This initial stage treats the data as images to extract the maximum amount of useful information after applying cleaning, filtering, and masking processes (i.e., image processing techniques). As input, the CRO²A pre-processing stage primarily requires simulated spatio-temporal datasets (see Fig. a and b) containing simulated GHG concentrations for a specific region (see Fig. c), generated using the resources described in Sect. .

The following description corresponds to the sequence of procedures within the yellow dashed line box in the flow chart of Fig. . The first transformations are applied sequentially to reveal relevant information and aid in the data cleaning process.

The variables of the selected tracer (the GHG to be analyzed) and the corresponding latitude and longitude coordinates delimiting the study region are defined and stored in matrices projected according to the Coordinate Reference System (CRS), here WGS84/pseudo-mercator.

In CRO²A, it is possible to select a subregion for analysis (i.e., an area within the boundaries of the input dataset without recompilation). This can be done graphically or manually by specifying only the minimal and maximal coordinate quadrants of the latitudes and longitudes.

Tracer data are converted into images in line with the proposed approach to leverage this representation. This is equivalent to treating the data as matrices allowing fast indexing, where each element specifies the color of a pixel in the image.

The row and column indices of the elements determine the centers of the corresponding pixels and, in turn, the associated concentration value.

The converted datasets are stored in a three-dimensional array whose dimensions define the image height and width and the number of images equals the length of the time vector. For example, from a (150×201×768) dataset, 768 frames or images (equal to the time vector length) can be obtained, as shown in Fig. a, each with 150×201 pixels (height and width, respectively).

In the pseudo-color plots, the logarithmic values of each matrix from the arrays are displayed as colored flat surfaces in the x–y plane. This procedure can be used for visualization and verification; however, each matrix can also be directly converted to a grayscale image without first converting it to a true-color RGB (Red–Green–Blue) image.

This approach reduces storage requirements, as only one channel (grayscale) is needed instead of three (RGB) to represent the same information. In addition to storage savings, this method can significantly reduce the processing time of the proposed optimal design scheme and ensures that the result is independent of the color space chosen for graphical representation. For these reasons, every matrix in the array is converted to a grayscale image, as shown in Fig. a, with values in the range [0, 1] corresponding to pixels from black to white, respectively.

Once this transformation is applied to each image, a datastore is created to hold the entire collection. The datastore enables rapid processing without exceeding available memory, particularly when handling a large number of images.

A further advantage of using datastores is that transformations can be applied to the entire collection simultaneously rather than individually in sequence.

To isolate pixels with intensities close to the target concentrations-defined by an instrument sensitivity or through contour analysis (as described below)-it is recommended to adjust image contrast by uniformly modifying pixel intensity values according to their values.

This adjustment shifts them toward either the bright or dark range, depending on the data and user-defined target, and is achieved through contrast stretching. Two contrast adjustment options are proposed: (i) logarithmic rescaling of pixel intensities according to concentration intensities (see Fig. b), and (ii) histogram equalization (see Fig. c).

In both cases, an optimal thresholding process is required, which can be considered a separate topic of study. The advantages and disadvantages of each method depend on the concentration values of interest, the size of the resulting dataset, and the associated processing time. Typically, the equalized version includes a larger proportion of the data, resulting in longer processing times than the scaled version. However, the latter should not be discarded; its use will depend on the particular application.

The differences between the two options are shown in Fig. , which presents the histogram transformations for versions of the same processed image. These figures highlight an implicit duality between the two transformations. Brighter pixels (intensity values close to 1) are more prominent in the equalized version (see Fig. c) than in the rescaled version (see Fig. b).

Such pixels are the focus of this procedure, though it is important to note that revealing more pixels increases the data volume and processing requirements, as already mentioned.

In this case, by default, contrast adjustment is performed by logarithmic rescaling, using the value obtained from contour analysis of the concentration intensities in each image as a threshold. Once the grayscale image contrast is adjusted, the image is binarized (see Fig. a) using the well-known Otsu method , which determines an optimal threshold corresponding to the minimal intraclass variance from a 256-bin histogram. Pixels with intensities above the global threshold are set to 1 (white), and those below are set to 0 (black). Both contrast adjustment and binarization define the search (solution) space for each image. These transformations isolate the information of interest-white pixel regions with intensity 1 in Fig. a – so that only target GHG concentrations are analyzed, while remaining areas are ignored as irrelevant or noise.

