An adaptive LIC based geographic flow field visualization method by means of rotation distance

ABSTRACT Geographic visualization is essential for explaining and describing spatiotemporal geographical processes in flow fields. However, due to multi-scale structures and irregular spatial distribution of vortices in complex geographic flow fields, existing two-dimensional visualization methods are susceptible to the effects of data accuracy and sampling resolution, resulting in incomplete and inaccurate vortex information. To address this, we propose an adaptive Line Integral Convolution (LIC) based geographic flow field visualization method by means of rotation distance. Our novel framework of rotation distance and its quantification allows for the effective identification and extraction of vortex features in flow fields effectively. We then improve the LIC algorithm using rotation distance by constructing high-frequency noise from it as input to the convolution, with the integration step size adjusted. This approach allows us to effectively distinguish between vortex and non-vortex fields and adaptively represent the details of vortex features in complex geographic flow fields. Our experimental results show that the proposed method leads to more accurate and effective visualization of the geographic flow fields.


Introduction
Visualization of flow fields is a crucial method for explaining and describing the spatiotemporal evolution of geographical processes such as atmospheric movement and ocean current movement (Sang, Gold, and Miller 2016;Zhang, Zhang, et al. 2020).The development of Earth system models and large-scale computing systems has led to the generation of increasingly complex flow fields from ecosystem simulations and climate analyses (Artés et al. 2016;Ziolkowska and Reyes 2016).Consequently, there is a need for practical visualization methods that can effectively represent geographic flow fields.
Geographic flow field data serve as a basis for visualization and can characterize a large number of typical, multi-scale, and complex spatial features within complex geographic processes (Zheng et al. 2021).These features are associated with flow field structures, including vortices, surges, and turbulence, which are closely related to the evolutionary laws of geographical processes (Tian, Cheng, and Chen 2020).Topological centroids in two-dimensional flow fields can be classified as vortex points, repulsive convergence points, attractive convergence points, saddle points, repulsive intersection points, or attractive intersection points, as shown in Figure 1 (Craig, De Kock, and Snyman 2001;Nishimura, Ogilvie, and Pangeni 2015).As a core and high-value feature, the extraction of vortices is crucial for the visualization of two-dimensional geophysical flow fields.
Numerous flow field visualization approaches have been proposed to present 2D vortex features, such as direct visualization, geometric visualization, and texture-based visualization (Zhang, Gong, et al. 2020;Liu et al. 2022).Among them, texture-based visualization is the most commonly used, which maps raw sampling data to dense textures, producing comprehensive and continuous visualization results that express the global spatial structure of the flow fields (Yu, Regenauer-Lieb, and Tian 2019).Recently, the vector field texture visualization method based on line integral convolution (LIC) (Forssell and Cohen 1995), which continuously expresses the global directional features of the vector field through the texture convolution of integral streamlines, has gained popularity in modern graphics hardware for its superior space-filling ability and high efficiency (Wang, Duckham, and Worboys 2016;Qin et al. 2019).However, existing LIC methods have limitations in terms of the quality of geographic flow field visualization.Firstly, they adopt the same convolution strategy for all sampling points and lack a dedicated flow field feature extraction mechanism (Höhlein et al. 2020), which leads to highly resembled important features such as vortices and saddle areas and blurred boundaries between features (Scheele, Yu, and Huang 2021).Secondly, LIC methods mainly rely on the resolution of the input texture and use uniformly sized integration steps (Prada et al. 2018;Rojo and Günther 2019), which can result in loss of detail when the texture resolution is too low, and dense and redundant low-value feature representations when the texture density is too high (Du et al. 2020).
To address the above problems, we propose an adaptive LIC-based 2D visualization method for complex geographic flow fields.Our contributions are as follows: (1) We propose a rotation distance framework for characterizing geographic flow fields.The rotation distance is calculated from the spatial position and vector direction of the flow field.It is better than flow field information entropy because it improves the extraction quality of the topological structure (especially vortex and saddle points) and addresses the lack of feature extraction in existing LIC-based visualization methods.(2) We present an adaptive LIC algorithm based on rotation distance, enabling multi-frequency noise textures and adaptive streamlines to present the high-value information in the geographic flow dataset.This approach allows for accurate and efficient visualization by scientifically displaying multi-scale geographic features in dense flow fields, reducing the problem of insufficient high-value information and redundancy of low-value information due to a single resolution.
(3) We apply our implementation to global 2D wind field simulation data provided by the NOAA (National Oceanic and Atmospheric Administration) and demonstrated the effectiveness and superiority of our method.
This paper is structured as follows.After briefly reviewing previous studies and summarizing the disadvantages of the traditional LIC algorithm in Section 2, Section 3 describes the concept and calculation framework of rotation distance in the geographic flow field as the core of this method.The adaptive LIC using rotation distance is specifically presented in Section 4, including constructing multi-frequency noise textures with rotation distance and LIC calculation based on adaptive integration step size.We experimentally verify the effectiveness of the method using NOAA's global 2D wind field simulation data and fully discuss the results and performance in Section 5. Finally, we summarize this paper and describe future research directions in Section 6.

