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Correction of Atmospheric Refraction Geolocation Error for High Resolution Optical Satellite Pushbroom Images
Correction of Atmospheric Refraction Geolocation Error for High Resolution Optical Satellite Pushbroom Images Ming Yan, Chengyi Wang, Jianglin Ma, Zhiyong Wang, and Bingyang Yu
Abstract
When an optical remote sensing satellite is imaging the Earth in-orbit, the propagation direction of the Line of Sight (LOS) will be changed because of atmospheric refraction. This will result in a geolocation deviation on the collinear rigorous geometric model for direct georeferencing, pushbroom images. To estimate and correct the atmospheric refraction geolocation error, the LOS vector tracking algorithm is introduced and a weighted mean algorithm is used to simplify the ISO standard atmospheric model into a troposphere and stratosphere, i.e., two layers spherical atmosphere. The simulation result shows the atmospheric refraction will introduce about 2 m and 7.5 m geometric displacement when the spacecraft is off-pointed view at 30 and 45 degree angle, respectively. For a state-of-theart high resolution satellite, the atmospheric refraction displacement shall be corrected. The method has been practiced in the DMC3/TripleSat Constellation to remove the atmospheric refraction geolocation error without ground control points.
Introduction
Satellite remote sensing image geolocation is about determining the correspondence between the pixel’s Cartesian coordinate and geodetic coordinate. The geometric transformation between these two systems can be expressed by a rigorous geometric model, which is established based on a collinear equation with interior orientation parameters from the imaging payload and exterior orientation parameters of the satellite platform (Crespi et al., 2007; Fan et al., 2011; Habib et al., 2007; Jeong and Bethel, 2010 and 2014; Jiang et al., 2013 and 2014; Leprince et al., 2007; Lussy and Greslou, 2012; Mahapatra et al., 2004; Müller et al., 2012; Pan et al., 2013; Poli and Toutin, 2012; Radhadevi et al., 2011; Tang et al., 2012; Toutin, 2004). The accuracy of a rigorous geometric model depends on the interior and exterior orientation parameters, such as the focal length, detector size, lens, detector line distortion of the imaging payload, and the ephemeris and attitude systematic error of satellite platform. Also, the atmospheric refraction and light aberration will have an impact on the rigorous geometric model estimation (Lussy et al., 2012; Oh and Lee, 2011). The distortions of the interior and exterior orientation parameters are generally corrected during satellite in-orbit geometric calibration and validation operation (Gruen and Kocaman, 2008; Leprince et al., 2008; Wang et al., 2014; Yastikli and Jacobsen, 2005). Greslou et al. (2008) have analyzed the Line of Sight (LOS) in the Earth Centered Earth Fixed (ECEF) coordinate system for the apparent deflection caused Ming Yan, Chengyi Wang, and Jianglin Ma are with the Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, 20 Datun Road, Chaoyang District, Beijing, 100101, P. R. China ([email protected]). Zhiyong Wang and Bingyang Yu are with the Twenty First Century Aerospace Technology Co., Ltd., 26 Jiancaicheng East Road, Haidian District, Beijing, 100096, P. R. China.
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by the relative velocity of the satellite platform and target to correct light aberration. The atmospheric refraction is rarely considered and corrected because they are specific to each acquisition location and information about the atmosphere. In this paper, we focus on atmospheric refraction and analyze how it can be better processed to reduce the geolocation error. The atmospheric refraction changes the LOS propagation direction of satellite imaging in-orbit, which results in the points of detector, projection center of imaging payload, and imaged object no longer following the rigorous collinear model. Several pioneering researches have confirmed the effect of atmospheric refraction on rigorous geometric model estimation. For example, Gyer (1996), Wang (2007), and Wei (2006) have found the atmospheric refraction will have an impact on aerial photogrammetry. While the aerial refraction correction formulas are valid up to the normal aircraft flying height, they are delivering the wrong results for a spaceborne image (Jacobsen, 2004). For atmosphere refraction effects on a space image, Jacobsen (2004) and Dowman (2012) gave the refraction correction formula for the nadir space image based on the 1959 ARDC standard atmospheric. Noerdlinger (1999) targeted on MODIS satellite data and researched the atmospheric refraction by developing an analytical method to calculate the angle of the refraction assuming a single layer of spherically symmetrical atmosphere. Saastamoinen (1972) expressed a simple atmospheric refraction angle calculation formula for radio ranging of low Earth orbit satellites. Oh and Lee (2011) simply extended the Saastamoinen model to express the constant related to the object’s terrain and satellite altitude. However, it rarely has the documentation to study the procedure and method regarding the atmospheric refraction correction for a high resolution optical satellite. This study, based on a ray tracking physical model, rigorously describes the LOS propagation direction of the satellite imaging sensor to the Earth surface object through the atmosphere. Meanwhile, the International Organization for Standardization (ISO) (1975) atmospheric model and the Owens (1967) optical refraction index calculation algorithm are used to calculate the atmospheric refraction index at any position. In the following sections, first we will describe the LOS ray tracking geometric algorithm, and then analyze the geolocation deviation introduced by atmospheric refraction. After that, we will reveal how we can use the deviation to correct the atmospheric refraction geolocation error in the ECEF coordinate system. Compared to previous research, more reasonable atmospheric refraction displacement estimation can be achieved. Because we not only take the latitude and altitude into account when calculating atmospheric refraction Photogrammetric Engineering & Remote Sensing Vol. 82, No. 6, June 2016, pp. 427–435. 0099-1112/16/427–435 © 2016 American Society for Photogrammetry and Remote Sensing doi: 10.14358/PERS.82.6.427
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index, we also track the LOS refracted direction in different atmospheric layers. Therefore, the proposed method is more conformal to the LOS propagation characteristics in the stratified atmosphere. By compensating the atmospheric refraction error in the rigorous collinear geometric model, the geometric positioning accuracy of satellite remote sensing image without ground control points can be improved. Since DMC3/TripleSat constellation was launched on 10 July 2015, this study has been continually experimenting with the three 1 m resolution optical satellites. The atmospheric refraction correction algorithm has been integrated into the DMC3/TripleSat rigorous geometric model, which is used to produce the Level-1A basic image products.
Atmospheric Refraction Geolocation Error
The stratified Earth’s atmosphere comprises mainly gas molecules, water vapor, and aerosols. When the Sun’s ray is reflected by ground objects through the atmosphere to the imaging detectors in-orbit, the propagation direction of reflection rays shall be deviated due to atmospheric refraction. Conversely, from the view of detectors of imaging payload, the atmospheric refraction results in a deviation of LOS from its original propagation direction and lead to an atmospheric refraction geolocation error. In the following, we will model the geolocation error with rigorous mathematic formulas. In Figure 1, suppose atmospheric refraction does not exist, the line dSP represents the LOS of one detector with view angle α, where Point d refers to the detector (the distance from the imaging detector to the linear array principal Point o is also expressed as d), Point S is the perspective center of optical imaging payload, and Point P is the intersection of LOS and the Earth’s ellipsoid. This line dSP also contains Point P0, which is the intersection of LOS and the top atmosphere. In order to describe the ray tracking geometric algorithm of LOS vector, it is assumed the Earth’s atmosphere is a single layer homogeneous spherical atmosphere only consisting of gas molecules and water vapor: h is the atmosphere thickness and n is the atmospheric refraction index (n >1). Point P1 is the intersection of LOS deviated by atmospheric refraction and the Earth ellipsoid. The line of d1SP1 (dashed line) is the atmospheric refraction bias compensated LOS. Also, in Figure 1, we define f as the focal length, H as satellite platform height, and R as the Earth’s mean radius. The ground surface distance between point P and P1 is the atmospheric refraction geolocation error that would exist in rigorous geometric model if atmospheric refraction is not considered. Next, we will calculate the atmospheric refraction geolocation error with a rigorous mathematic formula using symbols predefined in Figure 1. According to the sine law we have the following formulas:
R R+H = sin α sin(180 − β )
R+H ⇒ β = arcsin sin α R
(1) (2)
where α is the off-view angle, and β is the incident angle of to the earth ellipsoid without considering the atmospheric refraction error. In the same way, the incident angle of LOS to the top atmosphere can be calculated: LOS
R+H sin α i = arcsin R+h
(3)
where h is the atmosphere thickness. According to the Snell refraction law, the refraction angle r is as follows:
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Figure 1. LOS propagated in single layer spherical atmosphere. n sin r = sin i (4) sin i ⇒ r = arcsin (5) n where n is the refraction index, i is the incident angle, and r is the refraction angle. We define θ as ∠SOP, θ0 as ∠SOP0 as ∠P0OP1 and ∆θ as ∠P1OP. Since OP0 = R + h, OP = R, and the refraction angle r is known in ∆P0OP1, and θ1 can be calculated:
(R + h)sin i θ1 = arcsin −r R
(6)
Known as the off-view angle α, the incident angle i and β in ∆SOP0, θ0 can be calculated:
θ =β −α
(7)
(8) θ0 = i − α Combining the above equations, angle ∆θ (radian) and arc length are calculated.
