Another Fast and Simple DEM Depression-Filling Algorithm Based on Priority Queue Structure

Transcription

1 ATMOSPHERIC AND OCEANIC SCIENCE LETTERS, 2009, VOL. 2, NO. 4, Another Fast and Simple DEM Depression-Filling Algorithm Based on Priority Queue Structure LIU Yong-He 1, ZHANG Wan-Chang 2, and XU Jing-Wen 1 1 Key Laboratory of Regional Climate-Environment Research for Temperate East Asia, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing , China 2 Center for Hydro-Sciences Research, Nanjing University, Nanjing , China Received 20 April 2009; revised 27 June 2009; accepted 27 June 2009; published 16 July 2009 Abstract Some depression cells with heights lower than their surrounding cells may often be found in Grid-based digital elevation models (DEM) dataset due to sampling errors. The depression-filling algorithm presented by Planchon and Darboux works very quickly compared to other published methods. Despite its simplicity and delicacy, this algorithm remains difficult to understand due to its three complex subroutines and its recursive execution. Another fast algorithm is presented in this article. The main idea of this new algorithm is as follows: first, the DEM dataset is viewed as an island and the outer space as an ocean; when the ocean level increases, the DEM cells on the island s boundary will be inundated; when a cell is inundated for the first time, its elevation is increased to the ocean level at that moment; after the ocean has inundated the entire DEM, all of the depressions are filled. The depression-removing processing is performed using a priority queue. Theoretically, this new algorithm is a fast algorithm despite the fact that it runs more slowly than Planchon and Darboux s method. Its time-complexity in both the worst case and in an average case is O(8nlog 2 (m)), which is close to O(n). The running speed of this algorithm depends mainly on the insertion operation of the priority queue. As shown by the tests, the depression-filling effects of this algorithm are correct and valid, and the overall time consumption of this algorithm is less than twice the time consumed by Planchon & Darboux s method for handling a DEM smaller than cells. More importantly, this new algorithm is simpler and easier to understand than Planchon and Darboux s method. This advantage allows the correct program code to be written quickly. Keywords: digital elevation models, depression removing, priority queue, quick algorithm Citation: Liu, Y.-H., W.-C. Zhang, and J.-W. Xu, 2009: Another fast and simple DEM depression-filling algorithm based on priority queue structure, Atmos. Oceanic Sci. Lett., 2, Introduction Numerical hydrological models are powerful tools for researching the hydrological characteristics of watersheds, especially watersheds with large areas (Dodov and Foufoula-Georgiou, 2006; Gumbo et al., 2002; Wang and Yin, Corresponding author: ZHANG Wan-Chang, zhangwc@tea.ac.cn 1998; Wolock and McCabe, 2000; Yang, 2005). Before using these models, water drainage networks derived from digital elevation models (DEM) must be prepared so that runoffs can be calculated for each cell. Information about drainage networks is derived from the DEM data, including the flow direction from cell to cell, the water drainage networks, and the associated sub-watersheds (Paz and Collischonn, 2007; Turcotte et al., 2001). Many studies focused on extracting drainage networks have been published recently (Ahamed et al., 2002; Colombo et al., 2007; Jana et al., 2007; Jenson and Domingue, 1988; Planchon and Darboux, 2002; Turcotte et al., 2001; Wang and Yin, 1998). Commonly used DEM data for extracting drainage networks are squared regular grid based (Turcotte et al., 2001), while triangulated irregular networks (TIN) are also used in some applications (Tucker et al., 2001). Below, the DEM data are referred to as regular squared grid based DEM. When extracting drainage networks from any kind of DEM data, there is a common problem of existing depressed cells with elevations lower than any surrounding cells that have no flow direction linking them to the main river channel. This is because errors or noise exist in DEM data and the actual detailed topographic structure is generalized at a lower resolution by a DEM cell. Therefore, before determining flow directions, these depressions (known as pits in other publications) must be removed by modifying the cells' elevations. This paper discusses a depression-filling method that prepares valid DEM data for the approach of using an eight flow direction matrix (D8) because the D8 approach is a common method for deriving drainage networks from raster DEM data (Jenson and Domingue, 1988; McMaster, 2002; Tribe, 1992; Turcotte et al., 2001). Similar methods for other direction-calculating methods and other kinds of DEM data can be obtained by extending the method used for the D8 approach. Some efforts have been made to resolve this depression-removing problem. Ocallaghan does this by smoothing the DEM data; however, only tiny and shallow depressions can be removed, while larger and deeper depressions remain (Ocallaghan and Mark, 1984). The depression-filling method in a real sense was first used to study the roughness of soil, such as calculating a soil s depression storage capacity using DEM data (Moore and Larson, 1979; Onstad, 1984; Planchon and Darboux, 2002; Ullah and Dickinson, 1979). The idea of filling these de-