Vortex extraction
The vortex is one of the critical features of geographic flow fields with a typical spatial structure and needs to be accurately represented in the visualization process (Karimova and Gade 2016).Existing approaches to the study of vortex extraction around spatial structures can be divided into those based on the critical point method (Huang et al. 2022) and those based on the information entropy method (Deng et al. 2022).The critical point is the center of the vortex topology.Based on critical point theory, the coordinate positions of the candidate critical points are solved using bilinear interpolation for two-dimensional regular grid data of a global flow field.The types of these points are determined based on the Jacobian matrix (An et al. 2021), which in turn determines the topology and identifies the vortex.However, this method only enables the extraction of key points and lacks a distribution description of boundary information to continuously delineate between vortex and non-vortex regions.
In order to quantify continuous flow field features, the information entropy method has been proposed.Flow field information entropy is a statistical value of the probability of the vector direction distribution within the neighborhood of any flow field sampling point (Bulusu and Plesniak 2015;Shien-Tsung 2015).This method can effectively highlight information about the variation of the vector direction distribution within the flow field (Zhuang et al. 2016).A higher value of flow field information entropy within a region represents a higher degree of directional change, which is a sign of the emergence of a vortex (Young-Long 2019).However, the magnitude of the flow field information entropy is not substantially related to the topology of the vortex (Ni, Stoffelen, and Ren 2022).This is mainly reflected in the high information entropy at both vortex points and saddle points when it is difficult to support accurate extraction.In contrast, our vortex extraction by means of rotation distance emphasizes the difference in the neighborhood vector direction distribution of geographical flow field features, including vortices and saddles.

2D flow field visualization
Many studies on flow field visualization have been performed, and they can be divided into direct visualization, geometric visualization, and texture-based visualization (Zhou, Chen, and Gong 2016;Graser et al. 2019).Direct visualization methods are global and directly map flow field data through image forms such as color coding, dot icons, or particles (Tu et al. 2019).These methods are simple to draw, but their visual expression presents discretization.Geometric visualization methods express the flow field distribution by extracting the geometric shapes of the vector line or vector surface from the flow field (Chi and Gu 2022).Texture-based visualization methods include noise texture methods and texture advection methods (Hoarau and Christophe 2017).Noise texture methods involve filtering the noise texture of points in the direction of the flow field to produce a visual texture image of the flow field.The larger the noise is, the more blurred the filtered texture is and the more obvious the defects are when dealing with regions where the flow field changes more dramatically (He et al. 2019).Texture advection refers to a visualization method that moves the grid vertices or pixels in the texture along the direction of the flow field to characterize the motion of the vector field (Wang, Duckham, and Worboys 2016;Mandal and Goel 2022).Texture advection expresses the motion information of the flow field only through the deformation of the texture particles, which results in large image deformation when the velocity of the flow field varies greatly, leading to blurred images and poor visualization.
With the improvement of computing power, the Line Integral Convolution (LIC) method has been proposed as a texture visualization technique.It involves traversing all the sampling points in the flow field, generate front-to-back symmetric integrated streamlines along the flow direction of the sampling points, and perform the convolution operation on the input noise texture values at all positions on the streamlines using the selected convolution kernel (Liu et al. 2011;Han, Tao, and Wang 2018).The visualized images obtained by texture convolution of integral streamlines can express the global directional features of the flow field in detail, so LIC is the most widely used and improved method.For example, enhanced LIC performs contrast enhancement processing on LIC texture images through secondary convolution, high-pass filtering, histogram matching, and pseudo-color enhancement, which significantly improves the visual quality (Elhag and Alshamsi 2019).Researchers tend to pre-extract features as regions of interest prior to LIC processing (Baddock et al. 2007).(Tang et al. 2018) used the flow field information entropy to extract regions of interest for improved streamline generation, which could then visualize the flow field effectively.However, due to the diverse spatial features of geographic flow fields, which involve multi-scale and irregular distribution, it is difficult to visualize them accurately and effectively simply by superimposing information entropy and LIC.To address these challenges, a rotation distance-driven LIC-based geographic flow field visualization method is proposed.