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Δ θ = θ − θ0 − θ1 (9) P 1P = R •∆θ (10) If the characteristics of stratified Earth’s atmosphere and the atmospheric refraction index are known, the refraction deviation angle of each layer can be calculated by the iterative calculation of Equations 3, 5, and 6. Thus, the atmospheric refraction deviation angle ∆θ is calculated: Δθ = θ − θ 0 − θ1 − − θ j
(11)
where j is the number of layered atmosphere. The imager of a high resolution optical satellite is often composed of blue (0.45 - 0.52 μm), green (0.53 - 0.60 μm), red (0.63 - 0.69 μm), near-infrared (0.76 - 0.90 μm) multispectral bands and 0.45 - 0.8 μm panchromatic band (Jacobsen, 2011; Yan et al., 2013). When the monochromatic light passes through the Earth’s atmosphere, the refraction index depends on its wavelength. The shorter the wavelength is, the larger the atmospheric refraction index is. In the astronomical observation, Lipcanu (2005) and Stone (1996) adopted the Owens (1967) atmospheric refraction calculation algorithm to analyze the impact of atmospheric refraction on star observation. They found the atmospheric refraction result in a 57.5 micro-radian error of a viewed star at 45 degrees zenith angle under the conditions of 15 Celsius degree environment temperature and 760 mm Hg atmospheric pressure. When the atmospheric temperature t (Celsius), the atmospheric pressure Pa (Pascal) and the water vapor pressure Pw (Pascal) are known, the atmospheric refraction index is calculated according to the equations from 12 to 17:
(
)
+ 6487.31 + 58.058σ 2 − 0.7115 σ 4 + 0.08851σ 6 DW
where
(
)
(14)
PS = Pa − Pw (15) T = 273.15 + t (16) σ = 1λ (17) In the above equations, PS (Pascal) is dry air pressure after water vapor pressure removed, T is the absolute temperature and σ is the wave number of monochromatic light with wavelength λ micron.
Atmospheric Temperature (t)
Atmospheric Refraction Index of Monochromatic Light
683939.7 4547.3 + DS (n − 1) × 108 = 2371.34 + 130 − σ 2 38.9 − σ 2
2.23366 710.792 7.75141 × 104 PW DW = 0.01 1 + 0.01PW 1 + 3.7 × 10−6 PW −2.37321 × 10−3 + − + T T2 T3 T
(12)
9.325 × 10−4 0.25844 PS DS = 0.01 1 + 0.01PS 57.9 × 10−8 − + (13) T T 2 T
The international standard atmosphere model (ISO 2533: 1975) is used to calculate the atmospheric temperature variation with altitude. The ISO 1975 atmosphere model defines that the mean temperature at sea level is 15 Celsius degree, the atmospheric pressure is 760 mm Hg and the Earth’s atmosphere is divided into eight layers. The air temperature of each layer linearly changes with altitude as the following formula: 15 − 6.5h 0 km ≤ h ≤ 11.019 km − 56 . 5 11 . 019 km < h ≤ 20.063 km −56.5 + (h − 20.063) 20.063 km < h ≤ 32.162 km (18) t = −44.5 + 2.8 ( h − 32.162) 32.162 km < h ≤ 47.35 km −2.5 47.35 km < h ≤ 51.413 km −2.5 − 2.8(h − 51.413) 51.413 km < h ≤ 71.802 km 71.802 km < h ≤ 86 km −58.25 − 2(h − 71.802) where t is the atmospheric temperature in Celsius degree. However, the mean temperature at sea level varies with geographic latitude, which makes the ISO 1975 standard atmosphere model to estimate temperature inaccurate in troposphere. In the study of global precipitation with longitude and latitude, Roper (2011) studied the earth surface mean air temperature by averaging over all longitudes from 1948 to 2009, which is shown in Figure 2. Some interesting conclusions can be drawn from Figure 2. Let angle φ be the latitude degree, the cos(φ) can be regarded as an independent variable linearly related to the Earth surface mean air temperature in the northern hemisphere when the latitude range is between 0 and 82 degrees. The same conclusion holds true for the southern hemisphere as the following formula shows:
78.08 cos φ − 48.2 −82° ≤ φ ≤ 0 tL = 52.07 cos φ − 26.4 0 < φ ≤ 82°
(19)
where tL is the Earth’s surface mean air temperature. Here we are only interested in air temperature in the troposphere of Equation 18 which depends on the Earth surface air temperature tL. Then, the troposphere air temperature at different altitude becomes:
t = tL +
(−56.5 − t ) h L
11.019
0 km ≤ h ≤ 11.019 km .
(20)
Atmospheric Pressure (Pa)
With the increase of altitude, atmospheric pressure smoothly decreases following an exponential function (Portland State Aerospace Society, 2004)): Figure 2. Relationship between mean Earth surface air temperature with latitude.
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Figure 4. Water vapor pressure changes with temperature.
Figure 3. Atmospheric pressure changes with altitude. gM
Lh Rmol L Pa = P0 1 − T0
Pw ≈ 3386.39 (0.00738t + 0.8072) − 0.000019 1.8t + 48 + 0.001316 (23) 8
(21)
g is the earth surface gravity acceleration (g = 9.80665 m/s2);
where Pw is water vapor pressure (Pascal), and t is Celsius degree. Figure 4 illustrates how water vapor pressure changes from the Earth’s surface air temperature 30 Celsius degrees to the top mesosphere temperature −87 Celsius degrees.
M is the molar mass of dry air (M = 0.0289644 kg/mol);
Atmospheric Refraction Index (n)
where Pa is the atmospheric pressure (Pascal); P0 is the atmospheric pressure at sea level (P0 = 101325 Pa);
Rmol is the gas constant (Rmol = 8.31447 J/(mol·K)); T0 is the standard temperature at sea level (T0 = 288.15 K); L is the temperature lapse rate (such as: L = 0.0065 K/m in troposphere). The temperature of tropopause and stratopause is constant; atmospheric pressure at any altitude in these two layers is estimated according to the distance from the point to the bottom of each layer as the following equation:
Suppose the atmospheric temperature is t, the air pressure is Pa, the water vapor pressure is Pw and the central wavelength of spectral band is known, the atmospheric refraction index in the stratified atmosphere can be calculated by using Equations 12 to 17. For the object points located on the Equator, latitude 30° and 60° in the northern hemisphere, and 30° in the southern hemisphere, Figure 5 shows the atmospheric
g Pa = P1exp − (h − h1 ) (22) RkgT0 where Rkg is the gas constant (Rkg = 287.05287 J/(kg·k)), h1 is the distance to the bottom of tropopause or stratopause and P1 (Pascal) is the atmospheric pressure at the altitude of h1. Figure 3 shows how the atmospheric pressure Pa changes with the altitude.
Water Vapor Pressure Pw
Water vapor pressure is only related to the atmospheric temperature, and the change with temperature can be represented by the following equation (Bosen, 1960): Figure 5. Atmospheric refraction indexes change with altitude in the troposphere.
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refraction index of 0.5 μm blue light through the troposphere. Table 1 lists the atmospheric refraction index at intervals of altitude from Earth’s surface to the stratopause. From Table 1 and Figure 5, we can conclude that the atmospheric refraction index is 1 along LOS from payload in-orbit to the top of stratosphere at 47.35 km altitude. In the troposphere, tropopause and stratosphere, the atmospheric refraction index increases with the decreasing altitude. Under the same atmospheric pressure and water vapor pressure, the atmospheric refraction index is negatively correlated with the atmospheric temperature. The result is that the higher the latitude is, the larger the atmospheric refraction index is. The atmospheric refraction index is almost unchanged for the same latitude regardless it is in the southern hemisphere or in the northern hemisphere. For example, the mean Earth surface air temperature at N60°, N30° and Equator is 0°, 20°, and 25° Celsius degrees, respectively, the atmospheric refraction indexes of N60° are larger than N30°, and N30° are larger than the Equator in troposphere. The atmospheric refraction indexes of N30° and S30° are almost equal. The atmospheric refraction index at Equator has the minimum value in troposphere. Table 2 lists the distribution characteristics of atmospheric refraction with regard to the atmosphere layer and altitude. Therefore, the ISO eight layer atmosphere model can be simplified into two layers: 1. Troposphere layer: geometric altitude is between 0 and 11,019 m; 2. Stratosphere layer (include tropopause): geometric altitude is from 11,019 m to 47,350 m. This simplification is based on the fact that the refraction index is 1 from the stratopause to mesopause. We will use the simplified two-layer atmosphere model to calculate the geolocation error as shown in Figure 6. The atmospheric refraction index in the troposphere and stratosphere are set to be n1 and n2 respectively, and i is the incident angle of LOS at the top of the stratosphere. D is the displacement on the ellipsoid surface because of LOS refracted in the troposphere and stratosphere. A weighted average algorithm is employed to calculate the atmospheric refraction index, which is calculated at each 1,000 m interval in the troposphere and 2,000 m interval in the stratosphere. The weighted average algorithm in the troposphere is described as follows:
Table 1. Atmospheric Refraction Indexes Change with Altitude at the Equator, N60°, N30° and S30° Altitude (m)
N60 Degrees
N30 Degrees
Equator
S30 Degrees
1000
1.000268639
1.000250154
1.000242816
1.000249707
3000
1.000221848
1.000209777
1.000204972
1.000209484
5000
1.000184809
1.000178003
1.000175255
1.000177835
7000
1.000155253
1.000152781
1.000151744
1.000152718
9000
1.000131491
1.000132614
1.000133037
1.00013264
11000
1.000112255
1.000116392
1.000118097
1.000116495
15000
1.000044223
1.000044223
1.000044223
1.000044223
20000
1.000020098
1.000020098
1.000020098
1.000020098
30000
1.000003847
1.000003847
1.000003847
1.000003847
40000
1.000000753
1.000000753
1.000000753
1.000000753
50000
1.000000003
1.000000003
1.000000003
1.000000003
60000
1.000000034
1.000000034
1.000000034
1.000000034
70000
1.000000014
1.000000014
1.000000014
1.000000014
80000
1.000000002
1.000000002
1.000000002
1.000000002
Table 2. Characteristics of Atmospheric Refraction Index
Layer Name
Base Altitude (m)
Top Altitude (m)
1
Troposphere
0
11019
2 3 4 5 6 7 8
Tropopause Stratosphere Stratosphere Stratopause Mesosphere Mesosphere Mesopause
11019 20063 32162 47350 51413 71802 86000
20063 32162 47350 51413 71802 86000 —
Layer Number
Atmospheric Refraction Index Varying with latitude and altitude. Varying only with altitude. The refraction index is 1.
