An Approach to Identify Urban Waterlogging on a Deltaic Plain using ArcGIS on CHD based Flow Accumulation Models

9 10 The gradient for any point on the land surface can be calculated using the digital-elevation 11 model. Some empirical correlations are available to determine the gradient of any points. A 12 few studies were conducted for hilly forest areas to determine the aspect and gradient of 13 various points using computational hydrodynamics (CHD) based techniques. On a plain 14 surface, the accuracy of such techniques was rarely verified. The application of such 15 techniques for a plain surface is also extremely challenging for its small slope. Therefore, the 16 prime objective of the present study is to find out an advanced technique to more accurately 17 determine the gradient of various points on a plain surface which may help in determining the 18 key areas affected by run-off, subsequent flow accumulation, and waterlogging. Here, 19 Kolkata city as a deltaic plain surface is chosen for this study. Upto 600 m × 600 grid sizes 20 are used on the DEM map to calculate the run-off pattern using a D8 algorithm method and 21 second-order, third-order, and fourth-order finite difference techniques of CHD. After finding 22 out the gradient, the run-off pattern is determined from relatively higher to lower gradient 23 points. Based on the run-off pattern, waterlogging points of a plain surface are precisely 24 determined. The results obtained from all the different methods are compared with one other 25 as well as with the actual waterlogging map of Kolkata. It is found that the D8 algorithm and 26 fourth-order finite-difference-technique are the most accurate while determining the 27 waterlogging areas of a plain surface. Next, true gradients of waterlogging points are 28 calculated manually to compare the calculated gradient points using each method. This is also 29 done to determine the relationship and error between the true and calculated gradient of 30 waterlogged points using various statistical analysis methods. The relationship between true 31 and calculated gradients is observed from weak to strong if the D8 algorithm is replaced by 32 the newly introduced fourth-order finite difference technique. Better accuracy and stronger 33 relationships can be achieved by using a smaller grid size.


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Urban migration and agriculture can be considered as major causes of landscape  These five types of methods were quantitatively compared by taking three consecutive 77 horizontal and three consecutive vertical grid cells. The differences between the true aspect 78 and the estimated aspect calculated using six methods as mentioned above were also 79 calculated. The basic purpose of the study was to visualize these methods, which have a 80 much smaller deviation than the true values of the slope and aspect. Skidmore (1989) 81 concluded that there was almost no difference between third order method and multi 82 regression models for calculating slope (gradient) and aspect. Between these methods, the 83 FD3O method i.e. the finite difference technique of third order was found to be most 84 accurate. 85 It is noteworthy to mention that using computational hydrodynamics to solve the partial 86 differential equation and thereby determining the run-off pattern in a plain surface was hardly 87 ever done earlier. The challenge to determine the flow accumulation and waterlogging 88 areas on a plain surface is more than a semi-hilly or hilly surface because of the much 89 smaller deviation of the elevations in between the consecutive grids. Here it is intended 90 to find the usability of the following methods to determine the potential flow 91 accumulation areas thereby the waterlogging areas on a plain surface. Like Skidmore,92 here first three methods are evaluated for a plain surface. The fourth-order finite-difference technique (FD4O) is a new method, which was 98 not used by Skidmore (1989) or others. The basic purpose of introducing this method is 99 to check whether a more accurate result can be obtained than the second or third-order 100 finite-difference technique.
101 Skidmore (1989) found the FD3O method to be the most accurate only for hilly 102 forest areas. From the literature reviews, it is clear that till now any accurate technique 103 was neither proposed nor verified to identify the waterlogging areas on a plain surface.

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To find out the waterlogging area, one needs to first calculate the gradient values. find out which method is more accurate on a plain surface. Out of these four methods, it 115 is also found out which method is the most suitable for a plain surface.     189 190 In this method, the gradient is calculated as the path to the highest drop from the centre cell to 191 the nearest eight cells as given in Fig. 2  Here ∆x is the smallest spacing between grid positions in the plain (x) direction, spacing 216 ∆y is the smallest distance between the grid points in the crosswise (y) path, and i and j 217 indices are must not the side-line columns or rows. Here, k is the elevation (z) wise index.

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For the points at the end of a row of columns, equations 4-5 obtained using a polynomial 219 technique with the second-order difference (Anderson 1995), have been used to calculate 220 gradient components.  where n = 1,2,3,….. The gradient (tan G) is then defined as For the points at the end of a row of columns, equations 9-10 obtained using the 238 polynomial technique with the third-order difference, have been used to calculate 239 gradient components. The gradient is calculated using equation 6 as described before. Again, similar to 243 the D8A method, gradient tan G is calculated for all the points in the zone.