2 NO. 4 LIU ET AL.: A FAST ALGORITHM FOR DEPRESSION FILLING IN DEM DATA 215 pressions by increasing the cell s elevation to the minimal elevation of the cells that are found on the boundary of the depression is recommended by Jenson and Domingue (1988) and Marks et al. (1984). Based on this idea, Jenson and Domingue presented their traditional algorithm. For applications with small DEM data, this algorithm can produce satisfying results. However, for applications with a larger DEM dataset, small depressions are often embedded in larger ones, and time-consuming scans performed over the entire DEM dataset must be done more than once in order to find the embedded or adjacent depressions. The time complexity of these scans is O(n 2 ). A common DEM file of about cells requires several hours to accomplish this task. Moreover, this kind of algorithm is very complex and difficult to implement. In 2002, Planchon and Darboux published their fast algorithm. This algorithm involved two stages; the first stage inundates all DEM with a thick layer of water, and the second stage drains the excess water. In the second stage, they used two methods, namely scanning the whole DEM grid and exploring cells via recursion. The former method s time complexity is O(n 1.5 ) and the latter method s time complexity is O(n 1.2 ) (Planchon and Darboux, 2002). Compared to other algorithms, this algorithm is so fast that it can decrease the time consumed from hours to seconds. Despite its fast speed and delicacy, the algorithm remains difficult to understand to some degree because it requires three subroutines and still involves recursive execution. This forces programmers to spend a great amount of time to grasp the main idea of the algorithm and test the program code. The present article describes an additional fast depression-filling algorithm that is more simple but somewhat slower than Planchon and Darboux s algorithm. This algorithm takes advantage of a commonly used data structure, the priority queue, its time complexity is close to O(n) and it is easy to understand. 2 The algorithm The idea behind this algorithm is to imagine the DEM as an island surrounded by an ocean. At the start, the ocean s level is below all of the DEM cells, and the ocean s level then increases gradually. The first inundated cells are those below the ocean s level and lie on the boundary of the DEM. When these cells are inundated, the next to be inundated are the neighboring cells. When a cell in a depression is inundated for the first time, its elevation value should be set to the ocean level at that moment. When all of the DEM are inundated, all of the depressions have been removed. Here, inundating means that the ocean s level is equal to or greater than the cell s elevation values. Each time, the ocean level should increase to the elevation value of the lowest boundary cells neighboring the ocean cells; there may be more than one such cell with the same elevation. In order to find these cells quickly, a priority queue is used. A priority queue (PQ) is a data structure support that follows two operations: (1) add an element to the PQ with an associated priority; (2) remove the element from the PQ that has the highest priority and return it. A good PQ should support quick element insertion and removal. Currently, most modern programming languages (Planchon and Darboux, 2002) have their own built-in PQ structures based on a self-balancing binary search tree. A insertion operation for such a PQ takes O(log(n)) time and the removal operation takes O(1) time. Using a priority queue, every cell neighboring an ocean cell (the boundary cell of the island) should be inserted into the PQ and its elevation value should be taken as the priority value associated with it. Each time a cell is removed from the PQ, it is used as a seed for exploring other island-boundary cells, and its corresponding priority is taken as the current ocean level. From this seed, all cells on the island boundary are explored recursively without regard to whether their elevations are lower or higher than the seed (with a height equal to the sea level). This exploration method, known as the flood-filling approach, is used in image processing and computer graphics. All of the found neighbor cells should be inserted into the PQ, and those cells with elevations below the ocean level are then set to the ocean level. Before processing, the imagined ocean surrounds the entire DEM, so that when the PQ is initialized all cells on the boundary of the DEM data set should be inserted into the PQ. Another data structure that we used is a two-dimensional integer-value or Boolean-value array with the same size as the DEM. This data structure is used to mark ocean cells with 0 and island cells with 1. The pseudo-code for this algorithm is presented in Table 1. No more than 25 lines of code are in this algorithm, which is only about half of lines of the code used in Planchon and Darboux s method. Moreover, the code is all written in one procedure, so it is very easy to under- Table 1 Pseudo-code of the algorithm. Procedure FillDepression(DEM) new PQ For row=0 to DEM.rowCount //insert first column and last column PQ.Insert(row, 0) PQ.Insert(row, DEM.colCount-1) For col=1 to DEM.colCount-1 //insert first row and last row PQ.Insert(0, col) PQ.Insert(DEM.rowCount-1, col) End For marks=new double[dem.rowcount][dem.colcount](1) //all elements initialized to 1 while PQ.length>0 cell=pq.remove() waterlevel=cell.elevation for n in neighboursof(cell) if marks[n]=0 continue If n.elevation<=waterlevel n.elevation=waterlevel End if PQ.Insert(n) End while End procedure

