Problem 57879. Route a hydrograph through a river section with the Muskingum method

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Problem statement </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that routes a hydrograph through a river section using the Muskingum method. The input to the function will be the inflow hydrograph </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"I(t)\"/></w:customXmlPr><w:r><w:t>I(t)</w:t></w:r></w:customXml><w:r><w:t>, the times </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"t\"/></w:customXmlPr><w:r><w:t>t</w:t></w:r></w:customXml><w:r><w:t> at which the hydrograph is reported, the routing parameters </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"K\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"X\"/></w:customXmlPr><w:r><w:t>X</w:t></w:r></w:customXml><w:r><w:t>, and the desired time step </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"dt\"/></w:customXmlPr><w:r><w:t>\\Delta t</w:t></w:r></w:customXml><w:r><w:t>. If the desired time step differs from the time step in the given inflow hydrograph, use linear interpolation to get the inflow hydrograph onto the new time base. Return the time base of the outflow hydrograph, the outflow hydrograph </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"O(t)\"/></w:customXmlPr><w:r><w:t>O(t)</w:t></w:r></w:customXml><w:r><w:t>, and a Boolean flag that says whether the time step is within the guidelines </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"2KX &lt;= dt &lt; 2K(1-X)\"/></w:customXmlPr><w:r><w:t>2KX \\le \\Delta t \\le 2K(1-X)</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Bonus points to anyone who can solve this problem using the FILTER command. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Background</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Hydrologic routing is used to predict the movement of a flood wave in a river. The Muskingum method is an example of hydrologic routing, in which the inflow </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"I\"/></w:customXmlPr><w:r><w:t>I</w:t></w:r></w:customXml><w:r><w:t> and outflow </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"O\"/></w:customXmlPr><w:r><w:t>O</w:t></w:r></w:customXml><w:r><w:t> are connected to the storage </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"S\"/></w:customXmlPr><w:r><w:t>S</w:t></w:r></w:customXml><w:r><w:t> in a channel section through a conservation of mass (or volume), which states that the rate of change of storage is the difference between the inflow and outflow:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"dS/dt = I - O\"/></w:customXmlPr><w:r><w:t>\\frac{dS}{dt} = I-O</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>This equation is written in discretized form as </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"(S2-S1)/dt = (I1+I2)/2-(O1+O2)/2\"/></w:customXmlPr><w:r><w:t>\\frac{S_2 – S_1}{\\Delta t} = \\frac{I_1+I_2}{2} - \\frac{O_1+O_2}{2}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where subscripts 1 and 2 denote the previous and current time steps, respectively. In applications, the inflow hydrograph (or flow as a function of time) is known, and we assume that everything at the previous time step is known. That assumption leaves two unknowns, the current outflow </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"O2\"/></w:customXmlPr><w:r><w:t>O_2</w:t></w:r></w:customXml><w:r><w:t> and the current storage </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"S2\"/></w:customXmlPr><w:r><w:t>S_2</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>In the Muskingum method, the storage, inflow, and outflow are further related by </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"S = K[XI + (1-X)O]\"/></w:customXmlPr><w:r><w:t>S = K[XI+(1-X)O]</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"K\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t> is a time scale related to travel time in the river section and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"X\"/></w:customXmlPr><w:r><w:t>X</w:t></w:r></w:customXml><w:r><w:t> is a weighting factor. Substituting this relation into the discretized form of conservation of mass and rearranging gives</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"O2 = C0 I2 + C1 I1 + C2 O1\"/></w:customXmlPr><w:r><w:t>O_2 = C_0 I_2 + C_1 I_1 + C_2 O_1</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"C0 = (dt - 2KX)/(2K(1-X)+dt)\"/></w:customXmlPr><w:r><w:t>C_0 = \\frac{\\Delta t – 2KX}{2K(1-X)+\\Delta t}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"C1 = (dt + 2KX)/(2K(1-X)+dt)\"/></w:customXmlPr><w:r><w:t>C_1 = \\frac{\\Delta t + 2KX}{2K(1-X)+\\Delta t}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"C2 = (2K(1-X)-dt)/(2K(1-X)+dt)\"/></w:customXmlPr><w:r><w:t>C_2 = \\frac{2K(1-X)-\\Delta t}{2K(1-X)+\\Delta t}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Notice that </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"C0+C1+C2 = 1\"/></w:customXmlPr><w:r><w:t>C_0+C_1+C_2 = 1</w:t></w:r></w:customXml><w:r><w:t>. With the assumption that the first value of the outflow equals the first value of the inflow, this equation for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"O2\"/></w:customXmlPr><w:r><w:t>O_2</w:t></w:r></w:customXml><w:r><w:t> can be used to compute the outflow hydrograph. The guideline for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"dt\"/></w:customXmlPr><w:r><w:t>\\Delta t</w:t></w:r></w:customXml><w:r><w:t> above ensures that the coefficients </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"C0\"/></w:customXmlPr><w:r><w:t>C_0</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"C1\"/></w:customXmlPr><w:r><w:t>C_1</w:t></w:r></w:customXml><w:r><w:t>, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"C2\"/></w:customXmlPr><w:r><w:t>C_2</w:t></w:r></w:customXml><w:r><w:t> are positive.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 882.7px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 441.35px; transform-origin: 407px 441.35px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.95px 8px; transform-origin: 64.95px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Problem statement </span></span></div><div style="block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 63px; text-align: left; transform-origin: 384px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.783px 8px; transform-origin: 383.783px 8px; unicode-bidi: normal; "><span style="">Write a function that routes a hydrograph through a river section using the Muskingum method. The input to the function will be the inflow hydrograph </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADUAAAAlCAYAAADiMKHrAAADL0lEQVRYR+1YO2uVQRBNekGIlSm0iEWEIELwAYKQJgEbkYCKWEctxEpBf4CCVsEiKlYWokIQsdJWEKM/QAsVbbRSEO31HNmR4bo7jy/JBwn3wuG+Zmf3zOPs7jc6sglfo5uQ08iQ1EbJap+ZOoGgfATeJINzFvaPgO/RcX2R4sJmABLLvvZhwE3gfDQgfZAioWPAXJaNsiexJ8DRCLEIqVk4mgdOAVvURL/w+T5w15iIY5eBnZnyaZCnr0XgkOcrQkrmYOk8UBOexOeHRvTH8N9n4CJwaxVZ0kOf4csn4Izlryupr3A65UTsOv4/DYyvESG6mQDeAyzl5y2/GVK34WShOKJDq0e6ZonjjjuZfYX/f1jzZ0h9gaPthdQVvF8zMnAZ/10FtjnZHHTB8toKHDR8U3iWgF3Ah5pdlJSkXXw0HRaDt+V9d6L02J/MkhcwWUvTLkpKosM1vgMsBWIJfQPuAGZDF8KU63vAZPm+H+/eBv0TNi+BagtESUkUOS93d2sTpfSyjG4Al5xMsT8ONGzO4feWanLcDqAqQlFSup88KZd+MhVqgAgjzz3QC5gMk2BU1x8hxfJ4rRbhNT+lnHtTlJT2b2VHx0HmqPZ2hJREnk5XAEuZaJMlpf17AiTEzDkipNgf7BO+In2SJSWlRAGKquWqSUm9k1SkpLKkfpeARdWS5nIQqLaClyld75GjEScU+Y8EQJQyGrA1EQqJOp15RyOZMCPpEnGe+DMneZYsTx7VcvUypfcRb6fX6sSSisgzTx7cdCMCFPbvkZJ6p8OoMtGW4sLN0Wp8OXloAWJlMHvVM11hJS3RlH+LlL4/RftJoukeOmGo/fNoNA1ErvwkTkLNcrVIaSnPyC2JydWDp+nWUUn3K0t1D+DeamHD082LEhRdkv8+10gxyhdKretBFIrHQPQW610SdaYo57S3yo5rkccDey1br6eqkUj8SCF4amQr4eqvacjfepNKPQVyGDKThwHvmNbLY+fwUyCDVMrHemdK1ikPJI/gh/CT1jI4PbYvUlwfr+FURe9WO5gwCor1KO6/BPdJKisKne2HpDqHrueBw0z1HPDO0/0BZ0i3Jokz6M8AAAAASUVORK5CYII=" alt="I(t)" style="width: 26.5px; height: 18.5px;" width="26.5" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.2167px 8px; transform-origin: 34.2167px 8px; unicode-bidi: normal; "><span style="">, the times </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">t</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 187.875px 8px; transform-origin: 187.875px 8px; unicode-bidi: normal; "><span style=""> at which the hydrograph is reported, the routing parameters </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">K</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; "><span style=""> and </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">X</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1667px 8px; transform-origin: 29.1667px 8px; unicode-bidi: normal; "><span style="">, and the desired time step </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="dt" style="width: 18.5px; height: 18px;" width="18.5" height="18"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 284.583px 8px; transform-origin: 284.583px 8px; unicode-bidi: normal; "><span style="">. If the desired time step differs from the time step in the given inflow hydrograph, use linear interpolation to get the inflow hydrograph onto the new time base. Return the time base of the outflow hydrograph, the outflow hydrograph </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="O(t)" style="width: 30px; height: 18.5px;" width="30" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 234.942px 8px; transform-origin: 234.942px 8px; unicode-bidi: normal; "><span style="">, and a Boolean flag that says whether the time step is within the guidelines </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="2KX <= dt < 2K(1-X)" style="width: 149px; height: 18.5px;" width="149" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; "><span style="">.