projection<\/a> of the flow.\u00a0Consider a parallel flow with an angle of 45 \u00b0 (or a multiple of 45 \u00b0).<\/p>\n![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/06\/hydro41-300x300.png?resize=300%2C300\")
<\/p>\n
In the image, pixel A is the top of the slope.
\nThe upstream flow surface will be calculated by a diagonal of pixels, contacting by their corners.\u00a0For the pixel located at the top and on the right, it will receive, according to the calculation the rain fell on the three pixels located diagonally, up to the top.
\nActually it should also include a part (half) of each pixel located to the left and right of these upstream pixels.\u00a0The upstream flow area is therefore underestimated by a factor of 2 when using the D8 method.
\nWhen using D8, the calculated upstream surfaces for a parallel flow are accurate only for flows that occur precisely in a cardinal direction (0 \u00b0, 90 \u00b0, 180 \u00b0, or 270 \u00b0), are underestimated by a factor of two for flows that occur precisely in a diagonal direction (45 \u00b0, 135 \u00b0, 225 \u00b0 or 315 \u00b0), and are below \u00a0the correct value by a factor between 1 and 2 for the flows that occur between the cardinal directions and the diagonals.<\/p>\n
For a divergent flow as the one we have in the semi-spherical DEM of the previous articles, results are quite different.\u00a0Let’s first see how the D8 method works in this case.<\/p>\n
![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/06\/hydro42-300x300.png?resize=300%2C300\")
<\/p>\n
In the previous image, we consider that the pixel A is the top of the mountain, whether spherical or conical.\u00a0Since we must get out of this pixel downwards, let\u2019s suppose the method decides the flow is towards the NE pixel.
\nThe arrows indicate the directions defined for the other pixels.\u00a0Now consider the pixel B. None of the upstream pixels (West, South West and South contiguous pixels) are flowing to it.\u00a0Therefore the method considers that there is no SCA, watershed, for this pixel.\u00a0It is only the rain that falls on the pixel that will feed the downstream pixel.
\nIn fact, its SCA is drawn on the following image:<\/p>\n
![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/06\/hydro43-300x300.png?resize=300%2C300\")
<\/p>\n
Actually, the pixel must \u00a0receive all the rain falling in the triangle formed by the top of the mountain and its two \u201clateral\u201d corners. \u00a0Instead, the D8 method allocates the pixel an SCA = 0.<\/p>\n
Let’s see what happens with the pixel to the left of B.<\/p>\n
![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/06\/hydro44-300x300.png?resize=300%2C300\")
<\/p>\n
In the case of an SCA substantially equal to that of the pixel B, we will obtain a computed SCA corresponding to the two pixels located towards the South West, thus SCA = 2.\u00a0This is due to the fact that this pixel is on a direction that is a multiple of 45 \u00b0 from the top.<\/p>\n
This phenomenon will replicate throughout the slope.\u00a0We will have pixels at the bottom of the slope that, as pixel B, will have virtually no SCA, and others that will accumulate all along the slope.\u00a0In the case of spherical or conical mountain flow, all the pixels at the bottom of the slope should have exactly the same SCA and the same cumulative flow.<\/p>\n
In the case of \u00a0a convergent flow, as would be the case of a conical crater, we find the same type of problem but with discrepancies between the privileged directions (0,45,90, …) and the others, but \u00a0less marked.<\/p>\n
In light of what has just been said, you can now better understand the result that we obtained in our other articles for the flow using the D8 method.<\/p>\n
![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/06\/hydro45-296x300.png?resize=296%2C300\")
<\/p>\n
The white\u00a0\u201cfeatures\u201d\u00a0\u00a0in the privileged directions are the pixels that have recovered the SCA from their upstream pixels, the black areas at the edge of the image correspond to the pixels that happen to be as our pixel B of the example.<\/p>\n
Rho8 method: Analysis of the Problems<\/strong><\/p>\nThe Rho8 method attempts to solve one of the D8 method problems, the deviation of the flow direction modelled towards a cardinal or diagonal direction.\u00a0This is the result of the arbitrary structure of the grid where one can only choose one of these 8 directions.<\/p>\n
The method used is based on chance and probabilities, which is never easy to explain.\u00a0We will first try the real explanation:
\nThe Rho8 method introduces a stochastic component in the D8 method, which\u00a0makes\u00a0it\u00a0possible to better maintain the flow direction along a given slope and direction.\u00a0As in the case of the D8 method, each pixel emerges in one of its eight neighbours.\u00a0The choice of the reception pixel among the neighbours is performed stochastically.\u00a0A probability p is assigned to one pixel, and a probability 1-p to a neighbouring pixel.<\/p>\n
The Probability Attribution System and the purpose of this method are illustrated in the following example: Consider a flat surface with a 30 \u00b0 slope (measured counter clockwise from the East).\u00a0 According to the D8 method, the flow will be towards the neighbouring pixel located at NE.\u00a0The flow direction will, therefore, have a 15 \u00b0 difference with the true direction of flow.\u00a0The Rho8 method assigns a probability p to the flow toward the NE neighbour and a probability of 1′-p to the flow toward the other possible neighbour, the east pixel.
