**1. Introduction**

The precision of remote space observations is important when investigating and monitoring various components of global ecological systems, such as marine, forestry, and climate environments [1–4]. Satellite data integration with external marine and other datasets is crucial in various applications of remote sensing techniques [5,6]. For climate and meteorological investigations, observations of clouds and precipitation on a global scale are usually performed using ground-based radar data and observations from geostationary satellites, due to their high temporal and moderate spatial resolution [7–9]. However, during data comparison and integration from these sources, the problem of parallax shift occurs [7,10], which is particularly observable for mid and high latitudes, and also for longitudes far from the sub-satellite point. Parallax shift is also important for cloud shadow determination, which is a significant issue for solar farms [11] and for flood detection [12]. Parallax

phenomena also have a significant impact on the comparison of data from low-orbit satellites from different sensors [13–16].

In terms of mathematical problem formulation, the parallax shift effect for the geostationary satellites is actually a special case amongst low-orbit satellites, and it is easier to investigate due to higher temporal resolution data acquisition and the fixed satellite position.

There have been several attempts to solve parallax shift for geostationary satellites. One of them was proposed by Roebeling et al. [7,17,18], and was based on liquid water path (LWP) value pattern matching. This approach was suitable for stormy events and other inhomogeneous cloud formations, however it usually failed to perform correction in the case of homogeneous spatial LWP distribution. Another attempt proposed by Greuell et al. and Roebeling [19,20] used a simplified geometric model, which assumes Earth to be locally flat, as well as a sort of a priori knowledge about cloud height above the Earth's surface. There were also attempts by Li, Sun, and Yu [12] to solve this problem using a spherical model. Finally, there is the Vicente et al./Koenig method [21,22] based on the geometric properties of parallax shift phenomena, incorporating an ellipsoid model of Earth. The Vicente et al./Koenig method will be presented further in this paper.

There are two methods proposed by the author in this paper, which are based on the same assumptions as the Vicente et al./Koenig method. The first is an augmented version of the mentioned method. This augmentation reduces the correction error to centimeters. The second method proposed by the author is an original work which is based as before on a priori knowledge of cloud height, a geodetic equation of an ellipsoid, and numerical methods for solving the equation set. This method allows the correction error to be reduced to almost zero (assuming Earth to have an ellipsoidal shape).

#### **2. Nature of Parallax Shift Problem and Vicente et al.**/**Koenig Method**

### *2.1. Problem Description*

A parallax shift error in satellite observations occurs when the apparent image of the object is placed in the wrong location on the ellipsoid, considering the ellipsoid's normal line passing through the observed point. This geometric phenomena is particularly observable in geostationary and polar satellite observations due to the high angles of observations, particularly for edge areas of image scenes. In Figure 1, the problem is presented considering the case of a geostationary satellite. As a result, this phenomena causes pixel drift from the original position towards the edge of the observation disk. Consequently, the higher the cloud top layer is, the bigger the shift that occurs.

**Figure 1.** Parallax shift problem. The violet surface represents an image obtained from a geostationary satellite. The cloud top (T) is observed by the satellite as T' (on the violet surface). The result of the reprojection of point T' to ellipsoidal coordinates is I, which is not true the location of the cloud. The true location of the cloud is denoted as B, and from the perspective of the satellite sensor is observed as B' on the violet surface. The square (marked as 1) shows how parallax shift affects the satellite image, where T' is an image of the cloud top and B' is where the cloud top should be placed according to its geodetic coordinates. The scale of the cloud height is not preserved.

The position of the cloud top (T) in Cartesian coordinates can be formulated as follows [23]:

$$\begin{cases} \mathbf{x} = \begin{pmatrix} \mathcal{N}(\boldsymbol{\varrho}\_{\mathcal{S}}) + h \right) \cos \boldsymbol{\varrho}\_{\mathcal{S}} \cos(\boldsymbol{\lambda}\_{\mathcal{S}} - \boldsymbol{\lambda}\_{0}) \\ \mathbf{y} = \begin{pmatrix} \mathcal{N}(\boldsymbol{\varrho}\_{\mathcal{S}}) + h \end{pmatrix} \cos \boldsymbol{\varrho}\_{\mathcal{S}} \sin(\boldsymbol{\lambda}\_{\mathcal{S}} - \boldsymbol{\lambda}\_{0}) \\ \boldsymbol{z} = \begin{pmatrix} \mathcal{N}(\boldsymbol{\varrho}\_{\mathcal{S}}) \end{pmatrix} \mathbf{1} - \boldsymbol{c}^{2} \end{pmatrix} \tag{1}$$

