The vessel roll, pitch and yaw motions are observed in the right-handed level vessel co-ordinate system (LVCS) shown in Figure A1. Its origin is at the vessel's centre of motion and its x and y-axes are in a horizontal plane. The x-axis points in the direction of the vessel's forward motion. Vessel roll, pitch and yaw correspond to rotations about the x, y and z-axes, respectively. The direction of positive rotation is given by the curved arrows and is defined by the right hand rule, In calm weather, the LVCS coincides with the vessel co-ordinate system (VCS) which has its axes aligned with the vessel's stern to bow, port to starboard and vertical axes, respectively. We define roll, pitch and yaw in the LVCS as shown in Figure A1. However, roll, pitch and yaw are often defined in the VCS and may include sign conventions for the rotation directions based on naval traditions (Hare et al., 1995). We have chosen to develop our model in the LVCS for simplicity and as average results will be very similar.

　

　

In the LVCS, the effect of roll, pitch and yaw on the location of an object that is fixed to the vessel (e.g., a transducer) is described by the three rotation matrices (Hare et al., 1995; Harrington, 1987; Hearn and Baker, 1986; Wolfram, 2002):

　

　

where (ψ_x, ψ_y and ψ_z are the roll, pitch and yaw angles. Following a roll, pitch and yaw rotation the location of a transducer fixed to the vessel is given by,

　

　

where A₁₁ gives the location of the transducer on the vessel before rotation (Fig.A1) and A₁₂ is the new location (Fig.A2) and A₁₁, A₁₂ and P₁₁ are column vectors. The first 1 in the subscript indicates that we are using the LVCS. The position of the target remains unchanged at P₁₁.

　

　

To find the changed transducer view angle we now introduce the level transducer co-ordinate system (LTCS) that is obtained by translating the origin of the LVCS to the rotated transducer location A₁₂. The target position in the LTCS co-ordinate system is:

　

　

To account for the changed transducer view angle we now rotate the LTCS. This rotation differs from the previous one as we now rotate the co-ordinate system rather than a point in a fixed co-ordinate system. This is accomplished by the use of the inverse rotation matrices in the following equation:

　

　

where P₃₁ is the position of the target in the transducer co-ordinate system (TCS) shown in Figure A3, and R_x^-1, R_y^-1, and R_z^-1 are the inverse rotation matrices obtained by using the negative rotation angles -ψ_x -ψ_y and -ψ_z in equation A1.

　

To calculate the beampattern for the observed target we require the angular position of the target as seen in the TCS. It is obtained by using the (x, y, z) co-ordinates of P₃₁ in the following conversions.

　

　

Spherical co-ordinates are shown in Figure A4. The transformation of vector (x, y, z) from Cartesian to spherical co-ordinates is provided by:

　

　

　

Split-beam co-ordinates are shown in Figure A5. The transformation of vector (x,y,z) from Cartesian to split-beam co-ordinates is:

　

　

　

Although the target is fixed in an earth co-ordinate system, it will appear to move relative to the vessel. In the LVCS, the target position will be given by:

　

　

where (x_t , y_t , z_t) is the initial target position (x_v , y_v , z_v) and (v_x , v_y , v_z) are the vessel's initial position and velocity and s, w and h are vessel surge, sway and heave, respectively. All are measured in the right handed earth co-ordinate system that, at t=0, coincides with the LVCS, but with inverted y and z-axes. Thus, vessel heave will be positive when the vessel is on a wave crest. Generally x_t, y_t, x_v, y_v, z_v and v_z will be zero.