声压为 。灰色虚线表示环境噪声，与列车引起的振动频率范围内的结构噪声相比，环境噪声的幅度要小得多 。因此，分析可以忽略环境噪声的影响。如图6（a）所示，建筑楼层中心结构噪声随楼层的增加而降低的趋势如上图 所示。与上层建筑相比，地下室在低频下表现出更高的结构噪声，而从1楼到4楼的结构噪声在1 至4的范围内表现出一致性的趋势。相同面积的房间内的声压级大小由相应的振动决定。高频范围内振动的显著衰减导致高频结构噪声的降低。在频率 范围内，地下室的噪声明显高于上部结构的噪声。地下室没有内部装饰，导致声波在到达边界表面时吸收和显着反射，耗散主要依赖于空气阻尼。

Fig. 6 (b) compares the sound pressure levels of different room areas. The sound pressure levels remain consistent in the frequency band above , while smaller rooms exhibit lower sound pressure levels in the frequency band below . The reason is that the impact of room area on the attenuation of low-frequency noise is minimal, while highfrequency noise is significantly affected. This can be attributed to the lower vibration level due to larger floor dynamic stiffness in smaller rooms, resulting in lower levels of structure-borne noise at low frequencies. The magnitude of high-frequency floor vibrations in a small room is relatively diminished compared to that experienced in a larger room. However, due to the reduction in indoor space, there is an increase in the number of noise reflections within the room, resulting in comparable noise levels between both rooms.
图6（b）比较了不同房间区域的声压级。声压级在上面 的频段内保持一致，而较小的房间在下面的频段中表现出较低的声压级 。原因是房间面积对低频噪声衰减的影响最小，而高频噪声影响显著。这可以归因于较小房间中较大的地板动态刚度导致的较低振动水平，从而在低频下降低结构噪声水平。与大房间相比，小房间内高频地板振动的幅度相对减弱。然而，由于室内空间的减少，房间内的噪音反射数量增加，导致两个房间之间的噪音水平相当。

The sound pressure levels of different building components are illustrated in Fig. 6 (c) and (d), the position of the wall, serving as both a radiant and reflective surface for noise, exhibits the most pronounced sound pressure level. The sound pressure level at the central area of the floor is minimal. When considering the influence of structure-borne noise, it is imperative to comprehensively address its impact on both building structure and interior design. This includes meticulous consideration of interior decoration as well as the layout of different functional areas, ensuring a holistic approach towards mitigating such structure-borne noise emissions generated from the building structure vibrations.
图6（c）和（d）显示了不同建筑构件的声压级，墙体的位置既是噪音的辐射面又是反射面，表现出最明显的声压级。地板中心区域的声压级最小。在考虑结构噪声的影响时，必须全面解决其对建筑结构和室内设计的影响。这包括对室内装饰的细致考虑以及不同功能区域的布局，确保采用整体方法来减轻建筑结构振动产生的结构噪声排放。

To represent the relationship between building vibrations and structure-borne noise. The transfer functions between structureborne noise and vibrations at corresponding locations were calculated,
表示建筑物振动与结构噪声之间的关系。计算了相应位置的结构噪声和振动之间的传递函数 ，

(a) Comparison of vibration and sound
（a） 振动和声音的比较

pressure levels of different floors
不同楼层的压力水平

(c) Comparisons of vibration and sound
（c） 振动和声音的比较

(b) Transfer function on different floors
（b） 不同楼层的传递功能

Fig. 7. Comparisons of sound pressure levels and velocity levels.
图 7.声压级和速度级的比较。

which depicts both the transmission and attenuation effects and is expressed as:
它描述了透射和衰减效应，并表示为：

Fig. 7 (a) and (c) are the comparisons of velocity levels and sound pressure levels, and Fig. 7 (b) as well as (d) are their transfer functions. The gray solid lines in Fig. 7 (b) and (d) are from Eq. (2) and as a reference standard.
图7（a）和（c）是速度级和声压级的比较，图7（b）和（d）是它们的传递函数。图7（b）和（d）中的灰色实线来自方程（2），作为参考标准。