3 216 ATMOSPHERIC AND OCEANIC SCIENCE LETTERS VOL. 2 stand and implement. When a cell is removed from the PQ, its neighbors in eight directions are scanned, so the time complexity of the main procedure is O(8n) (n is the total number of cells in the DEM), while the insertion and removal operations in the priority queue take O(log 2 (m)) (m is number of cells at the island boundary, so it is very small compared to n). Here, the value of m varies over an entire depression-removal process, and it determines the time spent by each insertion. Each cell will be inserted into and removed from the priority queue only once, so the overall time complexity both in the worst cases and in average cases is O(8nlog 2 (m)) and is close to O(n). 3 Algorithm implementation using the C# language In popular languages like C++, Java, and Python, there is always a built-in priority queue. Actually, not every programming language has a built-in priority queue data structure available, so there is a need to write the code for a priority queue if none exists. In this section, a C# language implementation of the depression filling algorithm is described. In the Microsoft.Net Framework 2.0, there is no built-in priority queue available, but there is a generic structure named SortedDictionary (SD), which is a binary search tree with log 2 (n) retrieval, insertion or removal. Its elements are key/value pairs, and every key must be unique. The SD does not allow for multiple elements with the same key, but its elements can be defined as any collection type, such as a queue structure. The or- der of the cells with the same elevation in the priority queue is not important because the minimal value of those DEM cells neighboring the ocean is the only value required for the depression-filling algorithm. Thus, a SD structure that has a queue structure as its element value type is a good substitute for a real priority queue with no unique keys. In order to understand the pseudo-code easily, the insertion subroutine of the SD-based priority queue can be written as Table 2. When a cell is inserted, the program should first check whether the SD has a key equal to the cell elevation. If it does, the cell should be added to the queue with its key equal to the cell elevation; if not, then a queue object hould be created and push the cell into this new one, then add this queue object into the SD with the key equal to the cell elevation. When an element is removed from the SD, all cells with the same elevation are removed. With the SD substituting for the priority queue, the algorithm must be modified (Table 3). Table 2 Insertion method of the SD-based priority queue (pseudocode). Procedure SD.Insert(cell) If SD.Contains(cell.elevation) SD[cell.elevation].push(cell) Else SD[cell.elevation]=new Queue<Cell> SD[cell.elevation].push(cell) End if End procedure Table 3 The modified algorithm. Procedure FillDepression(DEM) SD=new SortedDictionary<double, Cell> For row=0 to DEM.rowCount SD.Insert( new Cell(row, 0) ) SD.Insert( new Cell(row, DEM.colCount-1) ) For col=1 to DEM.colCount-1 SD.Insert( new Cell(0, col) ) SD.Insert( new Cell(DEM.rowCount-1, col) ) End For marks=new double[dem.rowcount][dem.colcount](1) //all elements initialized to 1 while PQ.length>0 waterlevel=sd.firstkey() queue=sd[waterlevel] //get the element corresponding to the key of minvalue SD.Remove(waterlevel) while queue.count>0 for n in neighboursof(cell) if marks[n]=0 continue else marks[n]=0 end if if n.elevation<=waterlevel if marks[n]!=2 //marks value with 2 indicate that it is visited queue.enqueue(n) // but hasn t been marked as ocean marks[n]=2 end if n.elevation=waterlevel else SD.Insert(n) End if End while End while End procedure

4 NO. 4 LIU ET AL.: A FAST ALGORITHM FOR DEPRESSION FILLING IN DEM DATA Performance tests The algorithm in this paper can fill the depressions in DEM correctly (Fig. 1). The algorithm in this paper and the algorithm presented by Planchon and Darboux are implemented using the C# language within the framework of Microsoft.Net 2.0, and they are also tested using three differently sized DEM datasets with dimensions of , , and on a laptop with a 2.26 GHz Intel P8400 CPU and 768 MHz DDR-II memory. The extracted river nets of the first dataset depression-filled by these two algorithms are shown in Fig. 2. This shows that the results produced by the two algorithms are similar. In comparing the results of these two algorithms, both show parallel river lines in the flat area, but the patterns of these parallel river lines are different. This effect may be related to the processing order of cells that have the same height in the algorithms. Actually, the drainage networks in flat areas may behave very randomly, or can also be affected by human behavior. In this sense, neither of the extracted results from the two algorithms was more correct than the other. However, visually it seemed that the result extracted by the algorithm presented in this paper was more reasonable because the parallel river lines in the flat areas are not as dense as those extracted by Planchon s algorithm. The execution times of the two algorithms with the three datasets are shown in Table 4. The results indicate that these tests were not performed faster than the tests performed by Planchon and Darboux in 2001 because these new tests were executed by managed code under the Microsoft.Net Framework 2.0, which is slower than the unmanaged C-language program used by Planchon and Darboux. It appears that the execution time of the priority queue-based algorithm is similar to but slower than the time taken by Planchon and Darboux s algorithm. Compared to Planchon s algorithm, the cell numbers handled per millisecond by a priority queue-based algorithm decreased slowly as the size of the datasets increased, so this priority queue-based algorithm is not slow. In this case, it is important to note that the SD s element insertion op- Figure 1 A example of DEM with (a) depressions and (b) their counterpart whose depressions were removed using the algorithm presented in this paper. The highlighted cells are depressed cells. Figure 2 River nets extracted from the first dataset (size of ) depression-filled by the two algorithms. (a) the shadow effects of the DEM data, (b) depression-filled by Planchon and Darboux s algorithm, and (c) depression-filled by algorithm presented in this article. The other dataset is not presented here because it is very large and its extracted river channel cannot be shown clearly in this paper.