Once the binarized image collection is obtained, masking is applied to exclude all bodies of water or other undesired areas (when visible at the given resolution and dimensions), as these locations should not be part of the search space. In the illustrative example, the mask in Fig. b is used to analyze only CO₂ concentrations within the Grand-Est region.

Thus, the solution space for each image, initially defined through earlier transformations, is further restricted to account for the topographic conditions of the study area.

The masking procedure uses an equally binarized image with dimensions matching those of the dataset. Leveraging the binary values (1 and 0) that correspond to Boolean true and false, a logical AND operation is performed via a dot product between each image and the mask. This operation is inherent to computing systems and is computationally inexpensive compared with alternative methods.

As one of the final pre-processing steps, all resulting binarized images are accumulated (superimposed), producing a scoring matrix whose elements are the sums of pixel intensities at each position. The scoring matrix represents the spatial distribution of significant concentrations, expressed as the frequency of occurrence at each location. For example, for annual simulations (365 days), a matrix element with a value of 75 indicates that the location exhibits substantial (since they are of considerable importance, value, and therefore measurable) GHG concentrations for 75 % of the simulation period (approximately 273 d). The scoring matrix (Fig. ) depends directly on the pre-processed data and is used in the clustering process for weighting during both processing and post-processing stages.

Finally, before initiating the processing stage, the reference raster postings to geographic coordinates and the datastores are saved. The distance metric (squared Euclidean distance by default), maximum number of iterations (ite), repetitions (run), replicates (rep), and saturation value (sat) are also defined; some of these serve as stopping criteria for decision-making and optimization steps (their use is described later).

This intermediate iterative stage aims to identify similar or dissimilar groups by clustering the pixels detected in each image from the previous stage. The resulting groups are characterized according to the patterns that generated them (i.e., pattern recognition techniques are applied to the data). The following description corresponds to the sequence of procedures within the blue dashed box in the flow chart shown in Fig. .

The CRO²A processing stage requires the output images from the pre-processing stage (see Fig. a), the scoring matrix (see Fig. ), and the number of clusters under consideration (k), corresponding to the number of ground-based measurement stations.

Using the binarized images, the transformed dataset identifies pixels with an intensity of 1 (colored and marked pixels in Fig. ). For each image, these data points define the search space for the solution, as they represent the locations of the relevant concentrations (targets).

To apply the clustering process to these data points, the clustering algorithms must be initialized by defining starting points (i.e., a priori positions of ground-based measurement stations) that act as initial centroids in iteration zero. This initialization is commonly performed by selecting k points uniformly distributed within the search space, representing the simplest among several possible options.

Each clustering problem constitutes an optimization problem, and the present development assumes a global solution in both search space and values. To reduce the risk of converging to a local minimal, and given that the underlying algorithms (k-means and k-medoids) are sensitive to initial conditions, the starting points for clustering are defined using information from the marginal distributions of each image width (horizontal axis) and height (vertical axis). These are obtained as univariate histograms (see Fig. ). These marginal distributions show the vertical and horizontal densities of pixels (i.e., data points) per row and column, respectively.

This considerably reduces processing time and the number of iterations, as the starting points are typically located close to the highest concentrations. This positioning guides the search vectors during solution exploration, thereby requiring fewer iterations to converge to an optimal solution. The size of the starting point matrix is k×2, determined by the number of clusters tested (k) and the number of features-in this case, 2, corresponding to latitude (y-projection) and longitude (x-projection) for each point. In other words, the number of starting points equals the number of ground-based measurement stations in the monitoring network.

Once the pixels are identified, a primary grid is superimposed on the image, with the number of divisions in both height and width equal to k, resulting in k2 cells (see vertical and horizontal black lines in Fig. ). For each division in width and height, the maximum value of the corresponding portion of the marginal distribution is selected.

A secondary grid is then drawn, representing the locations most likely to host ground-based measurement stations (intersections of the green dashed lines in Fig. ).

Depending on the image characteristics, up to (k-1)2 starting point candidates can be obtained (i.e., the maximum number of internal intersections between the horizontal and vertical grid lines). However, even though the theoretical maximum is (k-1)2, some candidates are discarded if they do not belong to the search space (i.e., identified pixels). In some cases, certain cells may contain no identified pixels from the previous procedure, reducing the number of available starting points to less than (k-1)2.