The transfer function, in general, exhibits a negative correlation between value and frequency due to the heightened dissipation of higher frequency components when sound propagates through the air in enclosed spaces, which showed a similar trend found by in-site measurement in Ref. (Tao et al., 2022). The reason may be that the primary energy distribution of train-induced vibration predominantly occurs within the frequency range of 30 to , with only a negligible amount of energy distributed beyond this specific band. In the frequency range of , the influence of ambient noise leads to a relatively diminished disparity in the transfer function.
一般来说，由于声音在封闭空间中通过空气传播时，高频分量的耗散加剧，传递函数在值和频率之间表现出负相关，这显示出参考文献中现场测量发现的类似趋势（Tao et al.， 2022）。原因可能是列车引起的振动的一次能量分布主要发生在 30 到 的频率范围内，只有微不足道的能量分布在这个特定频段之外。在 的频率范围内 ，环境噪声的影响导致传递函数的视差相对减小。

As shown in Fig. 7(b), the transfer function of the upper rooms exhibits a resonance peak at , attributed to the room's natural frequency (Fahy and Walker, 2018), in contrast to the basement's lower natural frequency below due to its significantly larger area; hence this resonant peak is not depicted in the figure. Another transfer function difference amplification peak occurs at , corresponding to the
如图7（b）所示，上层房间的传递函数在 ，归因于房间的固有频率（Fahy和Walker，2018），与地下室 的较低固有频率形成鲜明对比，因为它的面积明显更大;因此，图中没有描绘出这个共振峰。另一个传递函数差分放大峰出现在 ，对应于

Fig. 8. Noise and vibration transfer functions for different components in rooms.
图 8.房间内不同组件的噪声和振动传递功能。

vibration velocity level. The reason is the elastic building resonance where the foundation and roof vibrate and amplify in an anti-phase ( and Tao, 2024; Auersch, 2010), which can be calculated by the following equation:
振动速度水平。原因是弹性建筑共振，其中地基和屋顶在反相中振动和放大（ 和Tao，2024;Auersch，2010），可以通过以下公式计算：

where is the anti-phase resonance frequency of the elastic building; is the building height, which equals to is the wave speed of concrete, which equals to , and the equals to and equals to are the Young's modulus and density of the concrete, respectively.
其中 为弹性建筑的反相共振频率; 是建筑高度，等于 是 混凝土的波速，等于 ， 等于 和 等于 分别是混凝土的杨氏模量和密度。

As shown in Fig. 7(c) and (d), the noise levels near the walls and pillars are increased. The reason is that the wall surface can be regarded as a robust reflection boundary when sound waves propagate in its vicinity (Bernhard et al., 1987; Wrobel, 2002; Wu, 2002), which meaning that the sound waves are reflected near the boundary and subsequently superimposed, thereby augmenting the energy density of sound pressure locally. Due to their dual function as both reflective and radiating surfaces, an upward shift of the transfer function was occurred. An inverse correlation between the spatial distribution of the indoor vibration wave field and the indoor sound field, indicating that regions characterized by high noise levels exhibit lower vibrations, while areas with low noise levels demonstrate an opposite trend. It is crucial to consider both structural design for controlling building vibrations and interior design for managing sound fields to comprehensively enhance occupants' comfort. The standards should further propose the estimation of noise near the room wall.
如图7（c）和（d）所示，墙壁和柱子附近的噪音水平增加。原因是当声波在其附近传播时，壁面可以被视为一个强大的反射边界（Bernhard等人，1987;Wrobel，2002 年;Wu，2002），这意味着声波在边界附近被反射并随后叠加，从而在局部增加了声压的能量密度。由于它们既是反射面又是辐射面的双重功能，因此发生了传递函数的向上移动。室内振动波场的空间分布与室内声场呈负相关关系，表明高噪声区域的振动程度较低，而噪声水平低的区域则表现出相反的趋势。关键是要考虑控制建筑振动的结构设计和管理声场的室内设计，以全面提高居住者的舒适度。这些标准应进一步提出对房间墙壁附近噪音的估计。

Noise transfer function and vibration transfer function for different components in rooms is defined as:
房间内不同组件的噪声传递函数 和振动传递函数 定义为：

