The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part Bl. Beijing 2008 
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The simulation of each laser point can be performed using the 
sensor equations in Eq. (1) and (2). The sensor equation without 
any error in Eq. (1) can be rearranged to 
PfV ~ P-GWP-NGP-LN + PgwPnG^GL n + t WG w ’ (3) 
where t is defined as (/ +/ ), indicating the translation 
vector between the GPS coordinate system and the LS 
coordinate system. This equations is further summarized as 
P w =u L r + P L , (4) 
where u L is defined as R GW R m R LN R 0L u 2 , indicating the unit 
vector along the direction of a transmitting laser pulse 
represented in the WGS84 coordinate system; P L is defined as 
R GIV R NG t C L + t WG ’ indicating the starting point of the pulse 
represented in the WGS84 coordinate system. 
Based on Eq. (4), the simulation of each laser point can be 
performed using the following procedures. 
1. The position of the platform at a particular time is computed 
using its flight path and velocity and then used to determine 
the starting position of a transmitted laser pulse ( p L ). 
2. With the assumption on the attitude of the vehicle, the 
direction of the transmitted laser pulse ( W; ) is calculated by 
setting the scanning angle at the time. 
3. Starting from the laser pulse starting point, a virtual line is 
generated with its direction. The intersecting points of this 
line with the given 3D models are found using a "ray 
tracing" algorithm. Among these points, only the first visible 
point from the sensor is selected as the true point ( p w ), 
where the laser pulse is reflected. Using this point with p L 
computed in step 1, the true range ( r ) is computed. 
4. Based on the sensor equation with the errors in Eq. (2) and 
the true range computed in step 3, the noisy point (/£) is 
then computed. 
Using the flight path given, the true value of sensor platform 
position (actually, the origin of the GPS coordinate system) at a 
particular time ( t ) is derived as 
where t s is the time of the platform at the position at p and V 
is the vector of the platform velocity. Here, the flight path 
information are given with two points, that is, the starting point 
(p ) and the ending point ( p ). It is assumed that the platform 
travels only in a straight line between these two points at the 
speed of v. In this case, the velocity ( V ) is expressed as 
We suppose that the description of the ground surface is given 
by a two dimensional surface function, as represented in Eq. (7). 
In this case, the surface function is expressed as a DEM or 3- 
dimensional point clouds. 
z = f(x,y) (7) 
The simulation determines the three dimensional coordinates of 
the reflected point, when a laser pulse is transmitted from the 
origin point (p L ) to the direction of the u L reflects from the 
surface z = f(x,y)- This point is determined by a ray-tracing 
algorithm to find the intersection point between the straight line 
starting from P L to the direction of u R and the surface 
expressed as z = /(x, y) • 
Figure 2. The basic concept of ray-tracing algorithm 
The basic concept of this ray-tracing algorithm is illustrated in 
Figure 2, being summarized as 
1. Search the minimum z value of DEM. 
2. Establish a virtual horizontal plane with the elevation of this 
value. 
3. Determine the intersection point between this horizontal 
plane the straight line. 
4. Using the horizontal coordinates of this point, compute its 
corresponding elevation on the DEM. 
5. Repeat step 2-4 until there is no change in the elevation 
value. 
These iterative processes finally derive the intersection point 
(Q) between the straight line and the surface. Using this point, 
the true range can be computed. By substituting this value for 
rand other specified bias values for aAR,„, A> Ar, 
NG LN 0 L 
, At , At , and A/™ in Eq. (2), we can determine the 
iAl NL N ’ l * 1 gn n ’ lxi wg w ’ tb n V 
observed coordinates of the laser reflected point, /£. 
3. EXPERIMENTAL RESULTS 
To demonstrate our approach, we generate the simulated 
LADAR data from the real DEM using the proposed method.