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 251.925px 8px; transform-origin: 251.925px 8px; unicode-bidi: normal; "><span style="">Bonus points to anyone who can solve this problem using the FILTER command. </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 8px; transform-origin: 40.8333px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Background</span></span></div><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.742px 8px; transform-origin: 368.742px 8px; unicode-bidi: normal; "><span style="">Hydrologic routing is used to predict the movement of a flood wave in a river. The Muskingum method is an example of hydrologic routing, in which the inflow </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">I</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; "><span style=""> and outflow </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">O</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.9583px 8px; transform-origin: 92.9583px 8px; unicode-bidi: normal; "><span style=""> are connected to the storage </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">S</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 97.25px 8px; transform-origin: 97.25px 8px; unicode-bidi: normal; "><span style=""> in a channel section through a conservation of mass (or volume), which states that the rate of change of storage is the difference between the inflow and outflow:</span></span></div><div style="block-size: 34.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.4px; text-align: left; transform-origin: 384px 17.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="dS/dt = I - O" style="width: 72.5px; height: 35px;" width="72.5" height="35"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 141.583px 8px; transform-origin: 141.583px 8px; unicode-bidi: normal; "><span style="">This equation is written in discretized form as </span></span></div><div style="block-size: 37.9px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.95px; text-align: left; transform-origin: 384px 18.95px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="(S2-S1)/dt = (I1+I2)/2-(O1+O2)/2" style="width: 156.5px; height: 38px;" width="156.5" height="38"></span></div><div style="block-size: 64px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32px; text-align: left; transform-origin: 384px 32px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; "><span style="">where subscripts 1 and 2 denote the previous and current time steps, respectively. In applications, the inflow hydrograph (or flow as a function of time) is known, and we assume that everything at the previous time step is known. That assumption leaves two unknowns, the current outflow </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="O2" style="width: 19px; height: 20px;" width="19" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.2333px 8px; transform-origin: 76.2333px 8px; unicode-bidi: normal; "><span style=""> and the current storage </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="S2" style="width: 15.5px; height: 20px;" width="15.5" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; "><span style="">. </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 249.317px 8px; transform-origin: 249.317px 8px; unicode-bidi: normal; "><span style="">In the Muskingum method, the storage, inflow, and outflow are further related by </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="S = K[XI + (1-X)O]" style="width: 143px; height: 18.5px;" width="143" height="18.5"></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; "><span style="">where </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">K</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 185.917px 8px; transform-origin: 185.917px 8px; unicode-bidi: normal; "><span style=""> is a time scale related to