Rotation distance
In the visualization of geographic flow fields, identifying flow features, especially multiscale eddies, is crucial (Hellsten et al. 2021).We propose a novel approach that uses rotation distance as a constraint index to extract vortex features and improve the quality of flow field visualization.

Definition of rotation distance
In flow field feature identification, the topology composed of sampling points and neighborhoods is important information to distinguish vortices and non-vortices (Domino, Sakievich, and Barone 2019).For the lack of flow field information entropy in topological structure expression, we define the rotation distance, which considers the data and position vectors.The counterclockwise angle between the position vector and the data vector of the sampling point in the flow field; when P 0 is the center point of the ideal vortex, its rotation angle is denoted as u I .
u = [tan −1 (P i , P 0 P i )|P 0 , P i ] (2) where the data vector P i is the flow field vector value at the neighboring sampling point P i , and the position vector P 0 P i is the position difference between P i and P 0 .

Calculation of rotation distance
This subsection presents a series of mathematical formulas for the specific calculation of the defined rotation distance.The calculation steps, as shown in Figure 2(b), for the rotation distance include the following: Step 1: Determine the calculation boundary of the rotation distance, including the sampling point P 0 and its neighborhood F N×N .
Step 2: Calculate the counterclockwise rotation angle between P 0 and sampling point P i (i = 0) in F N×N : where (x P 0 , y P 0 ) are the coordinates of the sampling point P 0 , (x P i , y P i ) are the coordinates of the neighborhood sampling point P i , F(x P i , y P i ) is the data vector of flow field F at P i , u(P i ) is the counterclockwise rotation angle between the data vector P i and the position vector P 0 P i , and Step 3: Preprocessing of u(P i ): considering that the counterclockwise included angle of the vortex center may be 270°or 90°, for the convenience of unified calculation, the value of u( Step 4: Calculate the rotation distance RD P 0 : calculate the distance between the counterclockwise rotation angles of the sampling points in the flow field and the counterclockwise rotation angle in the ideal vortex center point, as shown in Equation ( 7).
where n is the number of sampling points in F N×N and u I is the counterclockwise rotation angle of the center point in the ideal vortex.
Step 5: Normalization processing: Due to the heterogeneity of the flow fields, the numerical range of the corresponding rotation distance is different.Therefore, it is necessary to normalize the rotation distance, as shown in Equation ( 8).
where RD max is the maximum rotation distance in the flow field, RD min is the minimum rotation distance in the flow field, and RD ′ P 0 is the normalized value of the rotation distance at the sampling point P 0 .
In summary, the rotation distance is the statistic of the difference in counterclockwise rotation angle between the ideal vortex and global flow field considering the data vector and the position vector.Calculating the rotation distance requires only one parameter, the neighborhood radius N. Therefore, the rotation distance can accurately and efficiently identify and extract features such as vortices and non-vortices.

Overview
In this section, we introduce a 2D geographic flow field visualization method that leverages the proposed rotation distance to achieve high precision and time-efficient visualization.The method utilizes rotation distance as a quantitative indicator to extract the vortex and adjust the integration step size of the LIC algorithm.This enables fine and adaptive visual representation of the flow field areas with rich changing features.The method comprises the following components: (1) Multi-frequency noise texture construction constrained by rotation distance.The vortex region is extracted according to the rotation distance and used to constrain the construction of multi-frequency noise as the input texture of LIC, when the difference in texture resolution can characterize the difference in flow field information.
(2) Dynamic streamline generation adjusted by rotation distance.The rotation distance is the basis for setting the integral step size when generating the flow streamlines to improve the texture refinement of the vortex and reduce the calculation of the non-vortex.
(3) Adaptive LIC-based 2D flow field visualization.The texture of the sampling points on the streamline is convolved using ramp-function convolution to obtain the LIC grayscale texture and further color mapped to generate the 2D flow field (Figure 3).

