Re: Estimating parameters p and q in y'' + p y' + q y = Tide(t)
- To: mathgroup at smc.vnet.net
- Subject: [mg46284] Re: Estimating parameters p and q in y'' + p y' + q y = Tide(t)
- From: drbob at bigfoot.com (Bobby R. Treat)
- Date: Fri, 13 Feb 2004 21:56:48 -0500 (EST)
- References: <c0hhrh$lgd$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
There's no need to go so 'round about to get that into a notebook. Just copy and paste it into Mathematica. The front end asks if you want it to interpret the text; you say yes, and it's done. Bobby gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi) wrote in message news:<c0hhrh$lgd$1 at smc.vnet.net>... > Dear Math User friends: > I have two data sets; the first one corresponds to tide data, and the > second one corresponds to > water elevation data obtained from a groundwater monitoring well. The > tide affects the water level > inside the well. If we let the variable y(t) represent the height of > the water column inside the pipe, > and Tide(t) be a least square fit representation of our tide record, > with t representing time, > then we can form a Differential Equation: y'' + p y' + q y = Tide(t), > where Tide(t) acts as a forcing > function. Since I have a water elevation record; what I'm seeking is > to find a way to estimate > two values p and q, such that the solution y(t) to the above DE, > becomes a model that fits my > water elevation data; i.e. a model in the least square sense, showing > a correlation (of say) 0.95, > or above. The following is an unevaluated Mathematica notebook to > elaborate this question > with the aid of a specific example. Please copy the following text and > paste it into Wordpad, or > Notepad and save it as DE.txt Then change the name of this file to > DE.nb, (ignore the "are you > sure that you want to change extention name" message) and open the > new notebook using > Mathematica (version 5.0, or version above 5.0) as usual. Thank you > for your help! > > Start copying here: > (************** Content-type: application/mathematica ************** > CreatedBy='Mathematica 5.0' > > Mathematica-Compatible Notebook > > This notebook can be used with any Mathematica-compatible > application, such as Mathematica, MathReader or Publicon. The data > for the notebook starts with the line containing stars above. > > To get the notebook into a Mathematica-compatible application, do > one of the following: > > * Save the data starting with the line of stars above into a file > with a name ending in .nb, then open the file inside the > application; > > * Copy the data starting with the line of stars above to the > clipboard, then use the Paste menu command inside the application. > > Data for notebooks contains only printable 7-bit ASCII and can be > sent directly in email or through ftp in text mode. Newlines can be > CR, LF or CRLF (Unix, Macintosh or MS-DOS style). > > NOTE: If you modify the data for this notebook not in a Mathematica- > compatible application, you must delete the line below containing > the word CacheID, otherwise Mathematica-compatible applications may > try to use invalid cache data. > > For more information on notebooks and Mathematica-compatible > applications, contact Wolfram Research: > web: http://www.wolfram.com > email: info at wolfram.com > phone: +1-217-398-0700 (U.S.) > > Notebook reader applications are available free of charge from > Wolfram Research. > *******************************************************************) > > (*CacheID: 232*) > > > (*NotebookFileLineBreakTest > NotebookFileLineBreakTest*) > (*NotebookOptionsPosition[ 21225, 425]*) > (*NotebookOutlinePosition[ 21931, 449]*) > (* CellTagsIndexPosition[ 21887, 445]*) > (*WindowFrame->Normal*) > > > > Notebook[{ > Cell[BoxData[ > StyleBox[\( (*\(\(*\)\(\ \)\(I\)\(\ \)\(have\)\(\ \)\(two\)\(\ > \)\(data\)\ > \(\ \)\(sets\)\); \ the\ first\ one\ corresponds\ to\ tide\ data, \ > and\ the\ second\ one\ corresponds\ to\n > water\ elevation\ data\ obtained\ from\ a\ groundwater\ > monitoring\ \ > well . \ \ The\ tide\ affects\ the\ water\ level\n > inside\ the\ well . \ \ If\ we\ let\ the\ variable\ y > \((t)\)\ \ > represent\ the\ height\ of\ the\ water\ column\ inside\ the\ pipe, \n > and\ Tide \((t)\)\ be\ a\ least\ square\ fit\ representation\ > of\ our\ > \ tide\ record, \ with\ t\ reperesenting\ time, \n > then\ we\ can\ form\ a\ Differential\ \(Equation : \ \ y''\ + > \ > p\ y'\ + \ q\ y\)\ = \ Tide \((t)\), \ > where\ Tide \((t)\)\ acts\ as\ a\ forcing\n > function . \ \ Since\ I\ have\ a\ water\ elevation\ record; > \ > what\ I' m\ seeking\ is\ to\ find\ a\ way\ to\ estimate\n > two\ values\ p\ and\ q, \ > such\ that\ the\ solution\ y \((t)\)\ to\ the\ above\ DE, \ > becomes\ a\ model\ that\ fits\ my\nwater\ elevation\ data; \ > i . e . \ a\ model\ in\ the\ least\ square\ sense, \ > showing\ a\ correlation\ \((of\ say)\)\ 0.95, \n > or\ above . \ \ The\ following\ example\ is\ an\ attempt\ to\ > clarify\ > \ my\ \(question : \n\(\(Here\ is\ the\ tide\ record\ \((the\ values\ > are\ \ > measured\ in\ decimal\ meters)\)\)\(:\)\)\)\ **) \), > FormatType->StandardForm, > FontFamily->"Arial"]], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(\(tide = {{1, > 0}, {2, \(-0.015\)}, {3, \(-0.07\)}, {4, \(-0.102\)}, {5, > \ > \(-0.152\)}, {6, \(-0.221\)}, {7, \(-0.259\)}, {8, \(-0.303\)}, {9, \ > \(-0.361\)}, {10, \(-0.407\)}, {11, \(-0.456\)}, {12, \(-0.496\)}, > {13, \ > \(-0.547\)}, {14, \(-0.613\)}, {15, \(-0.65\)}, {16, \(-0.662\)}, {17, > \ > \(-0.691\)}, {18, \(-0.733\)}, {19, \(-0.763\)}, {20, \(-0.807\)}, > {21, \ > \(-0.814\)}, {22, \(-0.822\)}, {23, \(-0.784\)}, {24, \(-0.762\)}, > {25, \ > \(-0.756\)}, {26, \(-0.755\)}, {27, \(-0.743\)}, {28, \(-0.714\)}, > {29, \ > \(-0.688\)}, {30, \(-0.672\)}, {31, \(-0.632\)}, {32, \(-0.609\)}, > {33, \ > \(-0.552\)}, {34, \(-0.475\)}, {35, \(-0.443\)}, {36, \(-0.373\)}, > {37, \ > \(-0.34\)}, {38, \(-0.293\)}, {39, \(-0.232\)}, {40, \(-0.214\)}, {41, > \ > \(-0.152\)}, {42, \(-0.112\)}, {43, \(-0.082\)}, {44, \(-0.072\)}, > {45, \ > \(-0.075\)}, {46, \(-0.073\)}, {47, \(-0.071\)}, {48, \(-0.072\)}, > {49, \ > \(-0.078\)}, {50, \(-0.109\)}, {51, \(-0.113\)}, {52, \(-0.144\)}, > {53, \ > \(-0.165\)}, {54, \(-0.181\)}, {55, \(-0.202\)}, {56, \(-0.21\)}, {57, > \ > \(-0.248\)}, {58, \(-0.284\)}, {59, \(-0.327\)}, {60, \(-0.363\)}, > {61, \ > \(-0.403\)}, {62, \(-0.432\)}, {63, \(-0.477\)}, {64, \(-0.522\)}, > {65, \ > \(-0.569\)}, {66, \(-0.584\)}, {67, \(-0.617\)}, {68, \(-0.631\)}, > {69, \ > \(-0.63\)}, {70, \(-0.626\)}, {71, \(-0.611\)}, {72, \(-0.622\)}, {73, > \ > \(-0.593\)}, {74, \(-0.567\)}, {75, \(-0.549\)}, {76, \(-0.525\)}, > {77, \ > \(-0.503\)}, {78, \(-0.48\)}, {79, \(-0.445\)}, {80, \(-0.406\)}, {81, > \ > \(-0.363\)}, {82, \(-0.326\)}, {83, \(-0.279\)}, {84, \(-0.246\)}, > {85, \ > \(-0.197\)}, {86, \(-0.145\)}, {87, \(-0.119\)}, {88, \(-0.061\)}, > {89, \ > \(-0.021\)}, {90, 0.026}, {91, 0.032}, {92, 0.059}, {93, 0.068}, {94, > 0.059}, {95, 0.08}, {96, 0.084}, {97, 0.065}, {98, > 0.044}, {99, \(-0.021\)}, {100, \(-0.014\)}, {101, > \(-0.053\)}, \ > {102, \(-0.068\)}, {103, \(-0.101\)}, {104, \(-0.145\)}, {105, > \(-0.212\)}, \ > {106, \(-0.26\)}, {107, \(-0.319\)}, {108, \(-0.346\)}, {109, > \(-0.378\)}, \ > {110, \(-0.438\)}, {111, \(-0.496\)}, {112, \(-0.55\)}, {113, > \(-0.599\)}, \ > {114, \(-0.64\)}, {115, \(-0.684\)}, {116, \(-0.708\)}, {117, > \(-0.735\)}, \ > {118, \(-0.757\)}, {119, \(-0.785\)}, {120, \(-0.797\)}, {121, > \(-0.786\)}, \ > {122, \(-0.781\)}, {123, \(-0.767\)}, {124, \(-0.746\)}, {125, > \(-0.72\)}, \ > {126, \(-0.686\)}, {127, \(-0.672\)}, {128, \(-0.648\)}, {129, > \(-0.634\)}, \ > {130, \(-0.593\)}, {131, \(-0.54\)}, {132, \(-0.512\)}, {133, > \(-0.466\)}, \ > {134, \(-0.425\)}, {135, \(-0.371\)}, {136, \(-0.308\)}, {137, > \(-0.266\)}, \ > {138, \(-0.219\)}, {139, \(-0.182\)}, {140, \(-0.149\)}, {141, > \(-0.114\)}, \ > {142, \(-0.107\)}, {143, \(-0.102\)}, {144, \(-0.08\)}, {145, > \(-0.1\)}, \ > {146, \(-0.127\)}, {147, \(-0.143\)}, {148, \(-0.179\)}, {149, > \(-0.175\)}, \ > {150, \(-0.181\)}, {151, \(-0.215\)}, {152, \(-0.231\)}, {153, > \(-0.264\)}, \ > {154, \(-0.323\)}, {155, \(-0.373\)}, {156, \(-0.431\)}, {157, > \(-0.481\)}, \ > {158, \(-0.525\)}, {159, \(-0.58\)}, {160, \(-0.629\)}, {161, > \(-0.674\)}, \ > {162, \(-0.726\)}, {163, \(-0.766\)}, {164, \(-0.825\)}, {165, > \(-0.845\)}, \ > {166, \(-0.864\)}, {167, \(-0.891\)}, {168, \(-0.889\)}, {169, > \(-0.895\)}, \ > {170, \(-0.883\)}, {171, \(-0.876\)}, {172, \(-0.821\)}, {173, > \(-0.818\)}, \ > {174, \(-0.802\)}, {175, \(-0.76\)}, {176, \(-0.731\)}, {177, > \(-0.696\)}, \ > {178, \(-0.672\)}, {179, \(-0.631\)}, {180, \(-0.592\)}, {181, > \(-0.548\)}, \ > {182, \(-0.495\)}, {183, \(-0.441\)}, {184, \(-0.38\)}, {185, > \(-0.333\)}, \ > {186, \(-0.265\)}, {187, \(-0.185\)}, {188, \(-0.155\)}, {189, > \(-0.116\)}, \ > {190, \(-0.101\)}, {191, \(-0.074\)}, {192, \(-0.029\)}, {193, > \(-0.005\)}, \ > {194, 0.001}, {195, 0.029}, {196, > 0.001}, {197, \(-0.019\)}, {198, \(-0.053\)}, {199, > \(-0.081\)}, \ > {200, \(-0.097\)}, {201, \(-0.152\)}, {202, \(-0.185\)}, {203, > \(-0.206\)}, \ > {204, \(-0.219\)}, {205, \(-0.261\)}, {206, \(-0.282\)}, {207, > \(-0.331\)}, \ > {208, \(-0.378\)}, {209, \(-0.455\)}, {210, \(-0.495\)}, {211, > \(-0.556\)}, \ > {212, \(-0.599\)}, {213, \(-0.639\)}, {214, \(-0.637\)}, {215, > \(-0.667\)}, \ > {216, \(-0.673\)}, {217, \(-0.695\)}, {218, \(-0.7\)}, {219, > \(-0.687\)}, \ > {220, \(-0.687\)}, {221, \(-0.683\)}, {222, \(-0.656\)}, {223, > \(-0.631\)}, \ > {224, \(-0.616\)}, {225, \(-0.578\)}, {226, \(-0.534\)}, {227, > \(-0.482\)}, \ > {228, \(-0.428\)}, {229, \(-0.345\)}, {230, \(-0.313\)}, {231, > \(-0.282\)}, \ > {232, \(-0.25\)}, {233, \(-0.218\)}, {234, \(-0.187\)}, {235, > \(-0.155\)}, \ > {236, \(-0.139\)}, {237, \(-0.099\)}, {238, \(-0.083\)}, {239, > \(-0.076\)}, \ > {240, \(-0.054\)}, {241, \(-0.047\)}, {242, \(-0.064\)}, {243, > \(-0.038\)}, \ > {244, \(-0.043\)}, {245, \(-0.06\)}, {246, \(-0.056\)}, {247, > \(-0.077\)}, \ > {248, \(-0.106\)}, {249, \(-0.134\)}, {250, \(-0.212\)}, {251, > \(-0.269\)}, \ > {252, \(-0.327\)}, {253, \(-0.386\)}, {254, \(-0.443\)}, {255, > \(-0.476\)}, \ > {256, \(-0.525\)}, {257, \(-0.608\)}, {258, \(-0.665\)}, {259, > \(-0.71\)}, \ > {260, \(-0.723\)}, {261, \(-0.756\)}, {262, \(-0.784\)}, {263, > \(-0.807\)}, \ > {264, \(-0.845\)}};\)\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(L = Length[tide]\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(plt1 = > ListPlot[tide, PlotJoined \[Rule] True, > PlotStyle \[Rule] RGBColor[1, 0, 0]]\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(\(initparams = {{a\_0, \(-0.31278\)}, {a\_1, \(-0.21078\)}, > {a\_2, \ > \(-0.12503\)}, {a\_3, \(-0.03388\)}, {a\_4, \(-0.34959\)}, {b\_1, > 8.093077}, {b\_2, 10.51904}, {b\_3, 2.014087}, {b\_4, > 7.077294}, {c\_1, \(-0.05472\)}, {c\_2, \(-0.05678\)}, > {c\_3, \ > \(-0.1125\)}, {c\_4, \(-0.06463\)}, {d\_1, \(-5.01326\)}, {d\_2, \ > \(-6.27323\)}, {d\_3, \(-1.67511\)}, {d\_4, \(-10.6115\)}, {v\_1, > 0.126851}, {v\_2, 0.135713}, {v\_3, 0.089487}, {v\_4, > 0.130158}, {w\_1, 0.049697}, {w\_2, > 0.049183}, {w\_3, \(-0.00844\)}, {w\_4, 0.046353}};\)\)], > "Input",\ > > FontFamily->"Arial"], > > Cell[BoxData[ > \(\(model\ = \ > a\_0 + Sum[a\_i*Sin[v\_i*t - b\_i], {i, 1, 4}] + > Sum[c\_i*Sin[w\_i*t - d\_i], {i, 1, 4}];\)\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(<< Statistics`NonlinearFit`\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > RowBox[{\(TIDE[t_]\), "=", > RowBox[{"Chop", "[", > RowBox[{"NonlinearFit", "[", > RowBox[{"tide", ",", "model", ",", "t", ",", "initparams", > ",", > FormBox[ > FormBox[\(AccuracyGoal \[Rule] 2\), > "TraditionalForm"], > "TraditionalForm"]}], "]"}], "]"}]}]], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(plt2 = > Plot[TIDE[t], {t, 1, 264}, PlotStyle -> RGBColor[0, 1, 0]]\)], > "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(Show[{plt1, plt2}, ImageSize\ \[Rule] \ 540]\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \( (*\(\(*\)\(\ \)\(Gravitational\)\(\ \)\(Constant\)\(\ > \)\(G\)\)\ = \ > 9.8\ m/sec\^2\ **) \)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \( (*\(*\)\(\ \)\(Tide\)\(\ \)\(acting\)\(\ \)\(as\)\(\ \)\(a\)\(\ > \ > \)\(Forcing\)\(\ \)\(Function\)\(\ \)\(term\)\(\ \)\(in\)\(\ > \)\(the\)\(\ \ > \)\(following\)\(\ \)\(Differential\)\(\ \)\(\(Equation\)\(:\)\)\ **) > \)], \ > "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > RowBox[{"(*", > RowBox[{\(\(\(*\)\(\ \)\(Using\)\(\ \)\("\<trial and > error\>"\)\(\ \ > \)\(we\)\(\ \)\(are\)\(\ \)\(using\)\(\ \)\(values : \ p\)\)\ = \ > 9.8\ \((buoyancy\ factor)\)\), ",", " ", \(q = 1\), ",", > " ", \(and\ two\ initial\ conditions; \ y[0] = 0\), ",", " ", > RowBox[{ > RowBox[{"and", " ", > RowBox[{ > SuperscriptBox["y", "\[Prime]", > MultilineFunction->None], "[", "0", "]"}]}], "=", > "1"}], ",", > " ", \(to\ set\ up\ our\ differential\ > \(\(equation\)\(:\)\)\)}], > "**)"}]], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > RowBox[{ > RowBox[{"solution", "=", > RowBox[{"NDSolve", "[", > RowBox[{ > RowBox[{"{", > RowBox[{ > RowBox[{ > RowBox[{ > RowBox[{ > SuperscriptBox["y", "\[DoublePrime]", > MultilineFunction->None], "[", "t", "]"}], > "+", > RowBox[{\((9.8)\), "*", > RowBox[{ > SuperscriptBox["y", "\[Prime]", > MultilineFunction->None], "[", "t", "]"}]}], > "+", \(y[t]\)}], "\[Equal]", \(TIDE[t]\)}], > ",", \(y[0] \[Equal] 0\), ",", > RowBox[{ > RowBox[{ > SuperscriptBox["y", "\[Prime]", > MultilineFunction->None], "[", "0", "]"}], > "\[Equal]", > "1"}]}], "}"}], ",", "y", ",", \({t, 0, L}\)}], > "]"}]}], > ";"}]], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(\(plt3 = > Plot[y[t] /. solution, {t, 0, L}, > PlotStyle \[Rule] RGBColor[1, 0, 1]];\)\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \( (*\(\(*\)\(\ \)\(Here\)\(\ \)\(is\)\(\ \)\(the\)\(\ > \)\(elevation\)\(\ \ > \)\(data\)\); \ \(\(i . e . \ > the\ water\ elevation\ data\ obtained\ from\ a\ > groundwater\ \ > monitoring\ well\ \((the\ values\ are\ measured\ in\ decimal\ \ > meters)\)\)\(:\)\)\ **) \)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > StyleBox[\(Elev = {{1, 