The analysis proceeds trying to find suitable solutions to cover every day the whole extension of the region of interest. There exist a slight difference between:
revisit time of every eventual target within the entire region.
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In the first case, if a satellite passes directly over the region of interest every day, it can “potentially” acquire every target by means of appropriate pointing, but not “all” target during one pass. This is different from the situation in which every target “is” effectively acquired every day. In the coverage of the entire region, are determinant the extension of the Tuscany and the swath width of instruments carried on-board. Supposing to divide the Tuscany in “strips”, each of them geometrically characterized by an horizontal dimension 𝑊𝑆𝑡𝑟𝑖𝑝 equal to the generic instrument swath width 𝑆𝑊, there is a number of strips 𝑁𝑆𝑡𝑟𝑖𝑝 (equation 5.4):
𝑁𝑆𝑡𝑟𝑖𝑝 ≅
𝐷𝑇𝑢𝑠𝑐𝑎𝑛𝑦
𝑊𝑆𝑡𝑟𝑖𝑝 (5.4)
where 𝐷𝑇𝑢𝑠𝑐𝑎𝑛𝑦 is assumed to be the maximum longitudinal dimension of
Tuscany. If each of these strips is specifically allocated to a single satellite of the constellation, this means that this satellite is devoted to observe “only” that strip and to acquire only that specific portion. So, the number of satellites 𝑁∗ needed to cover the entire region, at a first analysis, becomes
(equation 5.5):
𝑁∗ =𝐷𝑇𝑢𝑠𝑐𝑎𝑛𝑦
𝑊𝑆𝑡𝑟𝑖𝑝 =
𝐷𝑇𝑢𝑠𝑐𝑎𝑛𝑦
𝑆𝑤 = 𝑁𝑆𝑡𝑟𝑖𝑝𝑠 (5.5)
To allow each satellite to acquire exactly a specific strip with the highest image quality possible (quasi-nadir mode), satellites relative positions within the orbit plane have to be properly phased. This results in a constellation where satellites aren’t equally spaced within the orbit plane. Relative angular positions ∆𝜑𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 can be calculated according to equation 5.6: ∆𝜑𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 = [( 𝜏𝐸× 𝑆𝑤 2𝜋𝑅𝑒cos 𝜆) × 360 𝜏 ] (5.6)
where 𝜏 is the keplerian orbit period. Examples are provided by following figures, where 3 satellites, able to provide a swath width of about 60 km
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127 from an altitude of 554 km, are needed to cover the entire Tuscany region. Figure 6.9 shows the passage of the first satellite (Strip1, in red), devoted to acquire the most easterly strip. The swath width edges of the first satellite are indicated by the red lines.
Figure 6.9: The Strip1 satellite images the most easterly Tuscany side during one pass.
Figure 6.10 and Figure 6.11 show the successive passages of the second satellite (Strip2, in blue) and of the third satellite (Strip3, in white), devoted to acquire the central and the most westerly strip, respectively. The swath width edges of satellites are indicated by the blue and the white lines.
Figure 6.10: The Strip2 satellite images the central portion of Tuscany during one pass.
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Figure 6.11: The Strip3 satellite images the most westerly Tuscany side during one pass.
Figures show that Tuscany could not easily be geometrically approximated and also that orbit inclination is important to identify the real number of strips. Satellites in the example are placed on a near-polar orbit. As a reference, in the figures are reported 4 relevant points [66]:
N, the most northerly Tuscany point.
S, the most southerly Tuscany point.
W, the most westerly Tuscany point.
E, the most easterly Tuscany point:
The distance between points E and O is the longest distance within the entire region, about 210 km. Therefore, 𝐷𝑇𝑢𝑠𝑐𝑎𝑛𝑦 can be initially taken equal to 210 km at this first step of computation, when many different altitudes and instrument swath width capabilities have to be considered. In future computation steps, when the number of possibilities will be reduced, more precise graphical analysis will be conducted with AGI STK (Systemas Tool Kit) to really quantify 𝑁𝑆𝑡𝑟𝑖𝑝. This approximation allows to perform an easier initial estimate of number of strips without initially considering inclination. If, the eventual selected instrument is characterized by a swath width such that (equation 5.7):
Chapter 6 - Preliminary Mission Architecture Design
129 hence, it would be possible to divide the whole region in a single hypothetical strip and to acquire it with just one satellite. This concept is shown by the Figure 6.12 (where an inclination of 55° has been selected).
Figure 6.12: Acquisition of the entire Tuscany during a single pass from an inclination orbit.
An example of land division in strips is provided by RapidEye observation of the Germany (Figure 6.13). In this case, the repeat coverage of the entire country is performed every 5 days with 5 satellites. The less frequent repeat coverage is due to the larger dimensions of Germany respect to Tuscany and to the fact that the RapidEye constellation has to observe lands all over the world [27].
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Number of satellites for a daily repeat coverage of the whole Tuscany calculation
To cover the entire region in quasi-nadir mode, the constellation needs a number of satellites at least equal to the number of strips. If satellites are equipped with an instrument characterized by a swath width such that the Tuscany could be divided into 𝑛 strips, hence the constellation needs at least 𝑛 identical satellites adequately phased. If satellites are placed at an altitude such that the orbit nominal RC is higher than 1 day, 𝑛 satellites aren’t sufficient to observe the whole region every day. To have a daily full coverage of the whole region with an orbit nominal RC higher than 1 day, is required a number of satellites equal to the number of strips over the region during every day of the nominal cycle. The total number of satellite 𝑁𝑆
therefore becomes (equation 5.8):
𝑁𝑆 = 𝑅𝐶 × 𝑁𝑆𝑡𝑟𝑖𝑝𝑠 (5.8)
These results are valid in a first approximation for each altitude identified in the 1 to 5-days nominal RC range and generally for every value of RC, even if more accurate graphical analysis are needed to verify them. This highlights the need to select multispectral and hyperspectral instruments characterized by a proper combination of:
spatial resolution performance.
Swath width capabilities.
This combination would have to provide the required level of performance with a limited number of satellites. Equivalently to the daily revisit time number of satellites computation, this analysis is devoted to identify the best possible solutions to cover the entire region of interest every day. Depending on different users that could be involved, the coverage frequency could be relaxed from the 1-day performance to more extended temporal performance and targets of interest, if heterogeneously distributed within the region, could be properly selected . In this particular case, the total number of satellites could be easily determined by proportionally tailoring previous relations to the specific request.
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