Sequential filtering for surface wind speed estimation from ambient noise measurement

Many research results show that ocean ambient noise and wind speed are highly relevant, and the surface wind speed can be effectively inverted using ocean noise data. In most deep-sea cases, the ambient noise of medium frequency is mainly determined by the surface wind, and there is a conventional relationship between them. This paper gives an equation which shows this relationship firstly, and then a surface-wind inversion method is proposed. An efficient particle filter is used to estimate the speed distribution, and the results exhibit more focused close to the actual wind speed. The method is verified by the measured noise data, and analysis results showed that this approach can accurately give the trend of sea surface wind speed.


Introduction
Many earlier studies of ambient noise in the ocean have suggested that the action of wind on the surface often dominates all others over a broad range of frequencies (Knudsen et al., 1948;Wenz, 1962Wenz, , 1972. The close relationship between wind speed and noise level provides the basis for an effective remote sensing scheme in which wind speed may be derived from the recorded ambient noise using empirically determined calibrations. Empirically a simple linear relationship between the logarithm of the wind speed and the noise spectrum level (NSL) was reported by Piggott (1964) for shallow water (~40 m) and extended to deep water (5000 m) by Crouch and Burt (1972) and Shaw et al. (1978). Similar relationships between the wind speed and NSL have been reported by Evans et al. (1984),  and Lin et al. (2001) in the following decades. The potential of the WOTAN (Weather Observations Through Ambient Noise) technique to estimate oceanic winds from underwater ambient sound was thoroughly evaluated by Vagle et al. (1990).
In our study, we propose an approach that extends previous efforts, combining sequential Bayesian filtering (Gordon et al., 1993;Li and Kadirkamanathan, 2001) (specifically, particle filtering) to accurately estimate the wind speed from the ambient noise. Particle filtering has been employed in multiple applications in ocean acoustics with sig-nificant success (Zorych and Michalopoulou, 2008;Yardim et al., 2009;Jain and Michalopoulou, 2011). The advantages of particle filtering handling non-linear relations between observations and unknown parameters, complex noise mechanisms, and unknown and varying model order motivate our choice of such an approach for wind-speed estimation.
For the purpose, ambient noise data were obtained from a vertical line array at a deep ocean area. We compute the spectrums from the acoustic data first, and then select the center frequency of 3.15 kHz (bandwidth is about 730 Hz) to establish the relationship. Advantage for this choice of frequency is that the surface wind is the primary noise source with local and distant shipping noise virtually absent. Using particle filtering, we track the wind speed with the observation of ambient noise, and the probability density functions (PDFs) of wind are available at the output of filter. An outline of the process is illustrated schematically in Fig. 1.
The paper is arranged as follows: Section 2 provides a short introduction to particle filtering and the model developed for our problem. Section 3 presents the experiment result to demonstrate the performance evaluation of the proposed method. Conclusions are drawn presented in Section 4.
(2) z 1:k = fz i ; i = 1; 2; :::; kg where, f() is the possibly nonlinear function of the state x k-1 , v k-1 is a process noise sequence. h() is also the possibly nonlinear function, and w k is the measurement noise sequence.
In particular, we seek filtered estimates of x k based on the set of all available measurements up to time k.
From the perspective of Bayesian theory, the estimation problem is to construct the PDF by taking different values with the data z 1-k up to time k. It is assumed that the initial PDF of the state vector, which is also known as the prior, is available. Then, in principle, the PDF may be obtained in two stages: prediction and update. Suppose that the required PDF at time k-1 is available. The prediction stage involves using Eq. (1) to obtain the prior PDF of the state at time k p(x k jx k¡1 ) The probabilistic model of the state evolution is defined by the system Eq. (1) and the known statistics of v k-1 . At time step k, a measurement z k becomes available, and this may be used to update the prior (update stage) via Bayes' rule depends on the likelihood function defined by Eq. (2) and measurement noise w k . In the update equation, the measurement value z k is used to modify the prior PDF, and to obtain the posterior density of the current state.