Multi-frequency noise texture construction constrained by rotation distance
The texture resolution generated by general LIC is unified and cannot accurately represent the temporal and spatial variability of the geographic flow field.Therefore, we generate a texture with multi-frequency noise based on the rotation distance and use it as input for the texture convolution, which is equivalent to sampling a variable grid of flow field data.
With the ideal vortex as the simulated flow field, the construction of textures with multi-frequency noise is shown in Figure 4. First, each sampling point in the ideal vortex field is traversed, and its rotation distance is calculated according to Section 3.2 to obtain the rotation distance field.Meanwhile, the high-frequency white noise of the same size as the ideal vortex is generated.Immediately thereafter, the high-frequency white noise will be converted to multi-frequency noise by Gaussian filtering in the vortex region extracted from the rotation distance field.Finally, histogram equalization of multi-frequency noise is performed to improve the contrast of the noise texture.The Gaussian filter is shown in Equation ( 9), and the standard deviation of the Gaussian filter is shown in Equation ( 10 where G(x, y) is a two-dimensional Gaussian distribution function, s(P i ) is the standard deviation of the Gaussian filter applied at the sampling point P i , and s(V j ) is the unified value of the standard deviation of the Gaussian filter at all sampling points in the vortex region set V j .
For sampling points P i with a small rotation distance in the two-dimensional flow field, a lower standard deviation is adopted to obtain high-frequency noise, or Gaussian filtering is not carried out at the sampling point P i .For sampling points with a larger rotation distance in the two-dimensional flow field, a higher standard deviation is used to obtain low-frequency noise.By adapting the standard deviation base on the rotation distance, a multi-frequency noise texture can be generated that accurately captures the intricate features of the flow field.

Dynamic streamline generation adjusted by rotation distance
The integral step size in general LIC is fixed, leading to a competition between computational accuracy and computational efficiency.To solve this problem, we design an integral step size selection mechanism for adaptive vortex features.Specifically, a smaller integration step size is selected from vortex features with a smaller rotation distance, and a larger integration step size is selected from the non-vortex features with a larger rotation distance.The dynamic adjustment of the integral step-size is shown in Equation (11).
where D min [ (0, 1] is the minimum step-size of global integration, D max [ (0, 1] is the maximum step-size of global integration, RD ′ P i is the normalized value of the rotation distance at the integral point P i , and D i is the integral step-size at the integral point P i . Furthermore, we generate the streamlines using the fourth-order Runge-Kutta (Huq et al. 2019).Specifically, we let the dynamic step-size be involved in the calculation of the coordinates of the integration point with the formulas shown in Equation ( 12).
where P i = (x P i , y P i ) are the coordinates of the integration point, F(P i ) = (u P i , v P i ) is the vector value at P i , and P i+1 = (x P i+1 , y P i+1 ) are the coordinates of the next integration point.
For any sampling point, the smaller the rotation distance, the smaller the integral step size in the streamline calculation, and the finer the integral calculation.The larger the rotation distance is, the larger the integral step, and the coarser the integral calculation.By adapting the integral step-size based on the rotation distance, the generation of streamlines can effectively mitigate the impact of extraneous low-value information on visualization computation.

