The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part Bl. Beijing 2008 
LADAR system, this equation must be a function of the data 
acquired by the individual sensors. In order to derive the sensor 
equation by combining the data acquired by the individual 
sensors, it is necessary to establish the geometric and time 
correspondence of all individual sensors. The geometric 
correspondence means that each sensor should be defined with 
its position and attitude in a common coordinate system; and 
the time correspondence means that the time referred to the 
individual sensors should be synchronized. These 
correspondences can be established by defining the mutual 
geometric and time relation to individual sensors. 
A sensor equation for a general LADAR system without 
considering any errors associated with the system is derived as 
R W = R GW R NG ( R LN R 0L U z r + * NL n + *GN S ) + Vg*. > ^ 
where each variable is explained in Table 1. Here, 0, L, N, G 
and W are used as a subscript indicating of its corresponding 
coordinate system, such as the initial LS coordinate system, the 
LS coordinate system, the INS coordinate system, the GPS 
coordinate system, the WGS84 coordinate system, respectively. 
These coordinate systems are shown in Figure 1. In addition, R 
and t indicates the rotation matrix and the translation vector, 
respectively. They are introduced to establish a geometric 
relationship between two coordinate systems. 
Variables 
definition and description 
P w 
The true values of laser pulse’s reflected point 
based on the WGS84 coordinate system 
U z 
The unit vector (0,0,1) along the z-axis based on 
the initial LADAR coordinate system 
r 
The range from the starting point of the 
transmitting laser pulse to its reflected point on 
the target surface 
Rol 
The rotation matrix used for the transformation 
from the initial LADAR coordinate system to 
the LADAR coordinate system 
R ln 
The rotation matrix used for the transformation 
from the LADAR coordinate system to the INS 
coordinate system 
R ng 
The rotation matrix used for the transformation 
from the INS coordinate system to the GPS 
coordinate system 
R gw 
The rotation matrix used for the transformation 
from the GPS coordinate system to the WGS84 
coordinate system 
t NL„ 
The translation vector connected from the origin 
of the INS coordinate system to the origin of the 
LADAR coordinate system represented in the 
INS coordinate system 
t GN N 
The translation vector connected from the origin 
of the GPS coordinate system to the origin of 
the INS coordinate system represented in the 
GPS coordinate system 
t WG w 
The translation vector connected from the origin 
of the WGS84 coordinate system to the origin of 
the GPS coordinate system represented in the 
WGS84 coordinate system 
Table 1. 
The definition and description of the variables 
embedded in the sensor equation 
GPS Coordinate System 
Figure 1. The coordinate system referred to each sensor 
The sensor equation presented Eq. (1) does not consider any 
errors and hence produces the true point at which a laser pulse 
is reflected. However, every measurement acquired by a sensor 
must include some systematic and random errors. Thus we 
identified the error sources and derive their parametric models. 
The errors associated with LADAR measurements are largely 
categorized into two groups, that is, the individual sensor errors 
and the sensor integration errors. The first group includes the 
systematic errors associated with the individual sensor, for 
examples, INS drift errors, GPS bias errors, the LS range error 
and other errors. The second one is caused from the integration 
of different sensors in terms of their geometry and time. For 
examples, thee individual sensors (GPS, IMU, LS) are mounted 
at different positions at a platform. Hence, the primitive data 
acquired by each sensor are represented with respect to the 
coordinates system fixed to each sensor. To combine these data, 
one should transform them into a common coordinate system. 
Here, the transformation parameters composed of the 
translation vector and rotational matrix can include some 
systematic errors. In addition, the different acquisition 
frequency of the individual sensors and the deviation from the 
time referred to the individual sensors can cause some errors. 
More details on the error sources and their parameterization are 
presented by Schenk (2001) and Lee (2005). 
From the parameterization of the main ones among these 
various error sources, we can derive the sensor equation with 
errors as 
R W — R GW^R\G R NG (^^LN^LN^^0L^0L U z( r + Ar)... (2) 
t NL n A/NL N ^GN n ^GN n ) ^WG w ■*" ^WG w ^TB ’ 
where indicates the observed values of the true laser pulse’s 
reflected point p w , which includes all the systematic errors. 
The variables representing the systematic errors starts with the 
notation of A. For example, Armeans the range bias added to 
the true range r. 
2.2 Data Simulation 
The essential information required for simulation are the flight 
paths, the attitude of the platform, and the three dimensional 
geometric models of the terrain or targets such as a DSM.