## 2 Comparison of TRN Based on RA and IRA
### TRN Using IRA  
In order to overcome the disadvantages of RA described above, an IRA was developed [4]. The IRA outputs the slant range (\(R\)) to the closest point in the signal reflected from the zero Doppler  
Figure 4: Example of TRN using RA (real case): (**a**) Measured terrain profile; (**b**) DEM; (**c**) position correction value.  
Figure 3: Error examples of RA [12].  
region (line) and the look angle (\(\theta\)) in the direction of the transverse axis (Figure 5). The IRA uses pulse compressing to select signals whose Doppler value is zero. In order to obtain the angle of the first returning signal, two or more antennas are placed in the transverse direction of the flight vehicle, and an interferometry method is used [13]. Since the IRA outputs the look angle, unavailable in RA, it is possible to more precisely measure the relative horizontal and vertical positions from the aircraft to the closest point.  
An example of TRN using IRA is shown in Figure 6. This IRA can be used to determine not only the relative altitude of the closest point but also the horizontal relative position. In this case, MAD is calculated as follows.  
\[\begin{array}{l}MAD_{i,j}=\frac{1}{M}\sum\limits_{k=1}^{M}\left|h_{measured,k }-h_{DB}\big{(}\hat{\lambda}+i\Delta\lambda+p_{k,lon},\phi+j\Delta\varphi+p_{k, lat}\big{)}\right|\\ =\frac{1}{M}\sum\limits_{k=1}^{M}\left|\big{(}h_{horo,k}-\left|p_{k,alt}\right| \big{)}-h_{DB}(\hat{\lambda}+i\Delta\lambda+p_{k,lon},\phi+j\Delta\varphi+p_{k, lat}\big{)}\right|\end{array} \tag{2}\]  
where \(p_{k,alt}=p_{k,u}\) is the vertical relative position to the closest point, \(p_{k,(lon,alt)}=\left[p_{e}/R_{ew},p_{n}/R_{ns}\right]\) are the east and north direction relative position to the closest point, \(R_{ew}\) is the Earth's prime vertical radius of curvature, \(R_{ns}\) is Earth's meridional radius of curvature.  
Figure 5: Measurement for TRN using IRA [8; 12].  
Figure 6: Example of TRN using IRA: (**a**) Measured terrain profile; (**b**) DEM; (**c**) position correction value.  
Comparing DEM and the relative position up to the closest point, correction values of \(-0.001^{\circ}\) in latitude direction and \(+0.002^{\circ}\) in longitude direction can be obtained. The correction values are the same as those of the ideal TRN with RA, shown in Figure 2.  
In order to perform TRN, it is necessary to calculate the relative position \(p_{k}\) of the closest point. Even if the IRA outputs the correct range and angle to the TRN, if \(p_{k}\) is miscalculated, the TRN calculates the wrong navigation solution. Figure 7 shows an example of TRN error caused by a miscalculated \(p_{k}\). It is assumed that there are no errors in \(p_{k,all}\) and \(p_{k,lat}\), and that only \(p_{k,lon}\) is calculated incorrectly. According to Equation (2), if there is an error of \(p_{k,lon}\), the terrain database values at the position biased by the error are used for the MAD calculation. The errors of \(p_{k,lon}\) are accumulated in the MAD calculation and cause the TRN longitude error. In this case, correction values of \(-0.001^{\circ}\) in latitude direction and \(+0.001^{\circ}\) in longitude direction can be obtained. The longitudinal correction value is different from the longitudinal correction value shown in Figure 6. Therefore, \(p_{k}\) must be calculated accurately to obtain a correct TRN solution.