Adaptive LIC-based 2D flow field visualization
The LIC algorithm is essentially a filtering method.For a sampling point in the flow field, depending on the flow speed of particle tracking, integral steps along the forward and reverse directions generate a flow line of some length, including sampling data points.Then, the filter kernel is defined on the streamline, and the convolution operation is performed on the input noise texture to obtain the output texture value of the sampling point.
Based on this, we select sampling points on the dynamic streamlines and use the multi-frequency noise texture as the input texture value for the sampling points.Using the ramp function as the convolution kernel, an adaptive LIC algorithm is proposed, as shown in Equation ( 13) (Edmunds et al. 2012).13) where (2n + 1) is the total number of sampling points on the integration streamline, T in (P i ) is the input texture value of the i-th sampling point P i on the streamline, and T LIC (P 0 ) is the output texture value of the sampling point P 0 .
With the help of GPU parallel computing, simultaneous streamline integration and texture convolution are performed for each sample point in the 2D flow field data to obtain adaptive multiresolution LIC grayscale textures.Further linear synthesis of the rotation distance with the convolutional texture gray values according to linear color mapping, as shown in Equation ( 14) (Hoarau and Christophe 2017), was performed, and the color vector texture image as the 2D visualization results of the geographic flow field was obtained.
where R(P 0 ) (r,g,b) is the RGB result of the color mapping, R HSL (P 0 ) (r,g,b) is the RGB result of a colorspace conversion of (hue, sat, light), hue is determined by the rotation distance RD ′ P 0 , and b is the linear synthesis coefficient.A better texture effect can be achieved if b is 0.4.

Experiments and analysis
This section describes the dataset, the experimental setting, and the method's performance.First, the rotation distance of the experimental data is calculated, and the vortices are extracted based on the rotation distance distribution field.Then, the input textures are generated, and the settings of the integration step size are adjusted according to the constraints on the rotation distance to visualize the 2D wind field with the improved adaptive LIC algorithm.At the same time, this method is superior in determining the quality of vortices, the clarity of visualization, and the system's efficiency compared to conventional methods.

Experimental environment and dataset
The experimental environment of the algorithm is a 2.4 GHz processor PC with 8 GB RAM, and Visual Studio Code used under 64-bit Microsoft Windows 10.The implementation is based on JavaScript and GLSL, which is efficient enough to enable two-dimensional rendering.
The global 2D wind field simulation data provided by the National Oceanic and Atmospheric Administration (NOAA) of the United States are used as the experimental dataset that contains detailed information about the wind patterns across the surface of the Earth (https://www.ncei.noaa.gov/products/weather-climate-models/global-forecast).Specifically, the dataset is provided in the form of gridded data, with each grid cell representing a specific location on the Earth's surface.The wind field data is typically provided in units of meters per second, and includes information about both the speed and direction of the winds at each grid cell (see Figure 5).And the detailed specifications of the dataset are shown in Table 1.

Rotation distance calculation for the wind field
In this section, we calculate the rotation distance field of any sampled point in the wind field data based on the defined rotation distance.This includes (1) the calculation of rotation distance, (2) the expression of individual vortices, and (3) the identification of vortices from the overall wind field.
First, calculate the counterclockwise rotation angle according to Step 1, Step 2, and Step 3 of Section 3.2.Figure 6(a) shows the distribution of u for the vortex points and the saddle points, and it is clear that the differences are significant.In this case, the angle near the vortex point always remains approximately 270 degrees, but near the saddle point, it shows a large variation.Immediately thereafter, we followed Step 4 and Step 5 to calculate the rotation distance with neighborhood radius N = 51, and the distribution along the radial direction in any given vortex is shown in Figure 6(b).The rotation distance at the center point of the vortex is 0; outward along the radial direction, the rotation distance presents an approximately linearly increasing distribution until the rotation distance approaches 1.
Then, we followed Step 4 and Step 5 to calculate the rotation distance.Finally, we calculated the rotation distance and the flow field information entropy for the vector fields of the 2D wind field simulation data, as shown in Figure 8.Compared to the flow field information entropy results, which lack unclear boundaries and redundancy of spatial features, the rotation distance field is able to identify more obvious and accurate vortex and non-vortex regions.Therefore, the rotation distance proposed in our research performs better in the extraction and identification of features such as vortices in a geographic flow field.