\nAs you go down the slope, some pixels will flow to their NE neighbour, and the rest will spill over to the east \u00a0neighbour.\u00a0If the number of pixels that flow towards NE relative to the East neighbour respects the correct proportions (the expected proportion is p \/ (1 – p)), the resulting flux lines will have an overall direction of 30 \u00b0.\u00a0The value of p shall be such that the expected value of the flow path direction is equal to the angle of the slope (\u00ab\u00a0aspect\u00a0\u00bb).<\/p>\n
In more grotesque terms, if the conditions are maintained (length of the flow, etc.) the hazard causes the orientation to be equally modified in one direction as in the other and to get back, at the arrival, to the original orientation.<\/p>\n
Although this method provides (in mathematical expectation) the appropriate flow path direction, all other problems identified for method D8 persist.\u00a0But in addition, the Rho8 method introduces its own problems: chance does not guarantee reproducible results, and in locations with parallel flows, the adjacent flow paths are not parallel, but randomly deviated, which causes, often, a convergence of flows that should be parallel.\u00a0This happens in particular on flat surface portions, where the flux paths should remain parallel and where, instead, we find pixels with strong accumulations.<\/p>\n
Once two flow paths have merged because of their random deviation, there is no mechanism that can cause them to diverge again.\u00a0As you move down the slope, the errors increase as the flow becomes more and more concentrated.<\/p>\n
In the end, some pixels will have overestimated SCA values, while others that have been neglected by the randomly deviated flow streams will depict underestimated values.<\/p>\n
![\"\"](\"https:\/\/i0.wp.com\/www.sigterritoires.fr\/wp-content\/uploads\/2018\/06\/hydro46-300x300.png?resize=300%2C300\")
<\/p>\n
MFD method: Analysis of the Problems<\/strong><\/p>\nThe multi-directional methods attempt to solve the major limitation of the D8 method, i.e. the one-dimensional representation of the flow, by distributing fluxes from one pixel to all the neighbouring pixels that are located below it.\u00a0The rule used is to distribute the respective flows to those neighbours which change, according to the method used.<\/p>\n
We will not discuss the details of the\u00a0 MFD method which is based on the directional slope of the pixels, but, let\u2019s just say that in any possible scenario, each pixel will flow to three or four other pixels depending on the slope.<\/p>\n
Besides, it will receive the flow of three or four neighbouring upstream pixels, and this flow will not be complete (as in the D8 method), but partial.
\nOn an inclined plane, whether conical or circular plan, if we draw the upstream zone of each pixel, we can observe that it constitutes an inverted triangle, with the vertex situated on the pixel considered and the base on the upstream boundary of the plan.<\/p>\n
The SCA value calculated for a pixel (the contributory surface or upstream basin) is correct, if and only if, it is far enough from the edges of the area covered by the DTM.\u00a0If this is the case, the triangular zone is included in the DTM, otherwise the triangular region is incomplete and the SCA is underestimated.<\/p>\n
A parameter of the method (the convergence) makes it possible to offset to a certain degree this problem. For a mountain in the shape of a circular cone, a value of 1.1 has given very good results.\u00a0The worry is that this \u00ab\u00a0calibration\u00a0\u00bb is based on the geometric symmetry of the surface and the measurement will be compromised since the natural surface of the ground deviates from the symmetrical geometry of the surfaces used for the calibration of the method.<\/p>\n
Conclusion (if there is one …)<\/strong><\/p>\nWe will conclude here this series of articles, not because the subject can be considered closed, but on the contrary, because the subject can be considered definitely open.<\/p>\n
The purpose of these articles is not to provide the answers, but to ask the questions.\u00a0Once asked, these questions, we will approach the hydrological modelling of a field more humbly.\u00a0And, to warn everyone to continue his journey trying to answer the new questions that will, undoubtedly arise.<\/p>\n
<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
In the last three articles, we have discussed an example of application for each of the main methods for calculating the flow of a watershed.\u00a0Although they are quite self-explanatory, we must go through a little theory…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"give_campaign_id":0,"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"sfsi_plus_gutenberg_text_before_share":"","sfsi_plus_gutenberg_show_text_before_share":"","sfsi_plus_gutenberg_icon_type":"","sfsi_plus_gutenberg_icon_alignemt":"","sfsi_plus_gutenburg_max_per_row":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1260],"tags":[],"class_list":["post-5536","post","type-post","status-publish","format-standard","hentry","category-non-classe-en"],"aioseo_notices":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6XU0A-1ri","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/5536","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/comments?post=5536"}],"version-history":[{"count":0,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/posts\/5536\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/media?parent=5536"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/categories?post=5536"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sigterritoires.fr\/index.php\/wp-json\/wp\/v2\/tags?post=5536"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}