Component Floor
组件 地板

Component Floor
组件 地板

where (Component) and (Component) is the sound pressure levels and velocity levels of the location near the wall or column, respectively. (Floor) and (Floor) are the sound pressure level and velocity levels on the floor center, respectively.
其中 （Component） 和 （Component） 分别是靠近墙壁或柱子的位置的声压级和速度级。 （Floor） 和 （Floor） 分别是地板中心的声压级和速度级。
The noise and vibration transfer function on the 1 st floor is depicted in Fig. 8, showcasing variations in sound pressure levels and vibration velocity levels across different areas. Notably, the wall and column exhibit higher sound pressure levels within the frequency range of 16-200 Hz compared to the central region of the floor due to their dual role as both reflective and radiating surfaces. Two prominent peaks are observed at frequencies of and , indicating that the column and wall noise is significantly higher than the noise at the center of the floor at both frequencies. However, in terms of the vibration transfer function, it can be inferred that the peak frequency observed at these two locations is not attributed to the vibration transfer function. There are two potential explanations for this phenomenon: (1) the radiation intensity of the wall as a radiant surface is higher at frequencies of 20 and , but it decreases when transmitted to the board due to wavelength absorption by the upholstery; (2) the absorption of these two frequencies is relatively small at the walls and column, resulting in their specific reflection when the sound wave reaches the corresponding locations.
图 8 描绘了 1 楼的噪声和振动传递函数，显示了不同区域的声压级和振动速度级的变化。值得注意的是，与地板的中心区域相比，墙壁和柱子在16-200 Hz的频率范围内表现出更高的声压级，因为它们既是反射表面又是辐射表面。在 和 的频率处观察到两个突出的峰值，表明在两个频率下，柱子和墙壁的噪声明显高于地板中心的噪声。然而，就振动传递函数而言，可以推断在这两个位置观察到的峰值频率不归因于振动传递函数。对于这种现象有两种可能的解释：（1）作为辐射表面的墙体的辐射强度在20和频率下较高 ，但由于室内装潢吸收了波长，当辐射强度传输到电路板时会降低;（2）这两个频率在壁和柱上的吸收相对较小，导致当声波到达相应位置时会产生比反射。

To experimentally determine decay parameters, such as decay times in interconnected rooms, a 24 -inch balloon explosion is employed to generate a room impulse response (RIR) , which characterizes the acoustic behavior between the noise source and receiver. Time is treated as a discrete variable Schroeder (1965) introduced a method for integrating the squared RIR, which results in an accurate representation of sound energy decay function with high precision:
为了通过实验确定衰减参数，例如互连房间中的衰减时间，采用24英寸气球爆炸来产生房间脉冲响应（RIR）， 该响应表征了噪声源和接收器之间的声学行为。时间被视为离散变量 Schroeder（1965）引入了一种对平方RIR进行积分的方法，该方法可以精确地表示声能衰减函数 ：

where is the time maximum of the signal, and is the time. It can be converted to a decibel scale in terms of power for comparison, the following explanation is:
其中 是信号的最大时间，是 时间。在功率方面可以换算成分贝标度进行比较，解释如下：

where is the decay function in the decibel scale.
其中 是分贝标度中的衰减函数。

For the impulse signal, the energy curve (not power) of conversion to decibel scale is done by the equation:
对于脉冲信号，转换为分贝标度的能量曲线（不是功率）由以下公式完成：

which is the ratio between the impulse signal amplitude and the maximum amplitude at time in the signal.
这是脉冲信号幅度与信号中时间 最大幅度之间的比率。

In addition, the reverberation time is also calculated based on STFT. STFT is a widely recognized method for time-frequency analysis (He et al., 2024), renowned for its exceptional performance in analyzing transient and non-stationary signals. It has been extensively employed across various research domains for the extraction of signal features (Wan et al., 2023; Lizhong et al., 2023). The process of the timefrequency analysis by STFT can be defined as:
此外，混响时间也是根据STFT计算的。STFT是一种被广泛认可的时频分析方法（He等人，2024），以其在分析瞬态和非平稳信号方面的卓越性能而闻名。它已被广泛用于各个研究领域，用于提取信号特征（Wan 等人，2023 年;Lizhong 等人，2023 年）。STFT的时频分析过程可以定义为：

where denotes the input signal to be analyzed, and is the time discrete variable of STFT. is the window function, which reverses in time and has an offset of samples. The is a twodimensional function of time and frequency , and is equal to .
其中 表示要分析的输入信号， 是STFT的时间离散变量。 是窗口函数，它在时间上反转并具有样本偏 移量。是 时间和 频率 的二维函数， 等于 。

(a)

(b)

(c)