Adaptive LIC-based 2D wind field visualization
Based on the rotation distance calculation results, multi-frequency noise texture construction, dynamic streamline generation, and adaptive texture convolution were performed on the global 2D wind field simulation data according to the adaptive LIC algorithm steps proposed in Section 4. The relevant parameter settings are shown in Table 2, where the standard deviation of Gaussian filtering σ is set as 0.5, 1.5, and 4.0, the Gaussian filter with a standard deviation σ range of [0.5, 4.0] for images from (256 × 256) to (1024 × 1024) in size gives fast and effective low-frequency noise of good quality (Zhou et al. 2019).To ensure accuracy and efficiency, the adaptive step-size D i in streamline integration is taken in the range [0.25, 1.0].As color mapping tends to reduce the contrast of line integral convolved texture images afterwards, the linear synthesis coefficient β for color mapping by rotation distance is set to 0.4 (Zheng et al. 2022).
With the help of GPUs, simultaneous streamline integration and texture convolution for each sample point in 2D wind field data is performed.The rotation distance is further linearly synthesized with the convolved texture gray values to express the overall wind distribution in terms of wind field texture lines, while color mapping is used to highlight the feature of the vortices, as shown in Figure 9(a).In the streamline integration calculation, the integration step-size is adjusted according to the region of vortex and non-vortex divided by the rotation distance, adapting to the accurate texture representation and balancing the calculation accuracy and efficiency.Compared to the traditional uniform resolution (low and high resolution) LIC, as shown in Figure 9(b), the adaptive LIC method generates 2D visualization results of wind fields with moderate texture line distribution, high contrast, and clear visibility of changing vortices.In addition, it can effectively display the spatial and temporal features of wind fields.
In addition, in order to verify the adaptability and performance of the method proposed in this paper, adaptive LIC visualization of wind fields for random scenes was carried out for each of the three image sizes (256 × 256, 512 × 512, 1024 × 1024) and analyzed in comparison with the traditional LIC method, as shown in Figure 10.The conventional method has a dense distribution of texture streamlines with poor contrast, resulting in some variation features such as disturbed and blurred vortex points.In contrast, the adaptive LIC performs better, using rotation distance as the basis for the color mapping, with red and yellow implying significant wind direction and velocity variations in the area.At this point, the combination with texture lines can effectively present a hierarchy of wind field variations.
The detailed comparison of the visualization results between the proposed method and the traditional method is shown in Figure 11.The results show that for the vortex region, which is rich in flow field information (as shown in the red box), the wind field texture obtained by the traditional method at low resolution is more turbulent and blurred.When the resolution is increased, for nonvortex regions with less vorticity information (as shown in the black box), the wind field texture lines obtained by the traditional method are dense, single, and even redundant.Geospatially, wind fields are complex, and their texture lines vary greatly in density over a given spatial extent.
The visualization method of adaptive LIC solves this problem well, as shown in Figure 11(a-d), and is able to visualize the vortex feature of a 2D wind field in focus based on vortex volume information.
At the end of the research, we compare the wind field mapping performance of the method in this paper with the traditional method.As shown in Table 3, the adaptive LIC method shows a huge advantage in terms of efficiency when driven by CPU computation.

Discussion
This paper proposes an innovative method for visualizing 2D geographic flow field.The method involves designing a rotation distance framework that takes into consideration the structure and features of spatial variation in the flow field.Additionally, the paper proposes an adaptive LIC   algorithm by rotation distance for accurate and efficient visualization.There are more obvious advantages over existing traditional methods, as follows: (1) From a perspective of vortex extraction and expression, the rotation distance considers the flow field's complex structure and direction, enabling the clear extraction of vortex and non-vortex regions.It fully considers the topology of arbitrary sampling points, resulting in more accurate results.
(2) From a perspective of visualization performance, adaptive LIC based on rotation distance produces a clear vortex image with appropriate texture distribution that reflects the flow field's characteristics.It satisfies the demand for real-time mapping in a network environment while balancing the contradiction between visualization accuracy and efficiency.

Conclusion and future work
This paper proposes an adaptive LIC-based method that uses rotation distance to visualize 2D vector data of geographic flow fields accurately.The approach involves precise extraction and expression of vortex features and improved line integral calculation.The rotation distance is a novel concept that considers both informational and topological distance within the vector field, enabling identification and extraction of well-bounded vortex features.Using the vortex region as a constraint, multi-frequency noise textures are generated to accurately capture flow field features with high-value variation.Additionally, the rotation distance is also used as a reference for setting the integration step size in line integration, making the texture resolution adaptive to reduce the impact of other redundant, low-value information on the amount of visualization computation.This method offers clear advantages in handling complex spatial structures in geographic flow fields.Its capability to process global data and accurately present detailed information on wind field characteristics, such as vortices and turbulence, provides valuable support for meteorological hazard description and prediction, as well as other applications.Additionally, the method can be used to visually represent and analyze the spatial and temporal motion characteristics of common geographic flow fields, including oceanic, riverine, and glacial flow fields.
We also summarized the current limitations of our proposed methods and outlined promising future work.
(1) The uncertainty in neighborhood radius (N) setting.Although the effectiveness of rotation distance in distinguishing between vortex and non-vortex currents has been demonstrated, we have not explored the value of neighborhood radius N. Future work will focus on improving this aspect of our method.
(2) The deficiencies in the representation of particular feature.Our method focuses on the spatial structure of the vortex region.It is able to represent wind direction and wind speed information well but lacks particular information such as temperature and humidity.We plan to develop strategies to fuse data and optimize the mapping in order to express richer features of the geographic flow field.

