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Constellation-to-ground coverage analysis is an important problem in practical satellite applications. The classical net point method is one of the most commonly used algorithms in resolving this problem, indicating that the computation efficiency significantly depends on the high-precision requirement. On this basis, an improved cell area-based method is proposed in this paper, in which a cell is used as the basic analytical unit. By calculating the accuracy area of a cell that is partly contained by the ground region or partly covered by the constellation, the accurate coverage area can be obtained accordingly. Experiments simulating different types of coverage problems are conducted, and the results reveal the correctness and high efficiency of the proposed analytical method.

With the rapid development of science and technology, satellite technology plays an increasingly important role in mobile communications [

In satellite applications, ground coverage is a very important component. Constellation-to-ground coverage is accomplished by continuously or intermittently serving from various types of sensors loaded in constellation satellites. The coverage capacity of a constellation (i.e., the coverage rate and the coverage quality) over a long time period will be important issues in constellation applications. This kind of problem is called “the constellation-to-ground coverage problem.” The problem model ignores the ground features and considers the whole Earth’s surface to be spherical or ellipsoid without fluctuations. In most satellite applications, the model can support engineering needs.

The constellation-to-ground coverage problem is complicated. To date, many scholars have carried out relevant research on this topic. References [

The net point method is a traditional method for solving the constellation-to-ground coverage problem [

In this paper, an improved algorithm is proposed, which is called the cell area analytical method based on the net point method. On the basis of keeping the basic operation of the net point method, the cell area analytical method takes the cell as the core of the whole algorithm and calculates the precise coverage area of each cell by the spherical geometric relation. Then, the characteristics of the satellite’s ground coverage are obtained.

The basic organizational structure of this paper is as follows: the first chapter is the introduction, and the second chapter introduces the net point method and the analysis of its existing problems. In the third chapter, an improved cell method is proposed to analyze the flow of the algorithm. The fourth chapter is the simulation experiment, and the last chapter is the conclusion.

The basic steps of the net point method are shown in Figure

For a certain target region

For each point

Let the coverage region of the constellation be

Basic steps of the net point method.

In the classical net point division method, the coverage state of the net point is regarded as that of the corresponding net grid. That is, if a satellite constellation can cover the net point, it is considered that the constellation can cover the whole corresponding net grid; otherwise, it is considered that the constellation has no coverage for the corresponding net grid. This strategy is called the 0-1 judgment strategy. According to the principle of the algorithm and the statistics of the experimental results, if the precision of the results increases

From a numerical simulation point of view, the net point division method chooses the net point as the basic unit of analysis. However, the generated net points are just a set of single point on the sphere and have no any spatial structure, making the low-precision calculation results cannot provide any guidance for the high-precision calculation results. Hence, it is difficult to calculate iteratively.

This method exhibits a low computational efficiency and little reliability of the results. Thus, an improved method called the cell area analytical method (CAAM) is proposed in this paper. The improvement of the CAAM mainly focuses on two points: one is replacing the net point with a cell as the basic analytical unit, and the other is changing the 0-1 judgment strategy into the real intersection area calculation strategy.

In this section, we propose a new method for efficiently and exactly calculating the coverage area for the constellation-to-ground coverage problem, which is called the cell area analytical method.

For a spherical circle, the inner region is called a spherical disk. That is, the spherical circle is the boundary of the spherical disk. Denote the spherical disk by

As shown in Figure

Schematic diagram of the spherical arch region.

For the spherical circle

Denote the longitude and latitude of a point

The relationship between

The spherical sector with central angle

The spherical triangle with vertices

The spherical arch region in Figure

Therefore, we can obtain the area of the spherical arch only if the radius

The definition of cell in this paper is as follows:

A cell is the spherical region surrounded by two latitude lines and two longitude lines. Additionally, it has the same length of longitude interval and latitude interval.

The length of the longitude interval is called the cell width.

Denote a cell by

Suppose the longitude range and latitude range of

A spherical disk is defined as the spherical region surrounded by a spherical circle, and a spherical disk with index

The intersection area between a cell

If

If

If

If

For

In Figure

A typical case of a spherical disk intersecting a cell.

