# A Maximum Likelihood Estimation Method for Underwater Radiated Noise Power
## 1 Introduction  
With the rapid expansion of global shipping trade, ocean noise levels continue to rise annually [1, 2, 3, 4, 5]. Substantial research indicates that underwater radiated noise poses a significant threat to marine life, particularly marine mammals [6, 7, 8]. It can interfere with essential behaviors such as navigation, communication, and foraging. Prolonged exposure to elevated noise levels may also cause hearing damage and even cause fatalities in marine life. To mitigate these impacts of ocean noise pollution on marine ecosystems, the International Maritime Organization and various nations have implemented regulations limiting the radiated noise levels from marine vessels [9, 10, 11]. Consequently, measuring the radiated noise of ships and underwater vehicles prior to delivery is essential to assess whether their noise levels comply with design requirements and regulatory standards [12, 13, 14, 15, 16]. As a fundamental physical characteristic, the power of radiated noise provides the basis for calculating other quantities that describe the characteristics of underwater radiated noise, such as the source level, source band level, and source spectral level [17]. Due to cost and operational constraints, early measurements of underwater radiated noise predominantly employed single hydrophone systems, where the received sound pressure level was corrected via acoustic propagation loss to deduce the source level of theunderwater radiated noise [18; 19; 20; 21; 22]. However, as underwater radiated noise levels decline while the ocean ambient noise level increases year by year, the Signal-to-Noise Ratio (SNR) of data received by a single hydrophone has progressively declined, leading to a gradual inability to meet measurement accuracy requirements [23; 24; 25; 26]. To address this issue, methods based on hydrophone arrays have been developed that leverage the spatial directivity and array gain to reduce the influence of ambient noise on radiated noise measurements, thereby improving measurement accuracy [27; 28; 29]. Nevertheless, these methods typically assume plane-wave acoustic propagation, which differs from the actual shallow-water waveguide environment, leading to degraded performance on radiated noise power estimation in practical applications. Subsequently, a radiated noise measurement method based on conventional Matched-Field Processing (MFP) is proposed, which calculates the channel transfer function from the sound source to each element of the array via acoustic field models, exploiting multipath signals for improved accuracy [30]. However, this method remains susceptible to ambient noise effects under low SNR conditions. For such scenarios, a denoising method is used to estimate the signal covariance matrix prior to beamforming, effectively improving measurement accuracy [17; 31; 32]. Maximum Likelihood Estimation (MLE) is one of the most fundamental and widely used parameter estimation methods in statistics [33; 34]. Its core principle is to find the parameter value that maximizes the probability of the observed data. This paper applies it to the measurement of underwater radiated noise and proposes a maximum likelihood estimation method for underwater radiated noise power. Unlike conventional methods for radiated noise measurement based on array signal processing, the proposed method firstly establishes the probability density function of the received data under the assumption of white Gaussian noise. Subsequently, an estimation of the radiated noise power, which conforms to the maximum likelihood criterion, is derived and validated as the minimum variance unbiased estimation. Compared to conventional radiated noise measurement methods, the proposed MLE is the optimal estimation, providing the optimal estimation accuracy for the underwater radiated noise power. The rest of the paper is organized as follows. Section 2 introduces the receiving data model with a Vertical Linear Array (VLA). Section 3 reviews several common radiated noise power estimation methods based on beamforming.