Figure 1 .
Figure 1.Conceptual diagram of the topology of flow field features.(a) Vortex point.(b) Repulsive convergence point.(c) Attractive convergence point.(d) Saddle point.(e) Repulsive intersection point.(f) Attractive intersection point.

)
Definition 3.1: Rotation Distance (RD): the distance between the u distribution in the neighborhood F N×N of any sampling point P 0 in the flow field and the u I distribution of the ideal vortex center point, as shown in Figure 2(a).Definition 3.2: Ideal vortex: the vortex with regular vorticity formed by the motion of the flow field in the ideal state can be simulated by F(x, y) = (−y, x) in the XOY coordinate system.Definition 3.3: Sampling point (P 0 ) and neighborhood (F N×N ): any sampling point in the flow field and the N × N neighborhood associated with it.Definition 3.4: Counterclockwise rotation angle (u):

Figure 2 .
Figure 2. Schematic diagram of the rotation distance.(a) Schematic diagram for concept of rotation distance.(b) Schematic diagram for calculation-steps of rotation distance.

Figure 3 .
Figure 3. Flow chart of the rotation distance-adaptive LIC 2D flow field visualization method.

Figure 4 .
Figure 4. Sample construction of rotation distance-constrained texture construction with multi-frequency noise.(a) Ideal vortex.(b) Rotation distance distribution field.(c) High-frequency white noise.(d) Multi-frequency noise.(e) Texture with multi-frequency noise.

Figure 5 .
Figure 5. Schematic diagram of the experimental dataset with both global view and grid cell details.
Figure 7(a) shows the results with neighborhood radius N = 19, 35, 51 and 81.In this process, regions with normalized rotation distance values of 0 are mapped as red, and regions with normalized rotation distance values of 1 are mapped as blue.Following the value of the rotation distance, the center, neighborhood, and boundary of the individual vortex can be finely expressed.It is worth noting that the value of N does not affect the extraction of vortex centers in terms of the boundary between vortex and non-vortex and that the vortex center becomes more focused as the value of N increases.Additionally, compared with flow field information entropy, as shown in Figure 7(b), the value of the flow field information entropy depends on the number of subsectors divided (m); the smaller the number of sectors, the more obvious the saw tooth phenomenon, and the greater the error of the flow field information entropy for the vortex extraction.Therefore, from the point of view of individual vortex expression, the rotation distance distinguishes well between vortices and non-vortex regions and plays a vital role in the subsequent visualization of the geographical flow field.

Figure 6 .
Figure 6.Results of counterclockwise rotation calculation.(a) Distribution of u near vortex points and saddle points.(b) Distribution of the rotation distance of the vortex.

Figure 7 .
Figure 7. Results of vortex expression.(a) Results of the rotation distance calculation.(b) Results of the flow field information entropy.

Figure 8 .
Figure 8. Results of wind field feature expression.(a)The vector field of wind field.(b) The flow field information entropy of wind field.(c) The rotation distance distribution field of the global wind field.

Figure 9 .
Figure 9. Results of global wind field 2D visualization.(a) Adaptive LIC 2D flow field visualization.(b) 2D flow field visualization using low-resolution LIC and high-resolution LIC.

Figure 10 .
Figure 10.Comparison of the visualization on different image sizes.(a) Adaptive LIC 2D flow field visualization.(b) Traditional LIC 2D flow field visualization.

Figure 11 .
Figure 11.Detailed comparison of visualization results between adaptive LIC and traditional LIC.

Table 3 .
Comparison of performance between adaptive LIC and traditional LIC.