In the various styles of formulizing the Kalman filter, we used the Wiener-Hopf-Kalman integral equation[5].

　

 Consider that a random process x(t) generated by the system

　

　

 The observed signal is

　

 The functions w1 (t) and z1(t) in equation (14) and equation (15) are independent random processes (white noise) with identically zero means and covariance matrices

　

　

 Here, δ is the Direc delta function, Q1(t) and R1(t) are symmetric, nonnegative definite matrices continuously differentiable in t , and the superscript T denotes transposition.

　

 In this case, the Kalman filter can be expressed as follows

　

　

 In equation (17), Kalman gain K1(t) is given as follows.

　

V1(t) = E[e1(t)e1^T(t))]

　

where,

　

　

 Although a continuous time Kalman filter is used in the theoretical calculation for minimizing the influence of the measurement error in this research, since an actual measurement error is acquired discretely, a discrete time Kalman filter is used in the numerical calculations.

　

　

3.3 Verification of the Kalman filter

　

 To verify the effect of the Kalman filter, the comparison of an estimated value by the Kalman filter and the original value for the random data supposing a measurement result of horizontal displacement of the FPSO was performed.

　

 In Fig.4, the dotted line expresses an original value of a random data, the dashed line expresses disturbed value by the white noise shown in Fig.5, and the solid line expressed estimated value by the Kalman filter. From the result of Fig.4, although the original value largely fluctuated by mixing in white noise, it is possible to get the estimated value near the original value by using the Kalman filter