travel time in the river section and </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">X</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 156.75px 8px; transform-origin: 156.75px 8px; unicode-bidi: normal; "><span style=""> is a weighting factor. Substituting this relation into the discretized form of conservation of mass and rearranging gives</span></span></div><div style="block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="O2 = C0 I2 + C1 I1 + C2 O1" style="width: 152px; height: 20px;" width="152" height="20"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.0667px 8px; transform-origin: 19.0667px 8px; unicode-bidi: normal; "><span style="">where</span></span></div><div style="block-size: 35.2px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.6px; text-align: left; transform-origin: 384px 17.6px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="C0 = (dt - 2KX)/(2K(1-X)+dt)" style="width: 135px; height: 35px;" width="135" height="35"></span></div><div style="block-size: 35.2px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.6px; text-align: left; transform-origin: 384px 17.6px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQ4AAABGCAYAAAA9zXoQAAAPSklEQVR4Xu1dbei2yRT/73fy9ok2Ch9oWSQvYdVqUVoJ5S3pCXnZkqQVPsgnxAfJBy9Jm7QWISlCUbTyWtgtEkqJT0viO+dn71/P2fPMzDkz91zXfV33/1w1Pffzv+aal9/M/ObMmTNnrrvIJxFIBBKBTgSu64yf0ROBRCARuEjiyE6QCCQC3QgkcXRDlh8kAolAEkf2gUQgEehGIImjG7L8IBFIBJI4sg8kAolANwJJHN2Q7eaDt0lJPybhwbsp8XhBHyeffkjCTRIeeUjm9/LvtyS8J5DsRyXOUyS8yMT9u/z/ixK+KuEXh/evlH9fJ+FBKu5X5PfdEq6X8HpVBkb5mfz4kYSPSPjH4Y/I08ZFfu+S8OVAmU8aJYnjpPAvmvlPJfVnSXi/hA8vmtNpEydB6oGsSwQCeWKwiL+TeE84xMV3t0r4U+FbxvuPvHuzGegPl//fq8gD6TxXEYZODkT13cMf0E6fqcQLFn+9aEkc62G9Zk66Q/YMnDXLOCMvSBq/lvBXCZ+Q8GcJD5PwCgmvUhl8T36/OJChJo4a4UIaQNotYgGZfUrlVxtnIA201e7IPYkj0Jt2GIWdm0V/jfwYFX8xg2KgfHoCDhhQVyRAbI8sIbwsMfAeIuElErgE4DfPkB8/Vwk8Xn6XpAdGAQn90YlPXD8r8d5XyJOfA7O/SKAU9PYCfvgeyyuk9Vavolt7n8SxtRY5vjwcAFgv6/V+VFy3JeDgfPbxRbvAuv52CVEJoJUlpY0XSCToH0qPJtDS4NXfaCnBLi9ABNRhYLBHln46b+g4NH6vlv/fJQG6EZTLkt4EqJdNIoljWXxPkTrWyVDeYUD9QAJnPYjqGLA9Dzv/LFF6JnFgoN8sAYOw9nCA4r1XBz3QMaCZLv79nAQsh97ZgaGVeJ4p34Lg8He0C9Kr6VB62ugkcZM4TgL7YplSRL5TcoD4qwdDzyyPzv0FCVQUstMfW/CZxBEpi9b1eMu1f0uCdmkBEn7LgSxuk39bS51SebTOBDtc2FXBEgZ5YQeoN71InVeJk8SxCsyrZYKBCdEXW4YgCrtujxAAd2NKhfbEfa+iaxMHJQ7sfjy1MVBL0gGJ85jdDuoxgAvKACkDW7ZsHw+vzb5P4ths0wwV7G/y1T0S9A4CNfdIUIvgXgacgXu+8dJcmzgoMXh1sAMc9YD04S1vvPpCArzPRDqWfL08V3mfxLEKzKtkQuWeFcm1uI6CPEKCp4zTM/DMjr42cYBIYQDXkjaAiSZX3VgztrJbStJVOsYSmSRxLIHqadLEehpPafdEr7Uj2396Bva2MXtquyZxkEgjUsN/VSVAlNBHUN/h6Ua8+rPOjDcTTy/vxd6PEAf39V8upXqyBG75oZBYx/1Ewm8kYH/9+RJqVnOLVeoSJkypotbJrUGSJ3VQz9E741rpZrQpIoO9lbZVErfi6jKj/z5GwnslYNsYD7a1HzVYEe6gaKvWY+s2WJS5n/UQBxoDgIKRAQT2pu+Q8H0J1A5b898eTf7cml2u1CBqg8SfJKG2DIHYTpL3Oi9n4Ih0opHeCnGA+PCUDMNsz6AeBH+nvYU14PLwKvU2pgGFKAzesDuD5xgi2kyvjhKH3p4DK7eMYDTL9na8zQCzo4Jw58Tr3FYBWDv8pgf/iO1HC7o1lirQKUBqiJAGymq3TGnRqqU0SiKeboh11wZjsKfBxKqVpLNxXb27RohDEwEAjGwlkcWPXR+uDsgOM8RAwSDxFIDRWZRt1ztYItAtTRy9pGF3Pex2tZbSeiZBtokeKzUDswhum4vjEQfNensVRZy1zkIRtLlWu1ogdvxop9ZieU1/wRnYmknPgGFJ4oiQBvDSUoOWKrCEsEs9u/SK9GfW0UqANi1v7M3Ae7E0vML3auN1QbHufq1pqMUqckkTpsGXJ20QHmsQZiVCPQNjZwFiO/IA4cywclyKOEAAHygMfN0tIDl/UoI+MxKxrNUGcZ7OjkRE7Gy37NEzbbpLt4hDbyN5lnebruQZF45GWpAOos8NEpESpJU69NkOiO1Pl+CdB4nmi3hLEIfedv1VpTCPlb9jMFvfGZGBbCWFmn6C5UBb1PQrEYmvB8+TxW0Rh7bd9yzvTlaBS5yx3WIdhUIPBD1ZoM1vlDBzO302cfRgYJcilhBahm7azqO0K6J3E1vpaGJGe+1WB1gjDtsgW9ACb2Wrb3SAzv5Oz5bHpK3Fb92xoTfBQJ+xRGH5ZhKHHYQeBtQDYbmGJRhOEGv7CkjVcL7DZRni4aAgbJHgSU0/jPsd+SOO2Ov3Nh18h757i4SSW0Hg/w0JM/ydeBhMe18jDr32W0K7PlKBJI4R1Lb1DR35wEBwd85rtgXlaUtTIw49m/VaD562Rpl7IpAILI5AjTj0mm6JbbnFK5YZJAKJwHIIRIgjaiMQKSXWpR+XsAsX8JEKZZxE4DIiECGOGRIH9BOwoqO9/m61yZexk2SdEwGLQI04tNHLDB0HjG/gb5HpjhBHKkez/yYCG0GgRhz6QBSKGjG1RTxYHn5JQs1idE/E8TSpB1wD5JMIJAIXF9gw+QOBqBGHPRBVM6HVgPJEIHw11tzHH0McazfejyXD562daeaXCGwUgQfoOluWo9aBa8sijpZzreP2wGNPxPFGKe+jN9qIWaxEYG0EYM6Pu3j//3iH3EAe35SgL/aBRIFEcNUezjLAGu5fEiJ3TuyJONZumMwvEdgNAh5xsCK8/AYOUmheC6XpbyV8XQK8gEWcnCRxrNM1oKPazQXG60BSzQVYRW5mO3Ext5V9lDhmlTqJYxaS9XRwXOCHEnZ19mF5WKo5wLYIdkVRj2EnLOp2sk7i2E5bzCgJSAO3t+cM2ocmJGostWeeBO4rwc5iJ3HEGwx2JO+W8BwJPFUJ4zhYwmLAth6etHypROK1ioyPNODMFtcDYrnHZSFuiOeDg4Z3SsAdprULlnHyFCc5ey6HpgOclzXSjSO0TkwaE9rTrcgdrgA0cdZOwgLzd5g6Az8sxVt30a5Tw+NzocOmyM19Q7klccRg0w5YSl9EzfKtj0t0dOxWWf2Qjgdd0hucgd3rqpGEQaX3Yh0sBu9QLLvr1/IZw+P8IGDrzIeZA/N7JZzDcYiZ7guKjbM2cdAV4Yjl6FDvmvAR/T7Ab8LnJfxTAjxKXZGg/TB4XsZRFM85LuPcLT8gmdSIxVYLuiNcZuzNlhhsGDh49Iy9R+JAHSK3pJEQ4NXdc7RNswJIHhFl/4TuNT0Ja4Pl3aEzVIC1iIMdlmdVMIt+UEJ0N2aocpM+gic0LBNK/iO0hW3kvgztIAkY2DW1dqvv2cSwej3SBkRYOubR3r72ShwRZ8K83jEyWXHQRbGPdjH6O4XZgr7XN/p9TzzrhCsqDZfy4OVr1yja1yKOnopvKW5EaaZ9l3im+S0X+byGAkR1RQIknMiDgQFDtdLVj63vz4E4UD+Nv7Vw5hIzIg0SK7TRTRJGb28rYU6Cw1Kpdp9NpK0jcYAHHn3D4qjUgb6FYxfX6M2SONpNEdna1PoPb+bWnVxb4uo16W1SpKi7vt7rEXRtz4U4ahdNceaNLveIDb/z2jIyiBlnLeLgshrSFXQ1XEpHjozY+nCSK5JuEkdP85fjasVb65oCq8yDdAJ9ybcPDYzG5c5KtFS6o3g7OzbNcyEOqzfiUgCzJaS23is62E49UorXXmsRB3RdkBAgfeolS4+ko29tRL2KBJrE4TW5/54Sh+d+QM+MiAu7ga8dkq9p+r3cOfhHnEmfC3EAI+ox8BvYXi8Bd7beKiEqvWms4QFvpmf/NYiDeWjCi1z/oOut3WnYvveAs2pJHN7Q9N9zp6h1CNB2bihSZ2yFcsCMtOM5EYf1eI4ZFne21mxevFYFccxwYMV81iAO6mb0bXS9ynuWl1ejVMlzpMN5oF+m9zS08aQNYKLvqdEYHaP15gwx0o7nRBzAs6Uk7e2TwBUK51kK0qWJg/3Q9iW7NRvZWdJL6upkONLhehvhnOOD5XHGwbMP0NuGmA1vl4A7PPiMar2TOK5iqMXsyNZ4q18eg2sp3aWJA8tl2OWUdGy9t8dpKaW6S5jEMU5r6AzQUYAEvANlenanCKw7+uh6+pgOPkPi0GmMIjlDCWltF1CWY3ZFRnDVNwOMYuHdTVtKl1JFzdbIuy/Ypsm6N6XoJI6xJvYay6aqL+/m1pg1Xhrp6CMdnGU7F+IgjiBf7fZhlIyBD9oLT49tzKmIAxICyLcl9Wr7IY+cWI/mEjqJo584aN2Jg2mR28hK24U07oqYTLdKyO9H2nEGcfSjN/cLGs1hBwVWuC+UcJfKYgQXfL4n5Sh0O/dIaFmkWlOA2iSlJ7PmTt0osHObfz+p9ZIGaqY1/vbi414x0iJ1mbdj2RbARG+71ozsenoZiOMYpbXNaykdB5doEaWnXhrXpA7qQ9xrX5M44t0pShqIpw9IeWKinvl7lXrskN5WcKmWe5c4MBBuKIjoJX1SvJXvvyAa29wjmNbyWYo4epZUdsu6pJDnktrdik7iiHcpEMBDHZEQ6034YsVt6Hw8Ixy7ZdarLGwdwmvVbs/EQTIuidNWiuvVHXFXwTt3FO85V8mox4LTS59kFJE2Sn2xtHV73yEi9XDoI5BCrjGiS+Lwmuf+99x2hYUnzMRLzy2HWUpviUU7sR7ErphoMh89lLVX4iBptJYSWhndu+SAJIOnxyGS14uWkDho/IfdD5y6jTywTakdftMSCcgWjshvllB01ZDE4cOtlxpebLt2tNuVtVnM7rD07Ajw216zc73m7Zm1PAyWfK9tEloSgXW8FLWTIdHPXKYAj9nEYSekUcz14TfdV9H/bpRQdaWYxNGG3PP8Zb/mAERHeZME7f4PcTE74G4KLmUQD5IKrpjQM0EpbqukUUc+dKUHF4Y2P3SWrforBU53mDJbLIEPlIW4ga/kVhCSB+xuWu4KMHiATc82bGTQziaOnsmsVT69dNISB7ACFtVzPkkckWbffhwQAga9Z8G6/ZqcroRLYkhHPqjdzCXQydBK4jgZ9NMzjjgdmp7pGSUIqQ22OVqxfUbVm1uVJI65eJ46NYiX5+Kpe00sE7dOtJM4OgHbQXRIHlVt+A7Kv3YRE68BxJM4BkDbwSdQxv1Swl49da8JMZSCvd7T1izfJvNK4thks2ShEoFtI5DEse32ydIlAptEIIljk82ShUoEto1AEse22ydLlwhsEoEkjk02SxYqEdg2Av8D+CrNdJVBoswAAAAASUVORK5CYII=" alt="C1 = (dt + 2KX)/(2K(1-X)+dt)" style="width: 135px; height: 35px;" width="135" height="35"></span></div><div style="block-size: 35.