If the number of available candidates is greater than or equal to k, exactly k are randomly selected and ordered. To diversify the clustering process, a starting point array of size k×2×k can be created by assigning random permutations of the selected points. This strategy allows the clustering process to be replicated with a different ordering of starting points, to which the clustering algorithms are also sensitive.

If the number of candidates is less than k, those options are discarded and the starting points are instead selected uniformly at random within the search space.

Since the number of candidates for the first analyzed image in the illustrative example is greater than k=7, the randomly selected starting points are shown as black circles in Fig. . Each selected starting point can be verified to belong to the identified pixels.

Before starting the clustering process, if any ground-based measurement station is already installed in the analyzed region and the user wishes to include it, its location can be added to the dataset by applying a multiplicity concatenation equivalent to 2.5 % of the total number of images.

The clustering algorithm then partitions the dataset into k clusters and returns: a vector containing the cluster index of each data point; the k cluster centroid locations; the within-cluster sums of point-to-centroid distances; and the distances from each point to every centroid. In other words, the clustering process characterizes the data points identified in each image according to patterns defined by the distance relationships between each data point and the centroids. Such characterization can be used to identify the most appropriate cluster for each point, considering all possible clusterings that meet an optimal criterion. The optimal criterion is based on the distance between a pixel and its cluster centroid, the distances between centroids, and the scoring matrix value (weight) at the pixel location.

This stage operates under the hypothesis that the location of the ground-based measurement station (the cluster centroid) should be as close as possible to the source of GHGs (highest concentration) to ensure accurate measurements and improve subsequent data inversion, if required. This is the same hypothesis used to define the clustering starting points. The gradual formation of the seven clusters is shown through the changing colored groups of identified pixels.

In Fig. a, the processing results for the first image in the dataset are shown after 27 iterations. The seven clusters obtained (k=7) are displayed in different colors. The x–y projection of each cluster centroid is indicated by black triangles, representing the optimal locations for ground-based measurement stations according to the first image (see Fig. a).

The relationship between the randomly selected starting points and the pseudo-global centroids confirms the hypothesis of spatial closeness and convergence proposed through the use of marginal distributions.

It should be noted that the optimal clustering is obtained without requiring clusters to contain the same number of members; instead, it depends on the distances between centroids and members, as well as the scoring matrix values at the member locations. Maximizing inter-centroid distances produces well-separated clusters that can form, expand, contract, and exchange members dynamically. Centroid tendencies align with regions of highest GHG concentration and correspond to dense pixel areas in the scoring matrix.

In this case, initializing starting points using the proposed strategy results in a low number of iterations to reach an optimal solution, as outcomes tend to follow the stated hypothesis. Convergence is thus relatively fast, reducing computational cost and reinforcing the hypothesis. As a result of the processing stage, after clustering each image in the dataset, the resulting centroids (see Fig. b), referred to as local centroids, are stored for use in the post-processing stage. The centroids shown for the first image in Fig. a are elements of the cluster sets in Fig. b, or the set of local centroids. In total, 5376 local centroids are identified in Fig. b, as the illustrative example comprises 768 frames with k=7 clusters each.

This final stage performs a general cluster analysis, estimates clusters, determines centroids, and displays optimal results. In addition to pattern recognition techniques, optimization and operations research methods are applied to the results collected in the processing stage. The following description refers to the sequence of procedures within the dashed red box in the flow chart in Fig. .

As input, the CRO²A post-processing stage requires the output data from the processing stage (local centroids of the images in Fig. b) and, as in the processing stage, the scoring matrix and the number of clusters to be evaluated (k). Although the processed data from both stages appear similar, they represent different information. In the processing stage, the spatio-temporal distributions of GHG concentrations were used (sets of images such as those shown in Fig. a).

In the post-processing stage, the collection of local centroids from each image is used (a single image as shown in Fig. b).

Building on the learning from the processing stage, the cluster analysis here again considers the large dataset size; however, it now processes representative entities (local centroids) rather than individual data points analyzed previously.

Once the overall results of the new clustering process are obtained (global centroids; see Fig. a), i.e., after clustering the data from the processing stage, percentage scores for each ground-based measurement station location and for the monitoring network are calculated using the scoring matrix, as shown in Fig. b. Promising permutations of the global centroids are then used to further explore the search space. Alternative solutions are tested around the general solution, i.e., reclustering with different permuted and modified starting points. This approach is analogous to forcing the Pareto profile in a multi-objective optimization, but at low computational cost. After at least 25 repetitions, to ensure adequate statistical reliability, the clustering with the highest mean score for a specific value of k is considered the optimal solution (k*), provided it outperforms the general solution; otherwise, the general solution is retained due to the lack of improvement.