Step 1: Calculate the difference (delta_atm_ref(i)) of two atmospheric refraction indexes at every 1,000 m interval. Step 2: Calculate the difference (total_delta_atm_ref) of the atmospheric refraction index at the surface and top troposphere. Step 3: Calculate the weight of each interval:
weight(i) = delta_atm_ref(i)/ total_delta_atm_ref
Step 4: Calculate mean atmospheric refraction index (mean_ atm_ref) of troposphere. mean_atm_ref = sum[weight(i)·atm_ref(i)] where atm_ref(i) is calculated by Equations 12 to 17 at any altitude increased 1,000 m interval in troposphere. For the 0.5 μm blue light, the atmospheric refraction index is 1.000014132 (n2) in the stratosphere at N60°, N30°, S30° and the Equator. The atmospheric refraction index in the troposphere is 1.000199059 (n1) at N60° and 1.000187379 (n1) at the Equator. As for the atmospheric refraction index of different wavelength monochromatic light, Figure 7 shows
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Figure 6. Two layers atmospheric refraction model
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the atmospheric refraction index of blue (0.5 μm), green (0.55 μm), red (0.65 μm), and near-infrared (0.83 μm) monochromatic light through the troposphere at N30°. The shorter the wavelength, the larger of atmospheric refraction index value. In the following section, the atmospheric refraction geolocation error of the four monochromatic lights will be analyzed.
Atmospheric Refraction Geolocation Error
When the satellite view angle α is known, we can calculate the atmospheric refraction geolocation error on the ellipsoid surface with zero altitude at any latitude. Suppose the mean Earth radius R is 6,371 km, the altitude of satellite orbit H is 650 km, Table 3 lists the atmospheric refraction geolocation error at N60°, N30° and the Equator of blue (0.5 μm), green (0.55 μm), red (0.65 μm) and near-infrared (0.83 μm) monochromatic light in different view angle of satellite in-orbit. In Table Figure 7. Atmospheric refraction index changes with different wavelength in tropo3, α is the satellite off-pointing view angle, sphere at N30 degrees. Blu is blue band, Grn is green band, Red is red band, and NIR is near-infrared band. Table 3. Atmospheric Refraction Geolocation Error (Unit: m) From Table 3, we can clearly observe N60 Degree N30 Degree Equator α that the atmospheric refraction geolocation Blu Grn Red NIR Blu Grn Red NIR Blu Grn Red NIR error increases nonlinearly with the increas5° 0.24 0.24 0.24 0.24 0.23 0.23 0.23 0.23 0.23 0.23 0.22 0.22 ing view angle. This conclusion is not only 10° 0.49 0.49 0.48 0.48 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.46 confirmed by our proposed method, but also 15° 0.78 0.78 0.77 0.77 0.76 0.76 0.75 0.75 0.75 0.75 0.75 0.74 verified by Noerdlinger (1999). Also, three 20° 1.14 1.14 1.13 1.12 1.11 1.11 1.10 1.10 1.10 1.10 1.09 1.09 other conclusions as follows: 25° 1.62 1.62 1.60 1.59 1.58 1.57 1.56 1.55 1.56 1.56 1.55 1.54 1. Atmospheric refraction geolocation 30° 2.28 2.27 2.25 2.24 2.22 2.21 2.20 2.19 2.20 2.19 2.18 2.17 error is increased from the Equator 35° 3.26 3.25 3.23 3.21 3.18 3.17 3.15 3.13 3.15 3.14 3.12 3.10 to South and North Poles. 40° 4.82 4.80 4.78 4.75 4.71 4.69 4.66 4.64 4.66 4.65 4.62 4.60 2. Atmospheric refraction geolocation 45° 7.56 7.53 7.49 7.45 7.38 7.35 7.31 7.27 7.31 7.28 7.24 7.20 error is increased with the shorter 50° 13.03 12.98 12.91 12.84 12.73 12.68 12.60 12.54 12.61 12.55 12.48 12.42 of monochromatic light wavelength in the same latitude. While, this value of the paper’s proposed method than Noerdlinger, Jaedifference between blue, green, red, and near-infrared hong and Jacobsen’s methods. bands are negligible. We will only use the green (0.55 μm) light to calculate the atmospheric refraction index for high resolution optical satellite images direct georeExperiment on DMC3/TripleSat Constellation ferencing. The DMC3/TripleSat Constellation was launched on 10 July 3. Despite the fact that the atmospheric refraction geoloca2015 by a PSLV-XL launch vehicle from the Satish Dhawan tion error is not large, there is a need to correct this error Space Centre, Sriharikota launch site in India. The constellafor the state-of-the-art high resolution satellite, such as tion was successfully delivered into a sun-synchronous orbit WorldView-3 (GSD: 0.31 m), GeoEye-2 (GSD: 0.34 m), Geowith the local time of the ascending node of 10:30 at 651 km Eye-1 (GSD: 0.46 m) and WorldView-1/2 (GSD: 0.46 m). altitude. The three satellites were phased 120 degrees apart around the same orbit using their on-board propulsion sysCompared the atmospheric refraction geolocation error of tem. The DMC3/TripleSat Constellation satellites use the 450 the proposed method with the results of Noerdlinger (1999), kg SSTL-300S1 series platform, which provides 45 degrees Jaehong’s (Oh and Lee, 2011; Saastamoinen, 1972) and Jafast slew off-pointing and is capable of acquiring multiple cobsen’s (Jacobsen, 2004; Dowman et al., 2012) methods, the targets in one pass using multiple viewing modes. The high atmospheric refraction displacement is almost in the same resolution imager on board equipped with four Time-Delayedwhen the satellite was imaging inside the core of 20 degrees Integration (TDI) CCD arrays provides 1 m ground sampling around nadir. When the off-pointing angle is over 20 degrees, distance (GSD) in panchromatic and 4 m GSD in blue, green, the results of these methods are different. The authors believe red and near-infrared multispectral mode with a swath width the Noerdlinger’s single layer spherically symmetrical atmoof 24 km. The wide swath of the imager combined with agile sphere assumption will result in the atmospheric refraction off-pointing capability enable the DMC3/TripleSat Constellaindex underestimated in the troposphere. And the Jaehong tion to target anywhere on earth at least once a day. Table 4 and Jacobsen’s methods took the satellite altitude and the lists the capability of DMC3/ TripleSat Constellation. object’s terrain elevation into account to calculate the atmoThe DMC3/TripleSat Constellation linear pushbroom imspheric refraction constant. These three methods were all not ages are direct georeferenced by using the collinear rigorous considering the LOS path increased through the atmosphere geometric model. The rigorous geometric model fixes the relawhen optical satellite operated in off-pointing imaging mode. tionship between LOS vector in the camera coordinate system Therefore, the larger of satellite off-pointing angle, the larger with the ECEF coordinate system. The rigorous geolocation for 432
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Table 4. DMC3/TripleSat Constellation Capability Sensor
Panchromatic
Multispectral
Band (μm)
0.45 to 0.65
Blue: 0.44 – 0.51 Green: 0.51 – 0.59 Red: 0.60 – 0.67 Near infrared: 0.76 – 0.91
GSD (m)
1
4
Swath (km)
24
24
Focal Length (m)
6.6667
6.6667
Detector Dimension (μm)
10*10
40*10
Detector Number
6408*4
1602*4
Digitization (bit)
10
10
Max Off-pointing (degree)
45
Imaging Mode
Stripe, Scene, Along track stereo, Across track stereo and Area
Figure 8. Definition of interior orientation parameters with atmospheric refraction geolocation error compensated. each pixel is obtained by calculating the intersection point coordinate of the LOS with the earth ellipsoid in ECEF coordinate system, which can be described as follows without considering the influence of atmospheric refraction. X − X0 = µM Orbit 2ECEF M Body 2Orbit M Camera2Body Y − Y0 Z − Z0 ECEF
tan(ψ y ) (24) − tan(ψ x ) 1 Camera
where [tan(ψy) – tan(ψx) 1]TCamera represents the detectors LOS vector in the camera coordinate system. MCamera2Body represents the bias matrix of the camera coordinate system to the attitude control coordinate system. MBody2Orbit represents the rotation matrix from the attitude control coordinate system to satellite platform coordinate system. MOrbit2ECEF represents the rotation matrix from orbit coordinate system to ECEF coordinate system. μ is a scalar. [X0 Y0 Z0]T ECEF represents the satellite platform position interpolated by ephemeris at imaging instant. [X0 Y0 Z0]T ECEF represents the ECEF coordinate of the intersection point of LOS with the Earth ellipsoid. During the in-orbit geometric calibration campaign, the precision instrument alignment angles in MCamera2Body matrix and
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vector [tan(ψy) – tan(ψx) 1]T in camera coordinate system is periodically calibrated by referencing the global geometric calibration sites. From the analysis of atmospheric refraction effect on geolocation, we draw the conclusion that the atmospheric refraction will change the direction of LOS and result in the geolocation error on earth spheroid. In the procedure of DMC3/TripleSat Constellation direct georeferencing, the detector’s LOS vector refractive deviation is subtracted from the calibrated LOS vector in the camera coordinate system to correct the atmospheric refraction geolocation error. When the satellite operated in the roll and pitch imaging mode, the collinear rigorous geometric model of Equation 24 is rewritten as below: LOS
X − X0 tan( ψ y − ψ pitch _ atmos ) = (25) Y − Y µ M 0 ECEF 2Camera − tan( ψ x − ψ roll _ atmos ) Z − Z0 1 Camera ECEF where MECEF2Camera = MOrbit2ECEF MBody2Orbit MCamera2Body. (ψx,ψy) is the interior orientation element in the camera coordinate system, ψroll_atmos is the atmospheric refraction deviation angle represented in the camera coordinate system when the platform rotation roll angle around the X axis, ψpitch_atmos is the atmospheric refraction deviation angle represented in the camera coordinate system when platform rotation pitch angle occurs around the Y axis. The definition of the interior orientation parameters is shown in Figure 8. OXYZ is the camera coordinate system and oxy is the image coordinate system. u is the LOS vector in Equation 24. u′ is the LOS vector in Equation 25 by subtracting the atmospheric refraction deviation angle in along and across track direction. In the DMC3/TripleSat Constellation geometric model, the original points of the camera coordinate system, body coordinate system, and orbit coordinate system are in the same. The values of roll and pitch are interpolated from the exterior orientation attitude parameters. In the direct georeferencing of DMC3/TripleSat Constellation images, the atmospheric refraction geolocation error was transformed into the LOS deviation in the camera coordinate system. And the atmospheric refraction geolocation error correction procedure is divided into 10 steps: Step 1: Use Equation 24 to calculate the ECEF coordinate (X,Y,Z) of one pixel (x,y) by referencing WGS84 ellipsoid. Step 2: Input the NASA Shuttle Radar Topographic Mission (SRTM) global 90 m digital elevation data. Step 3: Transform the object’s ECEF coordinate (X,Y, Z) into latitude and longitude. Step 4: Interpolate the roll and pitch value from exterior orientation attitude parameters at the imaging instant. The roll and pitch values are considered as the view angle of the detector across and along track direction. Step 5: Use the simplified two layer atmosphere model and terrain elevation to calculate the refraction index in stratosphere and troposphere. In the troposphere, there is a need to calculate the refraction index between the terrain elevation and the top of the troposphere. Step 6: Use the Equations 3 to 9 to calculate the Δθroll and Δθpitch across and along track direction on Earth’s surface. Step 7: Interpolate the spacecraft position (X0, Y0, Z0) by using ephemeris parameters at the imaging instant.