where *<sup>N</sup>*(ϕ*e*) <sup>=</sup> *<sup>a</sup>* <sup>1</sup>−*e*<sup>2</sup> sin<sup>2</sup> <sup>ϕ</sup>*<sup>g</sup>* is the prime vertical radius of the curvature, *e*<sup>2</sup> = *<sup>a</sup>*2−*b*<sup>2</sup> *<sup>a</sup>*<sup>2</sup> is the square of eccentricity, *a* is Earth's semi-major axis, *b* is Earth's semi-minor axis, *h* is the cloud top height, ϕ*g*, λ*<sup>g</sup>* is the geodetic latitude and longitude, and λ<sup>0</sup> is the longitude above which the geostationary satellite is floating. In this case, Equation (1) models the cloud position on a tangent line at coordinates ϕ*g*, λ*<sup>g</sup>* (see Figure 2). This is a more precise model than the flat-earth model or the spherical model. Note that: all longitudes (λ*g*, λ*<sup>c</sup>* and λ*p*) are equal and the same. Subscripts are given to formally distinguish these values between corresponding latitudes that have different definitions (see Figure 2).

**Figure 2.** Three types of latitude: where ϕ*g* is the geodetic latitude, ϕ*c* is the geocentric latitude, ϕ*p* is the parametric latitude, P is the point of interest on the ellipsoid, and P\* is the image of the point of interest on a sphere. Based on figures from [23,24].

Pixel displacement in satellite view coordinates is defined as:

$$p\_{\rm disp}(h) = \sqrt{c\_y^2 \left(q\_5(h) - q\_5(0)\right)^2 + c\_x^2 \left(\lambda\_s(h) - \lambda\_s(0)\right)^2} \tag{2}$$

where *cx* and *cy* are constants that allow for sensor inclination angles to be converted to pixels or distance units in the satellite view space. Also, ϕ*s*(*h*) and λ*s*(*h*) are defined as:

$$\varphi\_s(h) = \tan^{-1} \frac{z(h)}{\sqrt{\left(x(h) - l\right)^2 + y(h)^2}} \tag{3}$$

$$\lambda\_s(h) = -\tan^{-1} \frac{y(h)}{x(h) - l} \tag{4}$$

where *x*(*h*), *y*(*h*), and *z*(*h*) are cloud top coordinates from Equation (1) as functions of h; *l* = *a* + *hs*– distance from center of Earth to satellite; *a* is the Earth's semi-major axis; *hs* is the distance from the surface of Earth to the satellite. In order to illustrate *pdisp*(*h*), the following analysis presented in Figure 3 was performed. Namely, depending on the geographical localization of the affected pixel and cloud top height, the absolute shift error in observations is expressed in Spinning Enhanced Visible Infra-Red Imager (SEVIRI) pixel units (In this case *cx* = *cy* = *hs* <sup>3</sup> *km*/*px* ). It is worth noting that in many cases, especially for observations of clouds over 5000 m, this can cause pixel shift in the SEVIRI instruments used for the purpose of this study.

**Figure 3.** Error in pixels caused by cloud height parallax effect for 5 chosen cites, assuming the observation is acquired by SEVIRI instrument at longitude of 0◦. Spatial resolution was assumed as 3 km/pixel.

As mentioned earlier, this effect hinders the comparison process between satellite and ground-based radar data [7]. An example is depicted in Figure 4.

**Figure 4.** Comparison of detected precipitation mask based on ground-based radar data (blue) and data from Meteosat Second Generation (red). A parallax shift is particularly visible for small storm clouds in the bottom-right corner. The height of the cloud tops reaches 12 km. The stormy event is dated July 24, 2015, 13:00 UTC. EuroGeographics was used for the administrative boundaries.