According to the characteristics of the intersection region, we compute the intersection area between the spherical disk and the cell by dividing this region into subregions. In Figure

The blue region

The yellow region

Since the upper and the lower boundaries of the cell are latitude lines, which are spherical small arcs in general,

Then,

Obviously, there are many other cases in which a spherical disk intersects a cell. There exists a total of 16 cases by analysis. Figure

Two typical cases of a spherical disk intersecting a cell.

Let

Additionally, by the conclusion in the former section, there are three types of relationships between

Type-I: only one circular arc in

Type-II: more than one circular arc in

According to the spherical geometry, a random spherical circular arc

We can use Equation (

However, if more than one circular arc in

If

Then,

By this process, we can obtain the intersection area between the grid and the region with arbitrary precision.

A constellation is a set of satellites that is denoted by

Then,

For a

For the computation of

Consider the situation shown in Figure

For white cells,

For red cells,

For blue cells,

For green cells,

The relationship between the cells and the constellation coverage region.

In constellation to ground-region coverage problems, a cell

In some combinations of

The result of

0 | 0 | 0 | 0 | |

0 | × | × | ||

0 | × | × | × | |

0 | × |

In Table

Then, for the set of

Computing the coverage rate is a common indicator in coverage problems. The coverage rate is defined as follows:

Generally,

According to the preview discussion, we can propose the cell area analytical method. The basic steps of this method are as follows:

Obtain the minimum latitude and longitude box of the region

Select a division precision

For every cell

According to Equation (

If the calculation accuracy does not satisfy the requirement, dividing every cell

Obtain the result of coverage rate. Because for any precision, there are some cells that

In this section, we analyze a Walker-Delta constellation with configuration parameters

For the cell area analytical method, the partition accuracy is the most important factor affecting the result precision. We define the cell precision as follows:

Cell precision is defined as the square root of the cell number in the

The computational results for the cell analytical method and the traditional net point method are depicted in Figures

Results of coverage rate with an increase of the cell precision for target-1.

The differences between the upper and lower bound of coverage rate for target-1.

Results of coverage rate with an increase of the cell precision for target-2.

Comparison of the cell counts from two algorithms with different precisions for target-2.

The computation results indicate that the boundary of coverage rate converges to the same value. Figures

Figures

Comparison of the computing time from two algorithms with different precision for target-1.

Comparison of the computing time from two algorithms with different precision for target-2.

Further results show that when the cell precision is low, the cell area analytical method needs to take longer computing time than the traditional net point method. That is because the cell area analytical method takes much longer time to compute a single cell than the numerical method to compute a signal net point. However, with the high-precision calculation requirement based on the increase of the cell partition precision, the computing time of the traditional net point method increases sharply and quickly exceeds than that of the cell analytical method. Additionally, in Figure

Figures

The coverage region of simulation for target-1.

The coverage region of simulation for target-2.

We can see that the cell area analytical method can solve all types of coverage regions, including regular regions and irregular regions. For ground regions with fewer boundaries, the efficiency of the algorithm is extremely high, but for ground regions with a high number of boundaries, the efficiency of the algorithm is slightly lower than that of regions with fewer boundaries but still very high compared with the traditional net point method.

In Figures

In this paper, based on the net point method, an improved method is proposed, which we call the cell area analytical method. Compared with the traditional net point method, the improved cell area analytical method can accurately compute the coverage area of the partial coverage cell. By a simulation experiment, we can see that the method has high computational efficiency, especially for regions with fewer boundaries.

However, the cell area analytical method still has some disadvantages; that is, it can only solve the instantaneous coverage problem, while the net point method can also solve the accumulative coverage problem and continuous coverage problem.

Additionally, this algorithm has room for improvement. That is, for ground regions with many boundaries, the spherical polygon area formula can be used to calculate the exact area of the intersecting regions, but we need to consider many complicated situations.

The data used to support the findings of this study are available from the first author upon request.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work is supported by the National Key R&D Program of China under Grant No. 2016YFB0501001, the National Natural Science Foundation of China under Grant No. 62006214, the 13th Five-year Pre-research Project of Civil Aerospace in China, the China Postdoctoral Science Foundation under Grant No. 2019TQ0291, the Aeronautical Science Fund under Grant No. 2018ZCZ2002, the Opening Fund of Key Laboratory of Geological Survey and Evaluation of Ministry of Education under Grant No. GLAB2019ZR04, and the Open Research Project of the Hubei Key Laboratory of Intelligent Geo-Information Processing (KLIGIP-2018B06 and KLIGIP-2019B07).