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Step 8: Calculate the distance from the imaged object (X,Y, Z) and spacecraft position (X0, Y0, Z0). Step 9: Calculate the value of ψroll_atmos and ψpitch_atmos. An example of calculation (radian) as following: ψ roll _ atmos =
(R + hdem ) •∆ θ roll
( X − X ) + (Y − Y ) + (Z − Z ) 2
0
2
0
(26)
2
0
where, R is the mean Earth radius; hdem is the object’s terrain elevation interpolated from SRTM 90 m digital elevation data. Step 10: Use Equation 25 to calculate the atmospheric refraction geolocation error corrected ECEF coordinate. In the DMC3/TripleSat Constellation in-orbit commissioning and early operation phase, the studied method was used to direct georeferencing the DMC3/TripleSat 1 m resolution images. Table 5 lists four sample DMC3/TripleSat Constellation images with atmospheric refraction geolocation error correction. Since the spacecraft off-pointing limit in the early phase, the four sample images were all imaged in roll off-pointing mode. In the future, we will investigate in the atmospheric refraction geolocation error in both roll and pitch off-pointing mode.
Table 5. Experiment Results of Four samples DMC3/TripleSat Constellation Images with the Atmospheric Refraction Geolocation Error Correction. The value of X_corr is the Atmospheric Refraction Geolocation Error Corresponding to the Pitch Off-pointing View, and the Value of Y_ corr is the Atmospheric Refraction Geolocation Error Corresponding to the Roll Off-pointing View. Data Date: 23 Sep, 2015 Image ID: D2000093VI
As the propagation direction of LOS from satellite payload in-orbit is deviated because of the atmospheric refraction, the points of the detector, projection center and ground object are no longer in a line. This is one reason why there are geolocation errors in the collinear rigorous geometric model estimation. The paper describes how to use the LOS tracking geometry algorithm, ISO standard atmospheric model, and Owens atmospheric refraction index calculation method to estimate the atmospheric refraction geolocation error and how to correct this error in the collinear rigorous geometric model. The proposed method is especially suitable for the high resolution agile optical satellite as images taken from a large off-pointing view angle is more susceptible to the atmospheric refraction. We also have applied the proposed method in direct georeferencing DMC3/TripleSat Constellation 1 m resolution images to improve the geolocation accuracy without ground points. However, we have to admit that atmospheric refraction is one element that has made collinear geometric model nonrigorous. Other elements, such as the light aberration and the transmission delay of LOS, can also introduce geometric errors in the collinear geometric model. Only when all the causes that lead to inaccurate estimation of the interior orientation and exterior orientation parameters are calibrated, can we obtain a real accurate collinear rigorous geometric model.
Acknowledgments
This work was partly supported by the National High Technology Research and Development Program of China (No. 2013AA12A303) and Major Project of High Resolution Earth Observation System (Civilian Part) (No. 65-Y40B01-900113/15). The authors also thank the anonymous reviewers for their construction comments and suggestions.
References
Bosen, J., 1960. A formula for approximation of the saturation vapor pressure over water, Monthly Weather Review, pp. 275–276.
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Geolocation Error (m): X_corr: 0 Y_corr: 0.31
Location: Guildford, UK Off-pointing (degree): Roll: -5.908 Pitch: 0 Date: 2 Oct, 2015 Image ID: D20000A3VI
X_corr: 0 Y_corr: 0.82
Location: Beijing, China Off-pointing (degree): Roll: -16.967 Pitch: 0 Date: 13 Oct, 2015
Conclusions
Browse Image
Image ID: D20000BEVI
X_corr: 0 Y_corr: -1.54
Location: Aden, Yemen Off-pointing (degree): Roll: 23.996 Pitch: 0 Date: 16 Oct, 2015 Image ID: D10000B1VI
X_corr: 0 Y_corr: 0
Location: Zhaoyuan, Haerbin, China Off-pointing (degree): Roll: 0.001 Pitch: 0
Crespi, M., F. Fratarcangeli, F. Giannone, and F. Pieralice, 2007. SISAR: A rigorous orientation model for synchronous and asynchronous pushbroom sensors imagery, The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XXXVI-1/W51, unpaginated CD-ROM. Dowman, I., K. Jacobsen, G. Konecny, and R. Sandau, 2012. High Resolution Optical Satellite Imagery, Whittles Publishing, 129 p. Fan, D.Z., C.B. Liu, and T. Wang, 2011. Building and Validation of rigorous geometric model of ALOS PRISM imagery, Acta Geodaetica et Cartographica Sinica, 40(5):569–574. Greslou, D., F. Lussy, and J. Montel, 2008. Light Aberration Effect in HR Geometric Model, The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XXXVII-B1:859–864. Gruen, A., and S. Kocaman, 2008. Optical Sensors High Resolution: Geometry Validation Methodology, Technical report submitted to ESA/ESRIN, 34 p. Gyer, M., 1996. Methods for computing photogrammetric refraction corrections for vertical and oblique photographs, Photogrammetric Engineering & Remote Sensing, 62(3):301–310. Habib, A., S.W. Shin, K. Kim, C. Kim, K.I. Bang, E.M. Kim, and D.C. Lee, 2007. Comprehensive analysis of sensor modeling alternatives for high resolution imaging satellites, Photogrammetric Engineering & Remote Sensing, 73(11):1241– 1251.
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International Organization for Standardization, 1975. ISO 2533: 1975 Standard Atmosphere, 108 p. Jacobsen, K., 2004. Issues and Methods for In-Flight and On-Orbit Calibration, Post-Launch Calibration of Satellite Sensors, ISPRS Book Series, 2:83–92. Jacobsen, K., 2011. Characteristics of very high resolution optical satellites for topographic mapping, Proceeding of the ISPRS Hannover Workshop - Commission I, WG I/4, unpaginated CDROM. Jeong, I.S., and J. Bethel, 2010. A study of trajectory models for satellite image triangulation, Photogrammetric Engineering & Remote Sensing, 76(3):265–276. Jeong, I.S., and J. Bethel, 2014. An automatic parameter selection procedure for pushbroom sensor models on imaging satellites, Photogrammetric Engineering & Remote Sensing, 80(2):171–178. Jiang, Y.H., G. Zhang, and X.M. Tang, 2013. High accuracy geometric calibration of ZY-3 three-line image, Acta Geodaetica et Cartographica Sinica, 42(4):523–529. Jiang, Y.H., G. Zhang, X.M. Tang, D.R. Li, W.C. Huang, and H.B. Pan, 2014. Geometric calibration and accuracy assessment of ZiYuan-3 multispectral images, IEEE Transactions on Geoscience and Remote Sensing, 52(7):4161–4172. Leprince, S., S. Barbot, F. Ayoub, and J.P. Avouac, 2007. Automatic and precise orthorectification, coregistration, and subpixel correlation of satellite images, application to ground deformation measurements, IEEE Transactions on Geoscience and Remote Sensing, 45(6):1529–1558. Leprince, S., P. Musé, and J.P. Avouac, 2008. In-flight CCD distortion calibration for pushbroom satellites based on subpixel correlation, IEEE Transactions on Geoscience and Remote Sensing, 46(9):2675–2683. Lipcanu, M., 2005. A direct method for the calculation of astronomical refraction, The Proceedings of the Romanian Academy, Series A, 6(2):1–7. Lussy, F., and D. Greslou, 2012. Pleiades HR in flight geometric calibration: Location and mapping of the focal plane, International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XXXIX-B1:519–523. Mahapatra, A., R. Ramchandran, and R. Krishnan, 2004. Modeling the Uncertainty in orientation of IRS-1C/1D with a rigorous photogrammetric model, Photogrammetric Engineering & Remote Sensing, 70(8):939–946. Müller, R., T. Krauß, M. Schneider, and P. Reinartz, 2012. Automated georeferencing of optical satellite data with integrated sensor model improvement, Photogrammetric Engineering & Remote Sensing, 78(1):61–74. Noerdlinger, P., 1999. Atmospheric refraction effects in Earth remote sensing, ISPRS Journal of Photogrammetry & Remote Sensing, 54:360–373. Oh, J., and C. Lee, 2011. Atmospheric correction and velocity aberration for physical sensor modeling of high-resolution satellite images, Journal of the Korean Society of Surveying Geodesy Photogrammetry & Cartography, 29(5):519–525.
