延迟期望输出对有限数据最小平方滤波误差的作用

THE INFLUENCE OF DESIRED OUTPUT LAG ON LEAST SQUARES FILTER ERROR OF FINITE DATA

1. 中国科学院系统研究所;2. 地质部北京计算中心

1. The System Institute of the Chinese Academy of Science;2. Beijing Computer Center, Ministry of Geology

给定有限输入x和期望输出d,存在唯一的m+1长的最小平方滤波因子h^m=(h₀^m,h₁^m,…,h_m^m),为简单起见,记上式为q_m²。一般q_m²不趋于0（m→∞）,只有当x为最小延迟时才有q_m²→0（m→∞）。这样就限制了最小平方滤波的作用。但在地震数据处理中,有时允许期望输出d延迟其一定出现位置,即以x和为输入和期望输出,当最小平方滤波因子长度为m+1时,对应的误差记为q_m²（S）。本文的结论有q_2m²（m）→0,（m→∞）。更进一步有其中k>1的任何实数,[km]为km的整数部分。

For a given limited input x and a given desired output d, there is anunique least squares filter factor,h^m=(h₀^m,h₁^m,…,h_m^m), whoselength is m+1 and satisfies,For simplicity, denote this expression as q_m². As a general rule, q_m²does not tend to zero as m→∞, unless x is minimum delay. Therefore, the application of the least squares filter has limitations. But during seismic data processing, the position where the desired output appears isspermitted to delay a centain distance, i.e. x and Tsd = (0,……,0,d₀,d₁……,d_p) act as the input and desired output respectively. Then, if the length of the least squares filter factor is m+1 taking the corresponding error as q_m²(s), then according to our conclusion, qm₂²(m)→0 as m→∞. Furthermore, q²(km)+ 1(m)→0, as m→∞. Where k is an arbitrary real number and k>1; [km] is the integer part of the number km.