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The next method is the finite difference of fourth order (FD4O) technique by which the 248 gradient is calculated. The first step is to calculate (δz/δx)i,j,k and (δz/δy)i,j,k which are 249 calculated by the following equations 11-12 derived using CHD and Taylor series technique.
For the points at the end of a row of columns, equations 13-14 obtained using the 253 polynomial technique with the fourth-order difference, have been used to calculate 254 gradient components. The gradient and run-off pattern are again calculated as described above.

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As per the research methodology described in the earlier section, first, we need to determine  The elevation data of various points of Kolkata is extracted from the software FishNET 273 is indicated below. Using the elevation data from Fig. 4, the gradient is calculated using the 274 D8A and FD2O methods.

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In the D8A method, initially, the 1200 m × 1200 m grid has been used. This is explained 276 using equation 1. In Fig. 4, the elevation data of Kolkata has been extracted from DEM. All
364 365 366 Fig. 11. Run-off pattern using FD2O method. estimate the gradient points. After determining the gradient points, a run-off pattern is also 380 calculated and shown in the next figure (Fig. 12). and (δz/δy)i,j,k, the gradient is calculated using the formula described in equation 6. The run-401 off pattern is next determined and shown in the next picture (Fig. 13).

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Before going to analysis of the flow pattern obtained using all four methods, the actual 411 scenario of the waterlogging areas of Kolkata is depicted below in Fig. 14

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The flow pattern from all these methods using 600 m × 600 m grids is further analyzed to 426 determine the flow accumulation pattern in Kolkata. To find out the same, the following 427 principle is followed. First, all the points in the grid system are given proper nomenclature. In 428 the next picture ( Fig. 15) a, b, c…aa, bb, etc. are denoted as rows and 1, 2 3… are progressive 429 numbers along each row for 600 m × 600 m grid (Fig. 15). Based on the flow direction as As an example, in Fig.16(a) flow is coming from nearest all eight (8) cells, then 437 relative flow accumulation potential is described as 8. If the next any cell to the nearest 438 eight-cell is also contributed to the above particular cell, the same is also added in the 439 overall tally while finding out water logging potential. In Fig.16  In Table 1, we have identified three types of waterlogging areas, mild, moderate and massive.

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These names are given according to our postulation that a particular centre cell will be called

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The accuracy of the above methods can also be evaluated by the gradient of the respective 495 points. Random points can be chosen over the DEM for the calculation of gradient points.

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However, for ease of identification, we have chosen the zones as mentioned in Table 2. The 497 gradient can be calculated by all methods using equation 6 whereas the true gradient value 498 can be calculated from the contour map. The calculated gradient of the selected zones already 499 calculated using the above methods is indicated in Table 3. True gradient magnitudes are

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The gradient of all the points then calculated using equation 6 as described before and the 512 ratio to the calculated gradient (C g ) and mean gradient values (C gm ) are calculated (Table 3).  Next, true gradient points are found out, which is the ratio to the difference between the 522 elevation of adjacent two contour lines (sloping downwards) and the difference in distance 523 between the points while calculated by drawing a bisector from the respective points as

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Values of true and calculated gradients as obtained based on the above are graphically 543 compared and shown below in Fig. 23. The values of the true gradient (Tg) and calculated 544 gradient (Cg) are normalized by their mean values Tgm and Cgm, respectively.  It can be seen that there are some errors in the magnitude of 0.01 between true and calculated 551 gradient. The same can be seen in the following bar chart (Fig. 24) for all the methods.  The true gradient is calculated manually and a 600 m × 600 m grid is used. Hence error 559 between these values is quite evident and cannot be ruled off. However, if we have 560 considered in a particular zone using 100 m × 100 m cell or even smaller, the chances of error 561 of these values can be further minimized. 564 565 It is important to mention the monotonic function before describing Spearman's correlation.

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A monotonic function is one that either never increases or never decreases as its independent 567 variable increases. The monotonic function can be three types.  where n= number of points for the variable in question.

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To find out the relation between the locations wise true and calculated gradients in 587 each method, the Spearman correlation coefficient has been calculated.         Table 8.

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The RMSE value was found minimum for the FD4O method.    As the accuracy level is increased from second-order to fourth-order finite-