Fig. 9. Frequency-dependent reverberation time acquisition methods (a) section of hot spot map at ; (b) measured impulse signal; (c) hot spot map of impulse signal based on STFT; (d) section of hot spot map at .
图 9.频率相关混响时间采集方法（a）热点图剖面图; （b） 测得的脉冲信号;（c）基于STFT的脉冲信号热点图;（d） 热点图部分 。

Fig. 9 illustrates frequency-dependent reverberation time acquisition methods based on STFT. The black and red lines in Fig. 9 (c) represent the top view of two sections in different directions, which are depicted in Fig. 9 (a) and (d) respectively, showcasing their temporal and spectral characteristics., respectively. Fig. 9 (b) represents the measured impulse signal. Schroeder integral (Schroeder, 1965) was employed to calculate the impulse energy loss, and linear regression was used to match the linear segment of the energy decay process, as shown in Fig. 9 (a). After finding the slope of the curve, the reverberation time in a certain frequency can be calculated by the following equation:
图9显示了基于STFT的频率相关混响时间采集方法。图9（c）中的黑线和红线分别表示两个截面在不同方向上的俯视图，分别在图9（a）和（d）中描绘，分别显示了它们的时间和光谱特征。图9（b）表示测得的脉冲信号。采用Schroeder积分（Schroeder，1965）计算脉冲能量损失，并使用线性回归匹配能量衰减过程的线性段，如图9（a）所示。找到曲线 的斜率后，可以用以下公式计算出一定频率下的混响时间：

where represents the time required for the room to dissipate of sound energy.
其中 表示房间声能耗散 所需的时间。

The aforementioned steps are repeated for each individual frequency in order to obtain a frequency-dependent reverberation time.
对每个单独的频率重复上述步骤，以获得与频率相关的混响时间。

Fig. 10 illustrates the decay function and corresponding frequencydependent reverberation time of different rooms. In comparison of Fig. 10 (a), (c) and (e), the basement demonstrates the longest reverberation time, while the room on the 5th floor exhibits the shortest reverberation time. However, there is little disparity in reverberation time between the rooms on the 1st and 5th floors according to Sabine's formula (Sabine, 1913):
图10显示了不同房间的衰减函数和相应的频率相关混响时间。与图10（a）、（c）和（e）相比，地下室的混响时间最长，而5楼的房间的混响时间最短。然而，根据 Sabine 公式（Sabine，1913 年），1 楼和 5 楼房间之间的混响时间几乎没有差异：

The reverberation time of a room is determined by its sound absorption area , average sound absorption coefficient and room volume , which collectively influence the length of the reverberation time. The general principle postulates a positive correlation between room size and reverberation time, indicating that larger rooms tend to exhibit longer reverberation times.
房间的混响时间由其吸声面积 、平均吸声系 数和房间体积 决定，它们共同影响混响时间的长短。一般原理假设房间大小与混响时间呈正相关，表明较大的房间往往表现出更长的混响时间。

Logistic regression (Li et al., 2022; Menard, 2002) was employed to match the reverberation time curve for a clear comparison, the following expression is as follows:
逻辑回归（Li et al.， 2022;Menard，2002）用于匹配混响时间曲线以进行清晰的比较，以下表达式如下：

where is the initial value to control the upper limit, is the final value to control the lower bound, is the center of the curve, and is a value that can control the slope of the curve.
其中 是控制上限的初始值， 是控制下限的最终值， 是曲线的中心， 是可以控制曲线斜率的值。

