TIME-SPACE MAPPING BASED ON FREE NET - TRILATERATION 
Eihan SHIMIZU, Dr.-Eng. 
Associate Professor 
Department of Civil Engineering, University of Tokyo, Japan 
Commission IV, Working Group 1 
KEY WORDS: Time-Space Mapping, Trilateration, MDS, GIS, Free Network Adjustment, Moore-Penrose Matrix 
ABSTRACT: 
Time-space mapping gives the distortion to a physical map such that the distances between any two points on the map 
are as consistent as possible with given time distances. If the time-space mapping procedure is integrated into GIS, it 
will be an attractive presentation tool for regional and transportation analyses. Time-space mapping procedure has 
been in general divided into the following two steps: i) multi-dimensional scaling (MDS) step which configures the 
points given the time distances between their points; ii) interpolation step which gives a mapping from the physical map 
to the time-space map based on the configured points by MDS. In conventional studies, the above two steps have 
been independently implemented. That is, the errors of points' configuration by MDS which are generally inevitable 
have been ignored in the interpolation step. This paper provides a possible time-space mapping procedure which is 
theoretically consistent with the least squares method by introducing the free network adjustment of trilateration. 
1. INTRODUCTION step, the points configured by the MDS have been assumed 
to have independent and equal positional error. However, 
Time-space mapping gives the distortion to a physical we cannot necessarily assert positively that such an 
map such that the distances between any two points on assumption holds for the configured points. It would be 
the map are as consistent as possible with given time reasonable that we regard the coordinates of the points 
distances. Time-space map shows visually an outline of as the random variables which are mutually dependent 
the transportation level of service. Two time-space maps and unequally distributed. Variance and covariance of 
before and after a certain transportation improvement the coordinates of configured points should be dealt with 
visualize impressively its impact. in the interpolation procedure. 
Time-space mapping procedure is basically a technique This paper provides a possible time-space mapping 
of the transformation of map coordinates. If the time-space procedure which is theoretically consistent with the least 
mapping procedure is integrated into GIS, it will be an squares method by employing the free network 
attractive presentation tool for regional and transportation adjustment of trilateration based on Moore-Penrose 
analyses. generalized inverse. The next chapter shows Torgerson's 
MDS which has been most frequently used in MDS 
There have been so far a few studies associated with applications and discusses its the problems as MDS 
the time-space mapping (Ewing and Wolfe, 1977; Shimizu, procedure for time-space mapping. Chapter 3 gives the 
1993; Spiekermann and Wegener, 1933). According to basic formulation of the least squares MDS. In Chapter 
these studies, the time-space mapping procedure is in 4, the least squares MDS based on the free network 
general divided into the following two steps: concept (Free Network MDS) is formulated. The following 
chapter shows the interpolation procedure using the points 
- Given the time distances between some points, provide configured by Free Network MDS. 
the configuration of the points on the time-space map. 
This procedure is interpreted as multi-dimensional scaling 
(MDS) into two-dimensional space. 2. TORGERSON'S MDS 
- Interpolate (or extrapolate) the other map elements which Let i (i =1,2,..,n) be the points between which the time- 
are portrayed on the physical map onto the time-space distances, £j, are given. Let the coordinates of the time- 
coordinate system by using the points configured by the space map be denoted by (uj, v;). Assume that the 
MDS as control points. time-distances are given to all pairs of the points, that is, 
the number of the observations is m=n(n-1)/2. 
In conventional studies the above two steps have been 
independently implemented. That is, in the interpolation Torgerson's MDS  (Torgerson, 1952) begins from the 
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996 
  
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