# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 3 Systematic Errors
### Propagation Path (Atmosphere)
#### 3.2.1 Ionospheric Advance
As the solar zenith angle ascends with the increase in latitude, the ionization level decreases, so the variation of \(TEC_{s}\) in the mid-latitude regions is smaller and less variable (27; 41). When the latitude approaches the polar regions, the ionization process becomes active again due to the aurora effect, but the ionization level in the high-latitude regions is not as strong as in the equatorial regions [85; 88]. In addition to geographic location that influences the ionization process in the ionosphere, other factors such as time of day, season, solar cycle, and geomagnetic activity are also strongly related to the \(\Delta TEC_{s}\)[97; 98]. A study was conducted to inspect the variation of TEC based on the observation data at Taoyuan, Taiwan (24.954\({}^{\circ}\)N, 121.165\({}^{\circ}\)E), from 2002 to 2014 [83]. According to their results, the following summarizes the variations of TEC: (1) The diurnal variation shows the lowest TEC happened at 5 a.m. (5.51 \(TECu\)) and the highest value between 2 p.m. and 4 p.m. local time (48.92 \(TECu\)); (2) the seasonal variation shows the lowest TEC between June and July (18 \(TECu\)) and between December and January (20 \(TECu\)), while the highest TEC is between March and April (33 \(TECu\)); and (3) the solar activity cycles, which happen around every 11 years, show the lowest TEC between 2008 and 2009 (10 \(TECu\)) and the highest TEC in 2002 (70-80 \(TECu\)). Although the observations in Taiwan from 2002 to 2014 cannot be generalized to the rest of the globe, the study provides the readers with a complete overview of different variations in terms of magnitude and fluctuations. As the InSAR technique is often applied to image pairs with short time intervals, the diurnal variation is crucial. In order to limit the influence of the ionospheric effect, image acquisition in the morning hours is more desirable. Consider Sentinel-1: descending orbiting images are preferable to ascending orbiting images because descending orbiting images are acquired in the early mornings and ascending orbiting images in the late afternoons [88]. Another study arranged TEC characteristics of 47 pairs of Sentienl-1 ascending nodes from 2016 to 2017, in which the study area covered about 3\({}^{\circ}\) of latitudinal extent from 22\({}^{\circ}\)N to 25\({}^{\circ}\)N in Taiwan [99]. In this study, about 38.3% of image pairs obtain \(|\Delta TEC_{s}|\) of less than 2 TECu, 36.2% of image pairs obtain between 2 and 5 \(TECu\), 17% of image pairs obtain between 5 and 10 \(TECu\), and 8.5% of image pairs possess \(|\Delta TEC_{s}|\) of more than 10 \(TECu\). The numbers shown here are for reference purposes to showcase a rough reasonable range of \(\Delta TEC_{s}\). About 75% of image pairs have TEC variation within 5 \(TECu\); thus, 0 to 5 \(TECu\) can be considered as a reasonable variation range for normal situations. However, it is clear that \(\Delta TEC_{s}\) is highly variated, as stated in the previous paragraphs. So, one should be aware that this variation can drift to a certain degree depending on the geographical location, latitudinal extents, acquisition time, etc. Table 3 organizes the impact of the spatiotemporal variation of 1 \(TECu\) at L-band, S-band, C-band, and X-band frequencies [28; 83]. The third row demonstrates how many phase shifts would be generated by 1 \(TECu\) variation at different SAR frequencies. The  
calculation can essentially be achieved with Equation (7) with a small modification, changing \(4\pi\) to \(2\pi\) so the unit will be \(2\pi\), i.e., numbers of phase cycles. The variation of \(1\)\(TECu\) can result in \(2.11\times 2\pi\) phase cycles at L-band, \(1.07\times 2\pi\) at S-band, \(0.5\times 2\pi\) at C-band frequency, and \(0.28\times 2\pi\) at X-band frequency [28; 83; 85]. The fourth row displays the quantity of zenith ionospheric advances resulting from \(1\)\(TECu\) in millimeters. The values are calculated with Equation (6). The sign is negative because an increase in TEC leads to a phase advance [86].