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.8px; text-align: left; transform-origin: 384px 17.8px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="C2 = (2K(1-X)-dt)/(2K(1-X)+dt)" style="width: 135px; height: 35.5px;" width="135" height="35.5"></span></div><div style="block-size: 66px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 33px; text-align: left; transform-origin: 384px 33px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35.3917px 8px; transform-origin: 35.3917px 8px; unicode-bidi: normal; "><span style="">Notice that </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="C0+C1+C2 = 1" style="width: 108px; height: 20px;" width="108" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 289.742px 8px; transform-origin: 289.742px 8px; unicode-bidi: normal; "><span style="">. With the assumption that the first value of the outflow equals the first value of the inflow, this equation for </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACYAAAAoCAYAAACSN4jeAAACeklEQVRYR+1XPSiFURi+d1f+JhYDg0EW+VmslMiGTEr5mUwUgwyiGCSDn2RUlMHIakKKUgwMFiai7DxPnaP3nnu+73vPjXvFvfX0Hd93znmf87x/Rzr1S3/pX8orVSQW6pmiYv9esQ4oMAw0AvVCjVuMV4GNUIXc+aEx1owN1oBW4BRYAfbMpiRLUiT6BLQD97kSDCG2BCOTxtAMnoseoxV4dw1UGXINeL7kQk5LjKr0JZCy9qcxWDB/LOM59VPEJKktGBlNMESXHpk5dGn1TxAbw6brwojGNZIYl1bm4s44VzJeHoASQ2wATxvocSK4xLTrMvaMIyZdyAzsUp5cqkxj30qsFhveiSNEZaFPuU28HBEfWjA+D42zKMXczeuwsbYm3WCuLbrvGNcolVa5Um5ON7YpT8y4fBZzQ9aqiH2IWZoSYafLGsZ3ISGQSIxt50zMCgneR6xj1eePNcwtL+wePcLV+xiP+1zti7Fc091dR4OymTPLy4AdoByYMAS9cegj5saJVjEZl64StvnLkiP7apbLo7JSusQ9uS8PZIPn1acbkFnM7xeAW6Bt9mfFcRQxaYin7/exMe9kQfWRilmasnbUikmZuXFUvyPhbYBti6emIW294762u2QV4biWxLg4BOzdagjjY3N8fpsHGPDuhTFOIfcbQ+bE55Gk+xiV4zWnF+Ct1f5I5go4EGRDCHEuQ2AW8N50k4iFGtPOt6EivZCxtlDEWFrmgMhrVCGIkcwl4Puf4Uu1fBMjqVcTt9J1TKImSTafxFhMB4Fd4E2wKsWY/TMjCfJFzL3VuknCMtQpX+aLmDZbCxZjaoJFxdRSmYlFxf6MYp8ieIIpmC0ETQAAAABJRU5ErkJggg==" alt="O2" style="width: 19px; height: 20px;" width="19" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 207.717px 8px; transform-origin: 207.717px 8px; unicode-bidi: normal; "><span style=""> can be used to compute the outflow hydrograph. The guideline for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAAkCAYAAAAOwvOmAAACMElEQVRYR+1Vuy4FURS9tyeCSqWgIFFoPBLRkmglXh/gUSgUJHwACZUoPKIWfICEQilePQUFBRWJ0LNWcnayZ+acmT1zm5uYSVbu3DN777PO2o9TrdThU61DTpWSlDUrpVL/Uqk3nHoH2LCePo9dkZoawQbnwA/QmGczq20RUtcIPuA2mMLviXUzq11eUh0I/KSCP+K927qZx24Sa/fAs/6WlxRVGXMBGmpUqw/+t8AocFGUVAscP4AD4AtYdoFu8DuYUy0SugS+gR7gsyipTTguAL0ugE5jP9bujMQYRw5EhahU5LGmjyq9AFcqCFM54aJ5g8f2Yv0cApJ2/Zm1OSSKWUnNw2ELGAck/1ITErwTL5GCDSi3ivV1983rYyXFYck6ineaHg+stTlDCjnjOOsi6mg/CynKfgz4ZpJ8k5itkoIUcixupvAUoH/isZCiGk0elSQYVWxzf7LUktuA5sHBm0VK6mYNQUL3HOtt15HKunp05wVVzSLFDhsGErMkprmkhMtpB5AaTJ1taaTkSslKCYloBUJqyfClPTt5JVR3aaT24TQDcFhmtbreMFQvuikSV4smGCIlw/IIxpY2Z0w9TH0XNQ85C1DJdqDZxU4oFiIlA+4djq8hmWPr7NAutRbvrgf3ncN3GjgDFoHE9RQipdvcyClhFi/mX2dBFflsA3u+4D5SusWLEhI/fVGLUlR/yaXbGz9rJNRKqpB/ScoqW6lUqZRVAatdWVNWpf4AMMxqJZfZ0UIAAAAASUVORK5CYII=" alt="dt" style="width: 18.5px; height: 18px;" width="18.5" height="18"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 111.892px 8px; transform-origin: 111.892px 8px; unicode-bidi: normal; "><span style=""> above ensures that the coefficients </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAoCAYAAACWwljjAAACVklEQVRYR+1XyysFURx290pYWVEsKErJo5SyYS3lUdYeC1kR/wBF1h4r2YiFskTZkUcKJTYWSnaU2PN9Nb86c8yZc87MvbcrM/V1Z+6cx3e+33NyZSV25UqMT1lGyGaRTKF/oVAVTjkEDAAtQI1y6i/cnwG3QAXQA3QD7zZl5L2PD5HIPDAFlAMXwBZwDDwFC07idyV4z7+OgH5XMhznSqgdY7eBRoAqLAJLho049iQgtYnfiXwTUjcgmcHg5HH7bODlODAC7OaTUD0Wu1FM4LpBH+YcAg2KOZ142Uz2EJiJi/nKT0KjgLND23xoGQNmg2PRVK2+p3WSRBsUp9CnYqo93A8n2cB3jokQw3dNWYyhyxAu+GUixMhg8uNFc9X6+kJS5iZCr1hQMvAj7puSbuA7z0ToW1mIGbnLd+Gk410I+YZ7FBeWHdUn6QLTwJU+2IVQWoVI5hlg0ZW8tIB7lp8OnZSJ0DkGdgbs0/qQBIgeqUwrL7p/mgjJCURR1xJANXYUJfj8FixSrUWqHDpE1ERIZGabwYstxZzFUTnnFGBXIJ2A1LQolaUShHw0LlOzyl8qJNgHrRtISR+ktyWyaRyhkI/aiitJHQBqTqIC10Al0AaMAR/ADKBncxdCIbI2QiIIFegFGK6qs9/heR9g1xhV1QtGyOI+xtcSHHEmS6RQUkJ5deqkJPR5UoZ095CwD3Whrj6UhlxcYmRybFb9rxiEJKfdY2Mp0uJbv3r0YhCiuvxYYI5ilPIjsg5YjUgTzt9laUzmNbdYCjmTygjZpMoU+nMK/QB7534p343IYgAAAABJRU5ErkJggg==" alt="C0" style="width: 18px; height: 20px;" width="18" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; "><span style="">, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="C1" style="width: 18px; height: 20px;" width="18" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; "><span style="">, and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="C2" style="width: 18px; height: 20px;" width="18" height="20"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; "><span style=""> are positive.</span></span></div></div></div>

Problem statement 
Write a function that routes a hydrograph through a river section using the Muskingum method. The input to the function will be the inflow hydrograph , the times  at which the hydrograph is reported, the routing parameters  and , and the desired time step . If the desired time step differs from the time step in the given inflow hydrograph, use linear interpolation to get the inflow hydrograph onto the new time base. Return the time base of the outflow hydrograph, the outflow hydrograph , and a Boolean flag that says whether the time step is within the guidelines .
Bonus points to anyone who can solve this problem using the FILTER command. 
Background
Hydrologic routing is used to predict the movement of a flood wave in a river. The Muskingum method is an example of hydrologic routing, in which the inflow  and outflow  are connected to the storage  in a channel section through a conservation of mass (or volume), which states that the rate of change of storage is the difference between the inflow and outflow:

This equation is written in discretized form as

where subscripts 1 and 2 denote the previous and current time steps, respectively. In applications, the inflow hydrograph (or flow as a function of time) is known, and we assume that everything at the previous time step is known. That assumption leaves two unknowns, the current outflow  and the current storage . 
In the Muskingum method, the storage, inflow, and outflow are further related by

where  is a time scale related to travel time in the river section and  is a weighting factor. Substituting this relation into the discretized form of conservation of mass and rearranging gives

where

Notice that . With the assumption that the first value of the outflow equals the first value of the inflow, this equation for  can be used to compute the outflow hydrograph. The guideline for  above ensures that the coefficients , , and  are positive.

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Problem 57879. Route a hydrograph through a river section with the Muskingum method

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Problem 57879. Route a hydrograph through a river section with the Muskingum method

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