Research On Random Fields Theory From Local Average Sampling And Its Applications In Wave Monitoring

This paper aims to generalize the theory of stochastic process from local averages to that of random fields with general weight function,whose application is to extend the random wave theory based on stochastic process to the one based on random fields.Furthermore,the existing significant wave height inversion algorithms for wave monitoring are optimized and improved,and finally the inversion efficiency and approximation accuracy are obtained.Since Shannon introduced the sampling theorem into the information field in1948,the sampling theorem has been widely concerned by mathematicians and information processing experts both in theory and in applications.The local averages of random signals are used to push the sampling theorem into the new research stage.On the other hand,with the influence of random field theory in the application gradually expanded,objectively,it is necessary to combine the random field with the local average in theory to establish a random field local average sampling model with general weight function.In 1992,Gr ¨ochenig combined with the predecessors' results,proposed a mathematical model to take the local average of the determined signal with a series of continuous weight function,and used the obtained average sampling to reconstruct the original signal successfully.Then the scholars Aldroubi,Butzer,Lei,Sun,Zhou and Song used the local average sampling to get a series of approximation and reconstruction results about deterministic signals.In 2006,Song introduced the local average sampling and reconstruction of random signals.In the field of stochastic process applications,in the early 1950 s,the waves of full growth was considered as stochastic process in weak sense,and the statistical characteristics of the waves were given through the stochastic process theory analysis under various circumstances.However,exactly speaking,the waves are random signals dependent on space and time variables,hence reasonable to describe the wave random movement with a random field.Moreover,the current wave algorithm based on X-band radar has not yet separated from 3-D FFT proposed by Young and others,failed to make a breakthrough in theory,but also did not realize and clearly put forward the radar signal sampling is a random field local average sampling.Based on the previous work,the local average sampling of the random signal is extended to the local average sampling of the random fields,the local average sampling model of the random wave motion is established,and the significant wave height inversion algorithm is improved.The innovation can be summarized as the following four points:1.The model is established for band-limited homogeneous random fields with the local average of general weight function,the explicit error upper bound of band limited homogeneous random field by local averages in the mean square sense is given.2.By using the local averages of the general weight function,it is proved that the local average sampling of the band limited homogeneous random field can converge to the original signal with probability 1.3.By using the local averages of the general weight function,the explicit error upper bound of the band limited nonhomogeneous random field by local average in the mean square sense is given.4.A random sampling model of stochastic waves based on random fields from local average is established,and the motion of the random wave is newly described.The significant wave height is extracted directly from the X-band radar image with the rotated empirical orthogonal function analysis.The polynomial regression model improved the accuracy and approximation between the SWH retrieved by radar and that of the measured with the buoy.