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Owens, J., 1967. Optical refractive index of air: Dependence on pressure, temperature and composition, Applied Optics, 6(1):51–59. Pan, H.B., G. Zhang, X.M. Tang, D.R. Li, X.Y. Zhu, P. Zhou, and Y.H. Jiang, 2013. Basic products of the ZiYuan-3 satellite and accuracy evaluation, Photogrammetric Engineering & Remote Sensing, 79(11):1131–1145. Poli, D., and T. Toutin, 2012. Review of developments in geometric modelling for high resolution satellite pushbroom sensors, The Photogrammetric Record, 27(137):58–73. Portland State Aerospace Society, 2004. A quick derivation relating altitude to air pressure, URL: http://www.psas.pdx.edu (last data accessed: 26 April 2016). Radhadevi, P.V., R. Müller, P. d'Angelo, and P. Reinartz, 2011. In-flight geometric calibration and orientation of ALOS/PRISM imagery with a generic sensor model, Photogrammetric Engineering & Remote Sensing, 77(5):531–538. Roper, L., 2011. Precipitation rate versus latitude and longitude, URL: http://www.roperld.com/science/ (last data accessed: 26 April 2016). Saastamoinen, J., 1972. Atmospheric Correction for the Troposphere and Stratosphere in Radio Ranging of Satellites, The Use of Artificial Satellites for Geodesy, Geophysical Monograph Series 15, 247–251. Stone, R., 1996. An accurate method for computing atmosphere refraction, Publications of the Astronomical Society of the Pacific, 108:1051–1058. Tang, X.M., G. Zhang, and X.Y. Zhu, 2012. Triple linear-array Imaging geometry model of Ziyuan-3 surveying satellite and its validation, Acta Geodaetica et Cartographica Sinica, 41(2):191– 198. Toutin, T., 2004. Review article: Geometric processing of remote sensing images: Models, algorithms and methods, International Journal of Remote Sensing, 25(10):1893–1924. Wang, M., B. Yang, F. Hu, and X. Zang, 2014. On-orbit Geometric calibration model and its applications for high-resolution optical satellite imagery, Remote Sensing, 6:4391–4408. Wang, Z.Z., 2007. Principles of Photogrammetry, Wuhan University Press, 237 p. Wei, Z., F.C. Hu, and B. Zhang, 2006. The influence of atmospheric refraction on aerial digital remote sensing, Journal of Remote Sensing, 10(5):651–655. Yan, M., B.Y. Yu, and Z.Y. Wang, 2013. Design of width of panchromatic spectra of high resolution optical remote sensing camera based on pansharpening, Spacecraft Recovery & Remote Sensing, 34(3):1–8. Yastikli, N., and K. Jacobsen, 2005. Influence of system calibration on direct sensor orientation, Photogrammetric Engineering & Remote Sensing, 71(5):629–633.
(Received 28 June 2015; accepted 10 December 2015; final version 12 January 2016)
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Abstract
When an optical remote sensing satellite is imaging the Earth in-orbit, the propagation direction of the Line of Sight (LOS) will be changed because of atmospheric refraction. This will result in a geolocation deviation on the collinear rigorous geometric model for direct georeferencing, pushbroom images. To estimate and correct the atmospheric refraction geolocation error, the LOS vector tracking algorithm is introduced and a weighted mean algorithm is used to simplify the ISO standard atmospheric model into a troposphere and stratosphere, i.e., two layers spherical atmosphere. The simulation result shows the atmospheric refraction will introduce about 2 m and 7.5 m geometric displacement when the spacecraft is off-pointed view at 30 and 45 degree angle, respectively. For a state-of-theart high resolution satellite, the atmospheric refraction displacement shall be corrected. The method has been practiced in the DMC3/TripleSat Constellation to remove the atmospheric refraction geolocation error without ground control points.
Introduction
Satellite remote sensing image geolocation is about determining the correspondence between the pixel’s Cartesian coordinate and geodetic coordinate. The geometric transformation between these two systems can be expressed by a rigorous geometric model, which is established based on a collinear equation with interior orientation parameters from the imaging payload and exterior orientation parameters of the satellite platform (Crespi et al., 2007; Fan et al., 2011; Habib et al., 2007; Jeong and Bethel, 2010 and 2014; Jiang et al., 2013 and 2014; Leprince et al., 2007; Lussy and Greslou, 2012; Mahapatra et al., 2004; Müller et al., 2012; Pan et al., 2013; Poli and Toutin, 2012; Radhadevi et al., 2011; Tang et al., 2012; Toutin, 2004). The accuracy of a rigorous geometric model depends on the interior and exterior orientation parameters, such as the focal length, detector size, lens, detector line distortion of the imaging payload, and the ephemeris and attitude systematic error of satellite platform. Also, the atmospheric refraction and light aberration will have an impact on the rigorous geometric model estimation (Lussy et al., 2012; Oh and Lee, 2011). The distortions of the interior and exterior orientation parameters are generally corrected during satellite in-orbit geometric calibration and validation operation (Gruen and Kocaman, 2008; Leprince et al., 2008; Wang et al., 2014; Yastikli and Jacobsen, 2005). Greslou et al. (2008) have analyzed the Line of Sight (LOS) in the Earth Centered Earth Fixed (ECEF) coordinate system for the apparent deflection caused Ming Yan, Chengyi Wang, and Jianglin Ma are with the Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, 20 Datun Road, Chaoyang District, Beijing, 100101, P. R. China ([email protected]). Zhiyong Wang and Bingyang Yu are with the Twenty First Century Aerospace Technology Co., Ltd., 26 Jiancaicheng East Road, Haidian District, Beijing, 100096, P. R. China.
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by the relative velocity of the satellite platform and target to correct light aberration. The atmospheric refraction is rarely considered and corrected because they are specific to each acquisition location and information about the atmosphere. In this paper, we focus on atmospheric refraction and analyze how it can be better processed to reduce the geolocation error. The atmospheric refraction changes the LOS propagation direction of satellite imaging in-orbit, which results in the points of detector, projection center of imaging payload, and imaged object no longer following the rigorous collinear model. Several pioneering researches have confirmed the effect of atmospheric refraction on rigorous geometric model estimation. For example, Gyer (1996), Wang (2007), and Wei (2006) have found the atmospheric refraction will have an impact on aerial photogrammetry. While the aerial refraction correction formulas are valid up to the normal aircraft flying height, they are delivering the wrong results for a spaceborne image (Jacobsen, 2004). For atmosphere refraction effects on a space image, Jacobsen (2004) and Dowman (2012) gave the refraction correction formula for the nadir space image based on the 1959 ARDC standard atmospheric. Noerdlinger (1999) targeted on MODIS satellite data and researched the atmospheric refraction by developing an analytical method to calculate the angle of the refraction assuming a single layer of spherically symmetrical atmosphere. Saastamoinen (1972) expressed a simple atmospheric refraction angle calculation formula for radio ranging of low Earth orbit satellites. Oh and Lee (2011) simply extended the Saastamoinen model to express the constant related to the object’s terrain and satellite altitude. However, it rarely has the documentation to study the procedure and method regarding the atmospheric refraction correction for a high resolution optical satellite. This study, based on a ray tracking physical model, rigorously describes the LOS propagation direction of the satellite imaging sensor to the Earth surface object through the atmosphere. Meanwhile, the International Organization for Standardization (ISO) (1975) atmospheric model and the Owens (1967) optical refraction index calculation algorithm are used to calculate the atmospheric refraction index at any position. In the following sections, first we will describe the LOS ray tracking geometric algorithm, and then analyze the geolocation deviation introduced by atmospheric refraction. After that, we will reveal how we can use the deviation to correct the atmospheric refraction geolocation error in the ECEF coordinate system. Compared to previous research, more reasonable atmospheric refraction displacement estimation can be achieved. Because we not only take the latitude and altitude into account when calculating atmospheric refraction Photogrammetric Engineering & Remote Sensing Vol. 82, No. 6, June 2016, pp. 427–435. 0099-1112/16/427–435 © 2016 American Society for Photogrammetry and Remote Sensing doi: 10.14358/PERS.82.6.427
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index, we also track the LOS refracted direction in different atmospheric layers. Therefore, the proposed method is more conformal to the LOS propagation characteristics in the stratified atmosphere. By compensating the atmospheric refraction error in the rigorous collinear geometric model, the geometric positioning accuracy of satellite remote sensing image without ground control points can be improved. Since DMC3/TripleSat constellation was launched on 10 July 2015, this study has been continually experimenting with the three 1 m resolution optical satellites. The atmospheric refraction correction algorithm has been integrated into the DMC3/TripleSat rigorous geometric model, which is used to produce the Level-1A basic image products.