Finally, CRO²A outputs the geographic coordinates of the k* ground-based measurement stations, their individual scores, and the score of the optimally selected monitoring network.

It should be noted that in clustering algorithms (both k-medoids and k-means), one of their characteristics is that the number of k clusters used to represent the dataset is predefined before the algorithm starts.

Consequently, the algorithm iteratively modifies the locations of the centroids of these predefined clusters k, based on the calculation of the mean, which reflects the density distribution of the dataset. Given this characteristic, a trend analysis is implemented using the basic sequential algorithmic scheme (BSAS) . This approach is proposed to address the persistent and subjective question of the minimum number of clusters required for adequate clustering (i.e., the number of ground-based measurement stations in the network and their corresponding locations in the region under analysis).

Through this analysis and batch processing, as illustrated in the first panel in Fig. , an interval of clusters is tested sequentially and incrementally (in the example, [2, 15]), providing information on the overall performance of the monitoring network as a function of the number of ground-based measurement stations.

The results from this analysis are fitted to a sigmoidal curve, and the procedure presented by is applied to determine the optimal threshold (k*). This threshold indicates the minimal number of ground-based measurement stations required for the monitoring network (see first panel in Fig. ). The sigmoidal function selected in this case is particularly useful for modeling artificial neural networks. It typically describes complex systems characterized by learning curves with rapidly ascending intermediate transitions from low levels to a saturation point at high levels after a significant increase in the independent variable. A specific case of the sigmoidal function is the logistic function, widely used in regression models. The logistic function representing the fitted model and its first two derivatives are used to calculate the optimal threshold (see the first and second panel in Fig. , respectively) according to the procedure proposed by , which makes use of intersections between certain straight lines (including the slope line at the midpoint of the logistic curve, obtained by means of the derivative in the second panel in Fig. ) to calculate the baroreflex threshold and saturation points. It should be noted that the performance represented is normalized; therefore, the vertical axes of this figure and its first two derivatives, shown in Fig. , are dimensionless.

Since the main metric is based on the scoring matrix, and in this matrix a value of 100 % means that at that location a ground-based measurement station will be in the middle of a signal of considerable intensity for the entire simulation time (depending on the dataset used).

Therefore, a performance score of one means that each and every ground-based measurement station is located at points where signals of considerable intensity are present at all times. In other words, a performance score of 1 means that, according to the optimality criteria of CRO²A, the location obtained cannot be improved.

Note the tendency towards saturation in Fig. , which, although analyzed in more detail later, coincides with one of the conclusions of the methodology proposed by , according to an incremental scheme, since no matter how much the number of ground-based measurement stations increases, after a certain number, they do not contribute significantly to an improvement of the system.

According to Fig. , the illustrative example developed through these three stages shows that the minimal number of monitoring towers for the masked region corresponds to the calculated threshold. The optimal value for the overall performance of the network is approximately 57.14 %, as shown in Fig. b. As presented in these sections, complete processing of these large datasets is achieved through the training and validation of a machine learning system capable of generating optimal designs for atmospheric GHG monitoring networks based on such datasets. This unsupervised machine learning scheme adapts proportionally and progressively to the amount of data available during processing, improving its performance even under uncertainty.

As noted at the outset, CRO²A is structured in three stages, each functioning as an analysis module. Partial results are saved after each stage, allowing processing to be paused and resumed later. It is also important to note that once pre-processing is completed, it need not be repeated for either a single monitoring tower configuration or for trend analysis.

Finally, Fig.  presents the algorithmic complexity analysis of processing times for the three stages of the CRO²A design scheme. Figure a shows the processing times as a function of the number of images in the pre-processing stage and their corresponding curve fitting. The linear relationship between these variables is notable. Despite this characteristic, for datasets with larger numbers of images, the pre-processing stage does not impose a higher computational cost due to the use of datastores. Figure b shows the processing times for all three stages as a function of the number of clusters (or ground-based measurement stations). The processing times for the pre-processing stage remain constant, as they are independent of the number of clusters (k). The most evident variations occur for 2<k<7, primarily due to hardware adjustments during GPU initialization. The processing times for the processing and post-processing stages differ by at least two orders of magnitude, owing to the large number of images handled in the processing stage of the design scheme.