The particle filter is based on the Monte-Carlo method, and it can transfer the integral into the sum computing. The PDF distribution can be expressed approximately as: is the random particles standing for the posterior density of , N s is the number of particles, and is the normalized weight. The posterior filtered density can be approximated as: As , the approximation approaches the true posterior density . Detailed derivation can be referred in Arulampalam et al. (2002).
Let v k be the unknown wind speed at the particular moment k, referred to as the state variable; let N k be the received ambient noise level at the moment, and it is known as the observation variable. The goal is to estimate v k based on the set of available measurement N k . We also want to estimate the PDF p(v k |N k ), which will enable us to obtain point estimates for v k as well as an understanding of the uncertainty in the estimation process. In order to solve our estimation problem, two equations are required (Jain and Michalopoulou, 2011): (a) a prediction equation to describe the transition of state variable from the moment k-1 to the next moment k, v k =F(v k-1 , z k ), and (b) an observation equation to relate the noise measurement N k with the state at the same time, N k =H(v k , q k ). Here, z k and q k are the errors consisting of additive, zero mean and normally distributed perturbations. From the prediction model, we obtain the state transition probability distribution p(v k |v k-1 ); the observation model leads to the likelihood function l(v k |N k ). By implementing a Sequential Importance Resampling scheme, particles are drawn with the help of the transition and observation equations and approximate PDF p(v k |N k ) at the moment k.
We consider the time period between the moment k-1 and k is short enough, and wind speed changes slowly. A simple state equation to predict the wind speed evolving is set up by applying a small perturbation . σ is the stress perturbations considered from the moment k-1 to the next moment k and selected empirically, and Norm(•) means normal distribution. Based on this model, the transition probability distribution necessary for the particle filter implementation can be written as: (8) As discussed previously, several studies have been carried out on the relationship between the ambient sound and wind speed showing a good agreement between the observed and ambient-sound-derived wind speeds. However, there are important differences in the various empirical relationships. Clearly, a specific algorithm needs to be established before ambient sound can be used to estimate wind XIAO Peng et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 74-78 75 independently for this ocean area. In the past it has been common to relate the pressure variance to the speed as a power law: where b and g are parameters determining the relationship; and is the slope of the spectrum on a logarithmic frequency axis and need to be determined. The observations of Farmer and Lemon (1984) at high wind speeds (up to 25 m/s) indicate that Eq. (9) fails at the frequency above a critical frequency f c (v), and it is a function of the wind (Vagle et al., 1990), lg f c = 1:9 ¡ 0:07v; (11) where f c is expressed in kHz, and v in m/s. For the frequency of 3.15 kHz, Eq. (9) holds for the wind speed below 20 m/s. The speed of 20 m/s is corresponding to Beaufort scale 8, which is quite strong and barely appeared unless extreme weather. Then, the observation equation can be expressed as:

Experiment results and evaluation
The experiment was performed in the South China Sea. A vertical Uniform Linear Array of 18 hydrophones spaced about 4 m was deployed with the topmost hydrophone 30 m below the surface. Two depthometers were fixed at the head and tail of the array to measure the actual depth, and the depth of each hydrophone was modified according to the data. The vertical array moved slowly due to the ocean surface wave and the subsurface ocean current during the experiment, but the moving speed is slow enough to be ignored compared with the wind speed. A GPS receiver was on the buoy float to track the location of the vertical array. The geometry of the experiment is shown in Fig. 2, and the depths of hydrophones are listed in Table 1. The hydrophones have a receiving voltage sensitivity of -170 dB re 1 V/μPa, which were calibrated before the experiment. The received ambient noise was pre-amplified with 6-18 dB gain (the gain was setup before the array being deployed) underwater and then stored in the digital data acquisition unit, and it was sampled at a rate of 8000 samples/s. The collection process has lasted for 60 hours.