Atmospheric Refraction Geolocation Error
The stratified Earth’s atmosphere comprises mainly gas molecules, water vapor, and aerosols. When the Sun’s ray is reflected by ground objects through the atmosphere to the imaging detectors in-orbit, the propagation direction of reflection rays shall be deviated due to atmospheric refraction. Conversely, from the view of detectors of imaging payload, the atmospheric refraction results in a deviation of LOS from its original propagation direction and lead to an atmospheric refraction geolocation error. In the following, we will model the geolocation error with rigorous mathematic formulas. In Figure 1, suppose atmospheric refraction does not exist, the line dSP represents the LOS of one detector with view angle α, where Point d refers to the detector (the distance from the imaging detector to the linear array principal Point o is also expressed as d), Point S is the perspective center of optical imaging payload, and Point P is the intersection of LOS and the Earth’s ellipsoid. This line dSP also contains Point P0, which is the intersection of LOS and the top atmosphere. In order to describe the ray tracking geometric algorithm of LOS vector, it is assumed the Earth’s atmosphere is a single layer homogeneous spherical atmosphere only consisting of gas molecules and water vapor: h is the atmosphere thickness and n is the atmospheric refraction index (n >1). Point P1 is the intersection of LOS deviated by atmospheric refraction and the Earth ellipsoid. The line of d1SP1 (dashed line) is the atmospheric refraction bias compensated LOS. Also, in Figure 1, we define f as the focal length, H as satellite platform height, and R as the Earth’s mean radius. The ground surface distance between point P and P1 is the atmospheric refraction geolocation error that would exist in rigorous geometric model if atmospheric refraction is not considered. Next, we will calculate the atmospheric refraction geolocation error with a rigorous mathematic formula using symbols predefined in Figure 1. According to the sine law we have the following formulas:
R R+H = sin α sin(180 − β )
R+H ⇒ β = arcsin sin α R
(1) (2)
where α is the off-view angle, and β is the incident angle of to the earth ellipsoid without considering the atmospheric refraction error. In the same way, the incident angle of LOS to the top atmosphere can be calculated: LOS
R+H sin α i = arcsin R+h
(3)
where h is the atmosphere thickness. According to the Snell refraction law, the refraction angle r is as follows:
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Figure 1. LOS propagated in single layer spherical atmosphere. n sin r = sin i (4) sin i ⇒ r = arcsin (5) n where n is the refraction index, i is the incident angle, and r is the refraction angle. We define θ as ∠SOP, θ0 as ∠SOP0 as ∠P0OP1 and ∆θ as ∠P1OP. Since OP0 = R + h, OP = R, and the refraction angle r is known in ∆P0OP1, and θ1 can be calculated:
(R + h)sin i θ1 = arcsin −r R
(6)
Known as the off-view angle α, the incident angle i and β in ∆SOP0, θ0 can be calculated:
θ =β −α
(7)
(8) θ0 = i − α Combining the above equations, angle ∆θ (radian) and arc length are calculated.
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Δ θ = θ − θ0 − θ1 (9) P 1P = R •∆θ (10) If the characteristics of stratified Earth’s atmosphere and the atmospheric refraction index are known, the refraction deviation angle of each layer can be calculated by the iterative calculation of Equations 3, 5, and 6. Thus, the atmospheric refraction deviation angle ∆θ is calculated: Δθ = θ − θ 0 − θ1 − − θ j
(11)
where j is the number of layered atmosphere. The imager of a high resolution optical satellite is often composed of blue (0.45 - 0.52 μm), green (0.53 - 0.60 μm), red (0.63 - 0.69 μm), near-infrared (0.76 - 0.90 μm) multispectral bands and 0.45 - 0.8 μm panchromatic band (Jacobsen, 2011; Yan et al., 2013). When the monochromatic light passes through the Earth’s atmosphere, the refraction index depends on its wavelength. The shorter the wavelength is, the larger the atmospheric refraction index is. In the astronomical observation, Lipcanu (2005) and Stone (1996) adopted the Owens (1967) atmospheric refraction calculation algorithm to analyze the impact of atmospheric refraction on star observation. They found the atmospheric refraction result in a 57.5 micro-radian error of a viewed star at 45 degrees zenith angle under the conditions of 15 Celsius degree environment temperature and 760 mm Hg atmospheric pressure. When the atmospheric temperature t (Celsius), the atmospheric pressure Pa (Pascal) and the water vapor pressure Pw (Pascal) are known, the atmospheric refraction index is calculated according to the equations from 12 to 17:
(
)
+ 6487.31 + 58.058σ 2 − 0.7115 σ 4 + 0.08851σ 6 DW
where
(
)
(14)
PS = Pa − Pw (15) T = 273.15 + t (16) σ = 1λ (17) In the above equations, PS (Pascal) is dry air pressure after water vapor pressure removed, T is the absolute temperature and σ is the wave number of monochromatic light with wavelength λ micron.
Atmospheric Temperature (t)
Atmospheric Refraction Index of Monochromatic Light
683939.7 4547.3 + DS (n − 1) × 108 = 2371.34 + 130 − σ 2 38.9 − σ 2
2.23366 710.792 7.75141 × 104 PW DW = 0.01 1 + 0.01PW 1 + 3.7 × 10−6 PW −2.37321 × 10−3 + − + T T2 T3 T
(12)
9.325 × 10−4 0.25844 PS DS = 0.01 1 + 0.01PS 57.9 × 10−8 − + (13) T T 2 T
The international standard atmosphere model (ISO 2533: 1975) is used to calculate the atmospheric temperature variation with altitude. The ISO 1975 atmosphere model defines that the mean temperature at sea level is 15 Celsius degree, the atmospheric pressure is 760 mm Hg and the Earth’s atmosphere is divided into eight layers. The air temperature of each layer linearly changes with altitude as the following formula: 15 − 6.5h 0 km ≤ h ≤ 11.019 km − 56 . 5 11 . 019 km < h ≤ 20.063 km −56.5 + (h − 20.063) 20.063 km < h ≤ 32.162 km (18) t = −44.5 + 2.8 ( h − 32.162) 32.162 km < h ≤ 47.35 km −2.5 47.35 km < h ≤ 51.413 km −2.5 − 2.8(h − 51.413) 51.413 km < h ≤ 71.802 km 71.802 km < h ≤ 86 km −58.25 − 2(h − 71.802) where t is the atmospheric temperature in Celsius degree. However, the mean temperature at sea level varies with geographic latitude, which makes the ISO 1975 standard atmosphere model to estimate temperature inaccurate in troposphere. In the study of global precipitation with longitude and latitude, Roper (2011) studied the earth surface mean air temperature by averaging over all longitudes from 1948 to 2009, which is shown in Figure 2. Some interesting conclusions can be drawn from Figure 2. Let angle φ be the latitude degree, the cos(φ) can be regarded as an independent variable linearly related to the Earth surface mean air temperature in the northern hemisphere when the latitude range is between 0 and 82 degrees. The same conclusion holds true for the southern hemisphere as the following formula shows:
78.08 cos φ − 48.2 −82° ≤ φ ≤ 0 tL = 52.07 cos φ − 26.4 0 < φ ≤ 82°
(19)
where tL is the Earth’s surface mean air temperature. Here we are only interested in air temperature in the troposphere of Equation 18 which depends on the Earth surface air temperature tL. Then, the troposphere air temperature at different altitude becomes:
t = tL +
(−56.5 − t ) h L
11.019
0 km ≤ h ≤ 11.019 km .
(20)
Atmospheric Pressure (Pa)
With the increase of altitude, atmospheric pressure smoothly decreases following an exponential function (Portland State Aerospace Society, 2004)): Figure 2. Relationship between mean Earth surface air temperature with latitude.
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Figure 4. Water vapor pressure changes with temperature.
Figure 3. Atmospheric pressure changes with altitude. gM
Lh Rmol L Pa = P0 1 − T0
Pw ≈ 3386.39 (0.00738t + 0.8072) − 0.000019 1.8t + 48 + 0.001316 (23) 8
(21)
g is the earth surface gravity acceleration (g = 9.80665 m/s2);
where Pw is water vapor pressure (Pascal), and t is Celsius degree. Figure 4 illustrates how water vapor pressure changes from the Earth’s surface air temperature 30 Celsius degrees to the top mesosphere temperature −87 Celsius degrees.
M is the molar mass of dry air (M = 0.0289644 kg/mol);
Atmospheric Refraction Index (n)
where Pa is the atmospheric pressure (Pascal); P0 is the atmospheric pressure at sea level (P0 = 101325 Pa);
Rmol is the gas constant (Rmol = 8.31447 J/(mol·K)); T0 is the standard temperature at sea level (T0 = 288.15 K); L is the temperature lapse rate (such as: L = 0.0065 K/m in troposphere). The temperature of tropopause and stratopause is constant; atmospheric pressure at any altitude in these two layers is estimated according to the distance from the point to the bottom of each layer as the following equation:
Suppose the atmospheric temperature is t, the air pressure is Pa, the water vapor pressure is Pw and the central wavelength of spectral band is known, the atmospheric refraction index in the stratified atmosphere can be calculated by using Equations 12 to 17. For the object points located on the Equator, latitude 30° and 60° in the northern hemisphere, and 30° in the southern hemisphere, Figure 5 shows the atmospheric
g Pa = P1exp − (h − h1 ) (22) RkgT0 where Rkg is the gas constant (Rkg = 287.05287 J/(kg·k)), h1 is the distance to the bottom of tropopause or stratopause and P1 (Pascal) is the atmospheric pressure at the altitude of h1. Figure 3 shows how the atmospheric pressure Pa changes with the altitude.