As shown in Fig. 10 (b), (d) and (f), the frequency-dependent reverberation time exhibits a curve that decreases from low to high frequencies, which is similar to the results obtained by NTi Audio XL2 Audio and Acoustic Analyzer and the omnidirectional DS3 Dodecahedron Speaker Set in Ref. (Kawata et al., 2023), as well as the wavelength trend of the plate (Hsu, 2023). The short wavelength of high-frequency sound waves leads to an increased number of attenuation cycles during the propagation, reflection, and absorption processes within indoor air. Consequently, the reverberation time of high-frequency sound waves is reduced. In general, the frequency-dependent reverberation time curves exhibit distinct peaks or troughs at certain frequencies in comparison to regression curves. This phenomenon can be attributed to the upholstery's ability to selectively absorb or amplify sound energy across different wavelengths (Kawata et al., 2023), resulting in a non-uniform smoothness of the reverberation time curve. As the room area decreases, there is a gradual decrease in the reverberation time curve. This phenomenon can be attributed to the increased likelihood of sound waves reaching and being reflected and absorbed by the boundaries in smaller rooms.
如图10（b）、（d）和（f）所示，与频率相关的混响时间呈现出一条从低频到高频递减的曲线，这与NTi Audio XL2音频和声学分析仪和参考文献中的全向DS3十二面体扬声器组（Kawata et al.， 2023）获得的结果相似，以及板的波长趋势（Hsu， 2023）. 高频声波的短波长导致室内空气中传播、反射和吸收过程中的衰减周期增加。因此，高频声波的混响时间减少了。通常，与回归曲线相比，与频率相关的混响时间曲线在某些频率下表现出不同的峰值或谷值。这种现象可归因于室内装潢能够选择性地吸收或放大不同波长的声能（Kawata 等人，2023 年），导致混响时间曲线的平滑度不均匀。随着房间面积的减小，混响时间曲线逐渐减小。这种现象可以归因于声波到达较小房间的边界并被反射和吸收的可能性增加。

The high efficiency of ANN makes it a widely adopted tool in engi-
ANN的高效率使其成为工程学中被广泛采用的工具。

(a) Energy decay of basement
（a） 地下室的能量衰减

(c) Energy decay of floor
（c） 地板的能量衰减

(e) Energy decay of floor
（e） 地板的能量衰减

(b) Reverberation time of basement
（b） 地下室的混响时间

(d) Reverberation time of floor
（d） 地板的混响时间

(f) Reverberation time of floor
（f） 地板的混响时间

Fig. 10. Energy decay and frequency-dependent reverberation time in different rooms.
图 10.不同房间的能量衰减和频率相关的混响时间。

Fig. 11. Structure of genetic algorithm-artificial neural network.
图 11.遗传算法结构-人工神经网络.

Table 1 表1

Training sample. 训练示例。

neering practice (Li et al., 2023b; Yegnanarayana, 2009), particularly for addressing estimation problems with great efficacy. However, the estimation accuracy of it may diminish when confronted with data exhibiting substantial variability. The accuracy of neural networks can be enhanced through the utilization of various optimization methods, such as Genetic Algorithm (GA) (Holland, 1992), Sparrow Search Algorithm (SSA) (Xue and Shen, 2020), Whale Optimization Algorithm (WOA) (Mirjalili and Lewis, 2016), and others. The GA is a computational model inspired by the principles of natural selection and genetic mechanisms in Darwinian evolution. It serves as a method for searching optimal solutions by simulating the process of natural evolution, which was employed to combine with the ANN to estimate the structure-borne noise in the case. The training samples for the ANN model were obtained by measuring sound pressure levels of structure-borne noise in octave center frequencies. The mathematical representation of the ANN model can be expressed as follows:
neering practice （Li et al.， 2023b;Yegnanarayana，2009），特别是对于非常有效地解决估计问题。然而，当面对表现出很大变异性的数据时，它的估计准确性可能会降低。通过利用各种优化方法，可以提高神经网络的准确性，例如遗传算法 （GA） （Holland， 1992）、麻雀搜索算法 （SSA） （Xue and Shen， 2020）、鲸鱼优化算法 （WOA） （Mirjalili 和 Lewis， 2016） 等。GA是一个计算模型，其灵感来自达尔文进化论中的自然选择和遗传机制原理。它是一种通过模拟自然演化过程来寻找最优解的方法，该方法与人工神经网络相结合，以估计案例中的结构噪声。ANN模型的训练样本是通过测量 倍频程中心频率中结构噪声的声压级获得的。ANN模型的数学表示可以表示如下：

where the vector represents the input parameters, which serve as the condition variables for measurements, as illustrated in Fig. 11. The is the output value, representing the sound pressure level of structureborne noise; is the number of the neurons in the input layer, representing the number of input parameters, while is the neurons of input layer; is the number of neurons in the hidden layer, while is the neurons of hidden layer; is the fitting parameter; and is the nonlinear activation function, which is defined as:
其中，向量 表示输入参数，用作测量的条件变量，如图 11 所示。是 输出值，表示结构噪声的声压级; 是输入层的神经元数量，表示输入参数的数量，而 是输入层的 神经元数量; 是隐藏层的神经元数量，而 是隐藏层的 神经元数量; 是拟合参数; 并且是非线性激活函数，其定义为：