The work ship sailed 100 km away from the buoy, and the engine was shut down during the experiment to avoid the ship noise interference. Because the ship was too far away from the buoy and the measured data from the ship were not convincing, hence, wind speed data were obtained from the National Centers for Environmental Prediction (NCEP), and they were assimilated for 0.1°×0.1° (latitude× longitude) of every hour with the precision 0.1 m/s at the height of 10 m height. Fig. 3 shows the wind speed evolved process during the 60 hours. The speed ranged from 3 m/s to 7 m/s, much smaller than 20 m/s, thus the linear logarithmic relationship of Eq. (4) can hold holds for this situation. The standard deviation (STD) for the wind speed of the month was calculated to determine σ. The STD result was about 2 m/s for the month, thus the stress perturbation used in the analysis is 2, which means z k~N orm(0, 2). The measurement precision of the buoy is within 3 dB, thus the measurement perturbation τ is q k~N orm(0, 3).
In previous studies, wind-generated ambient sound was measured at any depth. The noise levels of 18 hydrophones of different depths (1 kHz, 2 kHz and 3.15 kHz, the data length of 10 s within at 20 h duration) are shown in Fig. 4a, and the average spectrum level is plotted in Fig. 4b. The depth-dependence relationship is not obvious, and the dynamic range is within 3 dB for 3.15 kHz. Therefore the average level is used in the following estimation.
The data with interval of 10 s are truncated in every 5 minutes during the experiment period to calculate NSL and then estimate wind speed. Wind speed and the corresponding NSL are used, and the regression equation for the dataset is It was reported by Lin that G regressed to 16.86 and B regressed to 41.73. G falls between 20×(0.659±0.014) found by Lemon et al. (1984) and equals 16.76 by Evans et al. (1984), and it was about 14.8 according to Vagle's research (Vagle et al., 1990). Eq. (13) gives a high correlation coefficient and leads to the results consistent with the previous studies.
Plot of the wind speeds estimated by the conventional method is shown in Fig. 5a, while the PF technique estim-ated one is illustrated in Fig. 5b, and the differences between the estimated and NECP wind speeds is presented in Fig. 5c. From Fig. 5, we can see that the estimates match quite well with the real speed. Between 50-60 h, the estimates have a relatively large difference from the real wind speed, and this may result from a moving ship passing by the vertical array. The RMS error for conventional estimation is 1.4430 m/s, and 1.0225 m/s for the PF estimation. The particle filter gives better estimations, and the distribution of speeds is more around the NCEP wind speed. Be-   5. Speed estimation results. (a) Results estimated by conventional method compared with the NECP wind speed; (b) results estimated by particle fil-tering compared with the NECP wind speed; (c) difference in wind speed estimates and the NECP data; (d) PDF for moment 20 h of the experiment. XIAO Peng et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 74-78 77 sides, another advantage of the particle filter method is that the posterior probability distributions can also be computed. The PDF of estimation for moment k is plotted in Fig. 5d (k=20 h). The estimate of the PDF is about 5.1 m/s, and the true value is 5.30 m/s. There is relatively high uncertainty in the speed estimation. However, the PDF is centered at the correct wind speed with significant probability surrounding this value.

Conclusion
Knowledge of wind stress is of central importance in studies of the response of the ocean to atmosphere forcing, such as the generation of surface waves and drift currents, deepening of the wind-mixed layer, eddy motion and the wind-driven circulation. The surface wind speed can be effectively estimated using ocean noise data. In most deep-sea cases, the ambient noise of medium frequency (800 Hz-10 kHz) is mainly determined by the surface wind, and there is a conventional relationship between them. A surface-wind inversion method is proposed based on the relationship. An efficient particle filter is used to estimate the speed distribution, and the results are focused closer to the actual wind speed. The method is verified by the measured noise data, and the analysis results show that this approach can give more accurate sea surface wind speed than the conventional method does.