Water Vapor Pressure Pw
Water vapor pressure is only related to the atmospheric temperature, and the change with temperature can be represented by the following equation (Bosen, 1960): Figure 5. Atmospheric refraction indexes change with altitude in the troposphere.
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refraction index of 0.5 μm blue light through the troposphere. Table 1 lists the atmospheric refraction index at intervals of altitude from Earth’s surface to the stratopause. From Table 1 and Figure 5, we can conclude that the atmospheric refraction index is 1 along LOS from payload in-orbit to the top of stratosphere at 47.35 km altitude. In the troposphere, tropopause and stratosphere, the atmospheric refraction index increases with the decreasing altitude. Under the same atmospheric pressure and water vapor pressure, the atmospheric refraction index is negatively correlated with the atmospheric temperature. The result is that the higher the latitude is, the larger the atmospheric refraction index is. The atmospheric refraction index is almost unchanged for the same latitude regardless it is in the southern hemisphere or in the northern hemisphere. For example, the mean Earth surface air temperature at N60°, N30° and Equator is 0°, 20°, and 25° Celsius degrees, respectively, the atmospheric refraction indexes of N60° are larger than N30°, and N30° are larger than the Equator in troposphere. The atmospheric refraction indexes of N30° and S30° are almost equal. The atmospheric refraction index at Equator has the minimum value in troposphere. Table 2 lists the distribution characteristics of atmospheric refraction with regard to the atmosphere layer and altitude. Therefore, the ISO eight layer atmosphere model can be simplified into two layers: 1. Troposphere layer: geometric altitude is between 0 and 11,019 m; 2. Stratosphere layer (include tropopause): geometric altitude is from 11,019 m to 47,350 m. This simplification is based on the fact that the refraction index is 1 from the stratopause to mesopause. We will use the simplified two-layer atmosphere model to calculate the geolocation error as shown in Figure 6. The atmospheric refraction index in the troposphere and stratosphere are set to be n1 and n2 respectively, and i is the incident angle of LOS at the top of the stratosphere. D is the displacement on the ellipsoid surface because of LOS refracted in the troposphere and stratosphere. A weighted average algorithm is employed to calculate the atmospheric refraction index, which is calculated at each 1,000 m interval in the troposphere and 2,000 m interval in the stratosphere. The weighted average algorithm in the troposphere is described as follows:
Table 1. Atmospheric Refraction Indexes Change with Altitude at the Equator, N60°, N30° and S30° Altitude (m)
N60 Degrees
N30 Degrees
Equator
S30 Degrees
1000
1.000268639
1.000250154
1.000242816
1.000249707
3000
1.000221848
1.000209777
1.000204972
1.000209484
5000
1.000184809
1.000178003
1.000175255
1.000177835
7000
1.000155253
1.000152781
1.000151744
1.000152718
9000
1.000131491
1.000132614
1.000133037
1.00013264
11000
1.000112255
1.000116392
1.000118097
1.000116495
15000
1.000044223
1.000044223
1.000044223
1.000044223
20000
1.000020098
1.000020098
1.000020098
1.000020098
30000
1.000003847
1.000003847
1.000003847
1.000003847
40000
1.000000753
1.000000753
1.000000753
1.000000753
50000
1.000000003
1.000000003
1.000000003
1.000000003
60000
1.000000034
1.000000034
1.000000034
1.000000034
70000
1.000000014
1.000000014
1.000000014
1.000000014
80000
1.000000002
1.000000002
1.000000002
1.000000002
Table 2. Characteristics of Atmospheric Refraction Index
Layer Name
Base Altitude (m)
Top Altitude (m)
1
Troposphere
0
11019
2 3 4 5 6 7 8
Tropopause Stratosphere Stratosphere Stratopause Mesosphere Mesosphere Mesopause
11019 20063 32162 47350 51413 71802 86000
20063 32162 47350 51413 71802 86000 —
Layer Number
Atmospheric Refraction Index Varying with latitude and altitude. Varying only with altitude. The refraction index is 1.
Step 1: Calculate the difference (delta_atm_ref(i)) of two atmospheric refraction indexes at every 1,000 m interval. Step 2: Calculate the difference (total_delta_atm_ref) of the atmospheric refraction index at the surface and top troposphere. Step 3: Calculate the weight of each interval:
weight(i) = delta_atm_ref(i)/ total_delta_atm_ref
Step 4: Calculate mean atmospheric refraction index (mean_ atm_ref) of troposphere. mean_atm_ref = sum[weight(i)·atm_ref(i)] where atm_ref(i) is calculated by Equations 12 to 17 at any altitude increased 1,000 m interval in troposphere. For the 0.5 μm blue light, the atmospheric refraction index is 1.000014132 (n2) in the stratosphere at N60°, N30°, S30° and the Equator. The atmospheric refraction index in the troposphere is 1.000199059 (n1) at N60° and 1.000187379 (n1) at the Equator. As for the atmospheric refraction index of different wavelength monochromatic light, Figure 7 shows
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Figure 6. Two layers atmospheric refraction model
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the atmospheric refraction index of blue (0.5 μm), green (0.55 μm), red (0.65 μm), and near-infrared (0.83 μm) monochromatic light through the troposphere at N30°. The shorter the wavelength, the larger of atmospheric refraction index value. In the following section, the atmospheric refraction geolocation error of the four monochromatic lights will be analyzed.
Atmospheric Refraction Geolocation Error
When the satellite view angle α is known, we can calculate the atmospheric refraction geolocation error on the ellipsoid surface with zero altitude at any latitude. Suppose the mean Earth radius R is 6,371 km, the altitude of satellite orbit H is 650 km, Table 3 lists the atmospheric refraction geolocation error at N60°, N30° and the Equator of blue (0.5 μm), green (0.55 μm), red (0.65 μm) and near-infrared (0.83 μm) monochromatic light in different view angle of satellite in-orbit. In Table Figure 7. Atmospheric refraction index changes with different wavelength in tropo3, α is the satellite off-pointing view angle, sphere at N30 degrees. Blu is blue band, Grn is green band, Red is red band, and NIR is near-infrared band. Table 3. Atmospheric Refraction Geolocation Error (Unit: m) From Table 3, we can clearly observe N60 Degree N30 Degree Equator α that the atmospheric refraction geolocation Blu Grn Red NIR Blu Grn Red NIR Blu Grn Red NIR error increases nonlinearly with the increas5° 0.24 0.24 0.24 0.24 0.23 0.23 0.23 0.23 0.23 0.23 0.22 0.22 ing view angle. This conclusion is not only 10° 0.49 0.49 0.48 0.48 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.46 confirmed by our proposed method, but also 15° 0.78 0.78 0.77 0.77 0.76 0.76 0.75 0.75 0.75 0.75 0.75 0.74 verified by Noerdlinger (1999). Also, three 20° 1.14 1.14 1.13 1.12 1.11 1.11 1.10 1.10 1.10 1.10 1.09 1.09 other conclusions as follows: 25° 1.62 1.62 1.60 1.59 1.58 1.57 1.56 1.55 1.56 1.56 1.55 1.54 1. Atmospheric refraction geolocation 30° 2.28 2.27 2.25 2.24 2.22 2.21 2.20 2.19 2.20 2.19 2.18 2.17 error is increased from the Equator 35° 3.26 3.25 3.23 3.21 3.18 3.17 3.15 3.13 3.15 3.14 3.12 3.10 to South and North Poles. 40° 4.82 4.80 4.78 4.75 4.71 4.69 4.66 4.64 4.66 4.65 4.62 4.60 2. Atmospheric refraction geolocation 45° 7.56 7.53 7.49 7.45 7.38 7.35 7.31 7.27 7.31 7.28 7.24 7.20 error is increased with the shorter 50° 13.03 12.98 12.91 12.84 12.73 12.68 12.60 12.54 12.61 12.55 12.48 12.42 of monochromatic light wavelength in the same latitude. While, this value of the paper’s proposed method than Noerdlinger, Jaedifference between blue, green, red, and near-infrared hong and Jacobsen’s methods. bands are negligible. We will only use the green (0.55 μm) light to calculate the atmospheric refraction index for high resolution optical satellite images direct georeExperiment on DMC3/TripleSat Constellation ferencing. The DMC3/TripleSat Constellation was launched on 10 July 3. Despite the fact that the atmospheric refraction geoloca2015 by a PSLV-XL launch vehicle from the Satish Dhawan tion error is not large, there is a need to correct this error Space Centre, Sriharikota launch site in India. The constellafor the state-of-the-art high resolution satellite, such as tion was successfully delivered into a sun-synchronous orbit WorldView-3 (GSD: 0.31 m), GeoEye-2 (GSD: 0.34 m), Geowith the local time of the ascending node of 10:30 at 651 km Eye-1 (GSD: 0.46 m) and WorldView-1/2 (GSD: 0.46 m). altitude. The three satellites were phased 120 degrees apart around the same orbit using their on-board propulsion sysCompared the atmospheric refraction geolocation error of tem. The DMC3/TripleSat Constellation satellites use the 450 the proposed method with the results of Noerdlinger (1999), kg SSTL-300S1 series platform, which provides 45 degrees Jaehong’s (Oh and Lee, 2011; Saastamoinen, 1972) and Jafast slew off-pointing and is capable of acquiring multiple cobsen’s (Jacobsen, 2004; Dowman et al., 2012) methods, the targets in one pass using multiple viewing modes. The high atmospheric refraction displacement is almost in the same resolution imager on board equipped with four Time-Delayedwhen the satellite was imaging inside the core of 20 degrees Integration (TDI) CCD arrays provides 1 m ground sampling around nadir. When the off-pointing angle is over 20 degrees, distance (GSD) in panchromatic and 4 m GSD in blue, green, the results of these methods are different. The authors believe red and near-infrared multispectral mode with a swath width the Noerdlinger’s single layer spherically symmetrical atmoof 24 km. The wide swath of the imager combined with agile sphere assumption will result in the atmospheric refraction off-pointing capability enable the DMC3/TripleSat Constellaindex underestimated in the troposphere. And the Jaehong tion to target anywhere on earth at least once a day. Table 4 and Jacobsen’s methods took the satellite altitude and the lists the capability of DMC3/ TripleSat Constellation. object’s terrain elevation into account to calculate the atmoThe DMC3/TripleSat Constellation linear pushbroom imspheric refraction constant. These three methods were all not ages are direct georeferenced by using the collinear rigorous considering the LOS path increased through the atmosphere geometric model. The rigorous geometric model fixes the relawhen optical satellite operated in off-pointing imaging mode. tionship between LOS vector in the camera coordinate system Therefore, the larger of satellite off-pointing angle, the larger with the ECEF coordinate system. 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Table 4. DMC3/TripleSat Constellation Capability Sensor