5 218 ATMOSPHERIC AND OCEANIC SCIENCE LETTERS VOL. 2 Table 4 Execution time of the 2 algorithms. Planchon and Darboux s algorithm Priority queue-based Algorithm Execution speed ratio of Planchon s Cell number algorithm to the priority queue-based Execution time (ms) cells per ms Execution time (ms) Cells per ms algorithm eration takes the same O(log 2 (n)) time as the actual priority queue s insertion operation, but its removal operation also takes a time of O(log 2 (n)), which is slower than that of the actual priority queue s removal. Like the elements operated on by SD, the queue object s insertion and removal takes O(1) time. Thus, this SD-substituted priority queue is slower than an actual priority queue. The slowness of this algorithm compared to Planchon s algorithm is mainly explained by the priority queue, which has a time complexity of O(log(n)) for insertion and removal operation of the elements. 5 Discussion and conclusions The depression-filling algorithm presented by Planchon and Darboux is quite fast compared to other published methods (Planchon and Darboux, 2002). Despite its simplicity and delicacy, this algorithm is still difficult to understand due to the three complex subroutines and the recursive execution. The additional fast algorithm presented in this article uses a new approach: first, the DEM dataset is regarded as an island and the outer space as an ocean, and as the ocean level is increased step by step, DEM cells on the island boundary are inundated. When a cell is inundated for the first time, the elevation of this cell increases to the ocean level at that time. After the ocean has inundated the entire DEM island, all depressions are filled. All of the boundary cells near the water are pushed into a priority queue, and with each step the cell with the lowest elevation is removed from the priority queue, so that it is inundated first. Theoretically, this new algorithm is also a fast algorithm, despite the fact that it runs slower than Planchon and Darboux s method. Its time-complexity is O(8nlog 2 (m)), where n represents the total number of cells and m represents the number of cells at the island boundary. Because m is very small compared to n, the time complexity is close to O(n). In this paper, a C# language-implemented depression-filling method based on the algorithm presented in this paper is introduced. Because there is no real priority queue available in the Microsoft.Net Framework 2.0, the SD structure combined with a queue structure was used to realize the functions of a priority queue. The program code was modified for use in the SD-based algorithm. The tests were performed by depression-filling three differently sized DEM, and the results showed that the algorithm presented in this paper is fast, although slightly slower than Planchon s method. The running speed of this algorithm is mainly dependent on the insertion and removal operations of the priority queue. If a real priority queue is used instead of using the structure of a SD combined with a queue object, the algorithm may be more efficient. The overall time consumption of the algorithm was no more than twice that consumed by Planchon & Darboux s method. The PQ-based algorithm provides an alternative to Planchon & Darboux s fast algorithm. The elementary coding used by this new depressionfilling algorithm requires only 25 lines, which is about half the length of the code used by Planchon s method. Furthermore, the main code for this new algorithm can be written in one procedure, while the code in Planchon s method requires three or four procedures. The use of a priority queue structure makes the algorithm easier to understand for users and programmers and will save them time. The advantages of this priority queue-based algorithm are its simplicity and intelligibility, which make it very easy to write correct program code quickly. Acknowledgements. We are very much indebted to Hu Yong-Hong for his help in providing us with DEM datasets for the algorithm tests. This study was financially supported by the National Basic Research Program of China (Grant No. 2006CB400502), the Promotion of 100 Young Talent Scientist Project of the Chinese Academy of Sciences ( ), and the Special Meteorology Project (GYHY(QX) ). References Ahamed, T., K. G. Rao, and J. Murthy, 2002: Automatic extraction of tank outlets in a sub-watershed using digital elevation models, Agricultural Water Management, 57(1), Colombo, R., R. V. Vogt, P. 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