The bore-sight angles lead to discrepancies in overlapping areas. In order to reduce the correlations between the bore-sight angles and find an optimal configuration of the conjugate planar patches, it is essential to understand the effects of bore-sight angles. We assume that $\alpha $, $\beta $, and $\gamma $ are the initial angles in rotation matrix $RM$, while $\Delta \alpha $, $\Delta \beta $, and $\Delta \gamma $ are the bore-sight angle errors in roll, pitch, and heading direction, respectively. Because the bore-sight angles are very small, the geolocation equation can be described as follows when considering the effects of bore-sight errors: Display Formula
$X\u2192=RWRGRN[\Delta RMRMRL[00\rho ]+P\u2192]+X\u2192GPS,$(2)
where Display Formula$RM=[cos\u2009\gamma \u2212sin\u2009\gamma 0sin\u2009\gamma cos\u2009\gamma 0001][cos\u2009\beta 0sin\u2009\beta 010\u2212sin\u2009\beta 0cos\u2009\beta ][1000cos\u2009\alpha \u2212sin\u2009\alpha 0sin\u2009\alpha cos\alpha ],$(3)
Display Formula$\Delta RM=[1\u2212\Delta \gamma \Delta \beta \Delta \gamma 1\u2212\Delta \alpha \u2212\Delta \beta \Delta \alpha 1].$(4)
The bore-sight errors cause deviation of the target coordinates from the correct location. To better understand the error effects, we denote $RM$ as an identity matrix and analyze the coordinate bias in the IMU reference frame; therefore $RW$, $RG$, $RN$, $P\u2192$, and $X\u2192GPS$ are not taken into account. The $x$, $y$, and $z$ are the coordinates in the IMU reference frame, and $\Delta xm$, $\Delta ym$, and $\Delta zm$ are the coordinate offsets between the true location and the biased location, which can be calculated by Display Formula$[\Delta xmym\Delta zm]=\Delta RMRMRL[00\rho ]\u2212RMRL[00\rho ],$(5)
Display Formula$\Delta xm=\Delta \gamma \rho \u2009sin\u2009\theta +\Delta \beta \rho \u2009cos\u2009\theta \Delta ym=\u2212\Delta \alpha \rho \u2009cos\u2009\theta \Delta zm=\u2212\Delta \alpha \rho \u2009sin\u2009\theta ,$(6)
where $\theta $ is the scan angle. If $\Delta H=\rho \u2009cos\u2009\theta $ represents the vertical distance from the scanner to the laser footprint, Eq. (6) can be changed to Display Formula$\Delta xm=\Delta \gamma \Delta H\u2009tan\u2009\theta +\Delta \beta \Delta H\Delta ym=\u2212\Delta \alpha \Delta H\Delta zm=\u2212\Delta \alpha \Delta H\u2009tan\u2009\theta .$(7)
As shown in the above equations, $\Delta \alpha $ generates offsets along both the $z$ and $y$ axes, while $\Delta \beta $ and $\Delta \gamma $ only generate offsets along the $x$ axis. Therefore, $\Delta \alpha $ is not related to $\Delta \beta $ or $\Delta \gamma $ in terms of coordinate offsets. The effect of $\Delta \beta $ relates to the flight altitude and is independent of the target distribution in the strip swath. Two strips with the same altitude but opposite directions lead to equal but opposite offsets caused by $\Delta \beta $ for the same target. The effects of $\Delta \alpha $ and $\Delta \gamma $ relate to the scan angle. In a special case, when the target is at the nadir of the strip, the scan angle is zero, as are the effects of $\Delta \gamma $ and $\Delta \alpha $.