Panchromatic
Multispectral
Band (μm)
0.45 to 0.65
Blue: 0.44 – 0.51 Green: 0.51 – 0.59 Red: 0.60 – 0.67 Near infrared: 0.76 – 0.91
GSD (m)
1
4
Swath (km)
24
24
Focal Length (m)
6.6667
6.6667
Detector Dimension (μm)
10*10
40*10
Detector Number
6408*4
1602*4
Digitization (bit)
10
10
Max Off-pointing (degree)
45
Imaging Mode
Stripe, Scene, Along track stereo, Across track stereo and Area
Figure 8. Definition of interior orientation parameters with atmospheric refraction geolocation error compensated. each pixel is obtained by calculating the intersection point coordinate of the LOS with the earth ellipsoid in ECEF coordinate system, which can be described as follows without considering the influence of atmospheric refraction. X − X0 = µM Orbit 2ECEF M Body 2Orbit M Camera2Body Y − Y0 Z − Z0 ECEF
tan(ψ y ) (24) − tan(ψ x ) 1 Camera
where [tan(ψy) – tan(ψx) 1]TCamera represents the detectors LOS vector in the camera coordinate system. MCamera2Body represents the bias matrix of the camera coordinate system to the attitude control coordinate system. MBody2Orbit represents the rotation matrix from the attitude control coordinate system to satellite platform coordinate system. MOrbit2ECEF represents the rotation matrix from orbit coordinate system to ECEF coordinate system. μ is a scalar. [X0 Y0 Z0]T ECEF represents the satellite platform position interpolated by ephemeris at imaging instant. [X0 Y0 Z0]T ECEF represents the ECEF coordinate of the intersection point of LOS with the Earth ellipsoid. During the in-orbit geometric calibration campaign, the precision instrument alignment angles in MCamera2Body matrix and
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vector [tan(ψy) – tan(ψx) 1]T in camera coordinate system is periodically calibrated by referencing the global geometric calibration sites. From the analysis of atmospheric refraction effect on geolocation, we draw the conclusion that the atmospheric refraction will change the direction of LOS and result in the geolocation error on earth spheroid. In the procedure of DMC3/TripleSat Constellation direct georeferencing, the detector’s LOS vector refractive deviation is subtracted from the calibrated LOS vector in the camera coordinate system to correct the atmospheric refraction geolocation error. When the satellite operated in the roll and pitch imaging mode, the collinear rigorous geometric model of Equation 24 is rewritten as below: LOS
X − X0 tan( ψ y − ψ pitch _ atmos ) = (25) Y − Y µ M 0 ECEF 2Camera − tan( ψ x − ψ roll _ atmos ) Z − Z0 1 Camera ECEF where MECEF2Camera = MOrbit2ECEF MBody2Orbit MCamera2Body. (ψx,ψy) is the interior orientation element in the camera coordinate system, ψroll_atmos is the atmospheric refraction deviation angle represented in the camera coordinate system when the platform rotation roll angle around the X axis, ψpitch_atmos is the atmospheric refraction deviation angle represented in the camera coordinate system when platform rotation pitch angle occurs around the Y axis. The definition of the interior orientation parameters is shown in Figure 8. OXYZ is the camera coordinate system and oxy is the image coordinate system. u is the LOS vector in Equation 24. u′ is the LOS vector in Equation 25 by subtracting the atmospheric refraction deviation angle in along and across track direction. In the DMC3/TripleSat Constellation geometric model, the original points of the camera coordinate system, body coordinate system, and orbit coordinate system are in the same. The values of roll and pitch are interpolated from the exterior orientation attitude parameters. In the direct georeferencing of DMC3/TripleSat Constellation images, the atmospheric refraction geolocation error was transformed into the LOS deviation in the camera coordinate system. And the atmospheric refraction geolocation error correction procedure is divided into 10 steps: Step 1: Use Equation 24 to calculate the ECEF coordinate (X,Y,Z) of one pixel (x,y) by referencing WGS84 ellipsoid. Step 2: Input the NASA Shuttle Radar Topographic Mission (SRTM) global 90 m digital elevation data. Step 3: Transform the object’s ECEF coordinate (X,Y, Z) into latitude and longitude. Step 4: Interpolate the roll and pitch value from exterior orientation attitude parameters at the imaging instant. The roll and pitch values are considered as the view angle of the detector across and along track direction. Step 5: Use the simplified two layer atmosphere model and terrain elevation to calculate the refraction index in stratosphere and troposphere. In the troposphere, there is a need to calculate the refraction index between the terrain elevation and the top of the troposphere. Step 6: Use the Equations 3 to 9 to calculate the Δθroll and Δθpitch across and along track direction on Earth’s surface. Step 7: Interpolate the spacecraft position (X0, Y0, Z0) by using ephemeris parameters at the imaging instant.
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Step 8: Calculate the distance from the imaged object (X,Y, Z) and spacecraft position (X0, Y0, Z0). Step 9: Calculate the value of ψroll_atmos and ψpitch_atmos. An example of calculation (radian) as following: ψ roll _ atmos =
(R + hdem ) •∆ θ roll
( X − X ) + (Y − Y ) + (Z − Z ) 2
0
2
0
(26)
2
0
where, R is the mean Earth radius; hdem is the object’s terrain elevation interpolated from SRTM 90 m digital elevation data. Step 10: Use Equation 25 to calculate the atmospheric refraction geolocation error corrected ECEF coordinate. In the DMC3/TripleSat Constellation in-orbit commissioning and early operation phase, the studied method was used to direct georeferencing the DMC3/TripleSat 1 m resolution images. Table 5 lists four sample DMC3/TripleSat Constellation images with atmospheric refraction geolocation error correction. Since the spacecraft off-pointing limit in the early phase, the four sample images were all imaged in roll off-pointing mode. In the future, we will investigate in the atmospheric refraction geolocation error in both roll and pitch off-pointing mode.
Table 5. Experiment Results of Four samples DMC3/TripleSat Constellation Images with the Atmospheric Refraction Geolocation Error Correction. The value of X_corr is the Atmospheric Refraction Geolocation Error Corresponding to the Pitch Off-pointing View, and the Value of Y_ corr is the Atmospheric Refraction Geolocation Error Corresponding to the Roll Off-pointing View. Data Date: 23 Sep, 2015 Image ID: D2000093VI
As the propagation direction of LOS from satellite payload in-orbit is deviated because of the atmospheric refraction, the points of the detector, projection center and ground object are no longer in a line. This is one reason why there are geolocation errors in the collinear rigorous geometric model estimation. The paper describes how to use the LOS tracking geometry algorithm, ISO standard atmospheric model, and Owens atmospheric refraction index calculation method to estimate the atmospheric refraction geolocation error and how to correct this error in the collinear rigorous geometric model. The proposed method is especially suitable for the high resolution agile optical satellite as images taken from a large off-pointing view angle is more susceptible to the atmospheric refraction. We also have applied the proposed method in direct georeferencing DMC3/TripleSat Constellation 1 m resolution images to improve the geolocation accuracy without ground points. However, we have to admit that atmospheric refraction is one element that has made collinear geometric model nonrigorous. Other elements, such as the light aberration and the transmission delay of LOS, can also introduce geometric errors in the collinear geometric model. Only when all the causes that lead to inaccurate estimation of the interior orientation and exterior orientation parameters are calibrated, can we obtain a real accurate collinear rigorous geometric model.
Acknowledgments
This work was partly supported by the National High Technology Research and Development Program of China (No. 2013AA12A303) and Major Project of High Resolution Earth Observation System (Civilian Part) (No. 65-Y40B01-900113/15). The authors also thank the anonymous reviewers for their construction comments and suggestions.
References
Bosen, J., 1960. A formula for approximation of the saturation vapor pressure over water, Monthly Weather Review, pp. 275–276.
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Geolocation Error (m): X_corr: 0 Y_corr: 0.31
Location: Guildford, UK Off-pointing (degree): Roll: -5.908 Pitch: 0 Date: 2 Oct, 2015 Image ID: D20000A3VI
X_corr: 0 Y_corr: 0.82
Location: Beijing, China Off-pointing (degree): Roll: -16.967 Pitch: 0 Date: 13 Oct, 2015
Conclusions
Browse Image
Image ID: D20000BEVI
X_corr: 0 Y_corr: -1.54
Location: Aden, Yemen Off-pointing (degree): Roll: 23.996 Pitch: 0 Date: 16 Oct, 2015 Image ID: D10000B1VI
X_corr: 0 Y_corr: 0
Location: Zhaoyuan, Haerbin, China Off-pointing (degree): Roll: 0.001 Pitch: 0
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(Received 28 June 2015; accepted 10 December 2015; final version 12 January 2016)
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