## 2 Comparison of TRN Based on RA and IRA  
### TRN Using RA  
Figures 1 and 2 show examples of how TRN using RA finds its position. The TRN algorithm used in the example is a batch processing algorithm known as TERCOM [1]. Measuring the barometric altitude and RA altitude while the aircraft is flying along the trajectory, TERCOM can use the difference between the two altitudes to obtain the elevation of the ground directly below the aircraft (Figure 1). The latitude and longitude of the inertial navigation system are stored together with the elevation of the measured terrain as shown in Figure 2a. The DEM is embedded in the TRN system of the aircraft, so it has altitude information in terms of latitude and longitude for the trajectory as shown in Figure 2b. When the correction period of TRN is reached, it is possible to find the matching reference elevation set by comparing the measured terrain elevation set (Figure 2a) with the DEM (Figure 2b). Figure 2c shows the Mean Absolute Difference (MAD) reference matrix of the batch processing algorithm. The MAD  
algorithm which is used for correlating the measured terrain elevation file with each down track column of the reference matrix, is defined as follows [1].  
\[MAD_{i,j}=\frac{1}{M}\sum_{k=1}^{M}\lvert h_{measur,k}-h_{DB}(\hat{\lambda}+i \Delta\lambda,\hat{\phi}+j\Delta\varphi)\rvert \tag{1}\]  
where \(M\) is the number of samples in the measured terrain elevation file. \(h_{measured,k}\) and \(h_{DB}\) are the terrain measurement and database value. \(\hat{\lambda}\) and \(\hat{\phi}\) are estimated longitude and latitude and \(\Delta\lambda\), \(\Delta\varphi\) are the longitudinal and latitudinal cell size of the reference matrix. \(i\) and \(j\) are the column and row index for the reference matrix.  
The position correction value can be obtained by selecting the minimum MAD cell from the reference matrix. In the case, the position correction value is \(-0.001^{\circ}\) in the latitude direction and \(+0.002^{\circ}\) in the longitude direction.  
In drawing Figure 2, it was assumed that the RA can measure the relative altitude of the terrain in the direct downward direction. However, the actual RA cannot always measure the relative altitude to the terrain directly below the aircraft because the actual RA outputs the relative distance to the closest  
Figure 1: Measurement of TRN using RA.  
Figure 2: Example of TRN using RA (ideal case): (**a**) Measured terrain profile; (**b**) DEM; (**c**) position correction value.  
point among points in the beam footprint. As shown in Figure 3, if there are other peaks in the radar beam footprint, objects such as trees or buildings, or objects with high reflectivity, such as lakes or oceans, the RA will measure the relative distance from the antenna to those points (blue arrow), and will not measure to a point directly below the aircraft (black dash line). If the media of the ground are all the same, the closest points measured by RA will be in peaks or ridges in the radar beam footprint. In Figure 4, the peaks on the DEM are marked with a dashed red box, and the altitude will be measured when the aircraft follows its trajectory. In the case of Figure 4, correction values of 0.000\({}^{\circ}\) in the latitude direction and 0.001\({}^{\circ}\) in the longitude direction can be obtained. However, these correction values are different from the correction values shown in Figure 2. In other words, the RA error generates a TRN error. In order to reduce the RA error, there is a method of reducing the beam width of the RA by lowering the flight altitude as much as possible or making the angle of the beam width smaller by increasing the frequency of the radio wave used in the RA. However, even if these methods are used, the RA error does not fundamentally disappear, and the fact that TRN error exists does not change.