0. }, {2, 0.002}, {3, 0.006}, {4, 0.006}, > {5, > 0.008}, {6, 0.008}, {7, 0.008}, {8, 0.006}, {9, 0.007}, > {10, > 0.006}, {11, 0.009}, \n\ \ \ \ \ \ \ \ {12, 0.013}, {13, > 0.009}, {14, 0.004}, {15, 0.004}, {16, 0.004}, {17, > 0.003}, {18, 0.002}, {19, 0. }, {20, 0. }, {21, > 0. }, {22, \(-0.002\)}, \n\ \ \ \ \ \ \ \ {23, > \(-0.004\)}, \ > {24, \(-0.005\)}, {25, \(-0.005\)}, {26, \(-0.006\)}, {27, > \(-0.005\)}, {28, \ > \(-0.004\)}, {29, \(-0.006\)}, {30, \(-0.009\)}, {31, \(-0.008\)}, \n\ > \ \ \ \ > \ \ \ \ {32, \(-0.009\)}, {33, \(-0.009\)}, {34, \(-0.01\)}, {35, \ > \(-0.009\)}, {36, \(-0.009\)}, {37, \(-0.008\)}, {38, \(-0.01\)}, {39, > \ > \(-0.007\)}, {40, \(-0.009\)}, \n\ \ \ \ \ \ \ \ {41, \(-0.008\)}, > {42, \ > \(-0.007\)}, {43, \(-0.007\)}, {44, \(-0.005\)}, {45, \(-0.005\)}, > {46, \ > \(-0.004\)}, {47, \(-0.004\)}, {48, \(-0.004\)}, {49, \(-0.003\)}, \n\ > \ \ \ \ > \ \ \ \ {50, \(-0.003\)}, {51, \(-0.001\)}, {52, 0. }, {53, 0. }, {54, > 0.002}, {55, 0.002}, {56, 0.003}, {57, 0.003}, {58, > 0.003}, {59, 0.004}, \n\ \ \ \ \ \ \ \ {60, 0.004}, {61, > 0.004}, {62, 0.002}, {63, 0.002}, {64, 0.002}, {65, > 0. }, {66, \(-0.001\)}, {67, \(-0.002\)}, {68, > \(-0.003\)}, \ > {69, \(-0.004\)}, \n\ \ \ \ \ \ \ \ {70, \(-0.004\)}, {71, > \(-0.005\)}, {72, \ > \(-0.006\)}, {73, \(-0.007\)}, {74, \(-0.007\)}, {75, \(-0.008\)}, > {76, \ > \(-0.009\)}, {77, \(-0.011\)}, {78, \(-0.011\)}, \n\ \ \ \ \ \ \ \ > {79, \ > \(-0.013\)}, {80, \(-0.013\)}, {81, \(-0.012\)}, {82, \(-0.013\)}, > {83, \ > \(-0.013\)}, {84, \(-0.013\)}, {85, \(-0.011\)}, {86, \(-0.01\)}, {87, > \ > \(-0.011\)}, \n\ \ \ \ \ \ \ \ {88, \(-0.013\)}, {89, \(-0.005\)}, > {90, \ > \(-0.009\)}, {91, \(-0.005\)}, {92, \(-0.011\)}, {93, \(-0.007\)}, > {94, \ > \(-0.007\)}, {95, \(-0.006\)}, {96, \(-0.003\)}, \n\ \ \ \ \ \ \ \ > {97, \ > \(-0.002\)}, {98, 0. }, {99, \(-0.001\)}, {100, 0.001}, {101, > 0.002}, {102, \(-0.004\)}, {103, 0.003}, {104, 0.003}, > {105, > 0.004}, \n\ \ \ \ \ \ \ \ {106, 0.004}, {107, 0.004}, > {108, > 0.004}, {109, 0.005}, {110, 0.005}, {111, 0.004}, {112, > 0.003}, {113, 0.002}, {114, > 0.001}, \n\ \ \ \ \ \ \ \ {115, \(-0.001\)}, {116, > \(-0.002\)}, \ > {117, \(-0.003\)}, {118, \(-0.004\)}, {119, \(-0.004\)}, {120, > \(-0.004\)}, \ > {121, \(-0.005\)}, {122, \(-0.007\)}, \n\ \ \ \ \ \ \ \ {123, > \(-0.008\)}, \ > {124, \(-0.01\)}, {125, \(-0.011\)}, {126, \(-0.011\)}, {127, > \(-0.012\)}, \ > {128, \(-0.013\)}, {129, \(-0.014\)}, {130, \(-0.014\)}, \n\ \ \ \ \ \ > \ \ \ > {131, \(-0.013\)}, {132, \(-0.014\)}, {133, \(-0.014\)}, {134, > \(-0.014\)}, \ > {135, \(-0.015\)}, {136, \(-0.014\)}, {137, \(-0.014\)}, {138, > \(-0.015\)}, \n\ > \ \ \ \ \ \ \ \ {139, \(-0.014\)}, {140, \(-0.014\)}, {141, > \(-0.012\)}, \ > {142, \(-0.011\)}, {143, \(-0.011\)}, {144, \(-0.009\)}, {145, > \(-0.009\)}, \ > {146, \(-0.007\)}, \n\ \ \ \ \ \ \ \ {147, \(-0.007\)}, {148, > \(-0.006\)}, \ > {149, \(-0.004\)}, {150, \(-0.004\)}, {151, \(-0.004\)}, {152, > \(-0.004\)}, \ > {153, \(-0.003\)}, {154, \(-0.003\)}, \n\ \ \ \ \ \ \ \ {155, > \(-0.003\)}, \ > {156, \(-0.004\)}, {157, \(-0.004\)}, {158, \(-0.004\)}, {159, > \(-0.004\)}, \ > {160, \(-0.004\)}, {161, \(-0.004\)}, {162, \(-0.005\)}, \n\ \ \ \ \ \ > \ \ \ > {163, \(-0.006\)}, {164, \(-0.007\)}, {165, \(-0.007\)}, {166, > \(-0.008\)}, \ > {167, \(-0.01\)}, {168, \(-0.011\)}, {169, \(-0.011\)}, {170, > \(-0.013\)}, \n\ > \ \ \ \ \ \ \ \ {171, \(-0.015\)}, {172, \(-0.017\)}, {173, > \(-0.017\)}, \ > {174, \(-0.018\)}, {175, \(-0.018\)}, {176, \(-0.018\)}, {177, > \(-0.018\)}, \ > {178, \(-0.019\)}, \n\ \ \ \ \ \ \ \ {179, \(-0.02\)}, {180, > \(-0.021\)}, \ > {181, \(-0.021\)}, {182, \(-0.021\)}, {183, \(-0.021\)}, {184, > \(-0.022\)}, \ > {185, \(-0.019\)}, {186, \(-0.021\)}, \n\ \ \ \ \ \ \ \ {187, > \(-0.02\)}, \ > {188, \(-0.018\)}, {189, \(-0.017\)}, {190, \(-0.017\)}, {191, > \(-0.011\)}, \ > {192, \(-0.014\)}, {193, \(-0.011\)}, {194, \(-0.011\)}, \n\ \ \ \ \ \ > \ \ \ > {195, \(-0.01\)}, {196, \(-0.01\)}, {197, \(-0.005\)}, {198, > \(-0.008\)}, \ > {199, \(-0.008\)}, {200, \(-0.006\)}, {201, \(-0.004\)}, {202, > \(-0.004\)}, \n\ > \ \ \ \ \ \ \ \ {203, \(-0.006\)}, {204, \(-0.005\)}, {205, > \(-0.003\)}, \ > {206, \(-0.004\)}, {207, \(-0.003\)}, {208, \(-0.003\)}, {209, > \(-0.003\)}, \ > {210, \(-0.004\)}, \n\ \ \ \ \ \ \ \ {211, \(-0.004\)}, {212, > \(-0.004\)}, \ > {213, \(-0.005\)}, {214, \(-0.005\)}, {215, \(-0.006\)}, {216, > \(-0.008\)}, \ > {217, \(-0.008\)}, {218, \(-0.01\)}, \n\ \ \ \ \ \ \ \ {219, > \(-0.011\)}, \ > {220, \(-0.013\)}, {221, \(-0.013\)}, {222, \(-0.015\)}, {223, > \(-0.017\)}, \ > {224, \(-0.018\)}, {225, \(-0.018\)}, {226, \(-0.018\)}, \n\ \ \ \ \ \ > \ \ \ > {227, \(-0.018\)}, {228, \(-0.018\)}, {229, \(-0.02\)}, {230, > \(-0.02\)}, \ > {231, \(-0.019\)}, {232, \(-0.023\)}, {233, \(-0.02\)}, {234, > \(-0.021\)}, \n\ > \ \ \ \ \ \ \ \ {235, \(-0.02\)}, {236, \(-0.019\)}, {237, > \(-0.018\)}, {238, \ > \(-0.016\)}, {239, \(-0.016\)}, {240, \(-0.014\)}, {241, \(-0.013\)}, > {242, \ > \(-0.013\)}, \n\ \ \ \ \ \ \ \ {243, \(-0.012\)}, {244, \(-0.011\)}, > {245, \ > \(-0.01\)}, {246, \(-0.009\)}, {247, \(-0.009\)}, {248, \(-0.007\)}, > {249, \ > \(-0.005\)}, {250, \(-0.004\)}, \n\ \ \ \ \ \ \ \ {251, \(-0.004\)}, > {252, \ > \(-0.004\)}, {253, \(-0.003\)}, {254, \(-0.004\)}, {255, \(-0.005\)}, > {256, \ > \(-0.004\)}, {257, \(-0.005\)}, {258, \(-0.006\)}, \n\ \ \ \ \ \ \ \ > {259, \ > \(-0.007\)}, {260, \(-0.007\)}, {261, \(-0.008\)}, {262, \(-0.009\)}, > {263, \ > \(-0.01\)}, {264, \(-0.011\)}, {265, \(-0.011\)}, {266, \(-0.012\)}, > \n\ \ \ \ > \ \ \ \ \ {267, \(-0.013\)}, {268, \(-0.017\)}, {269, \(-0.017\)}};\), > FormatType->StandardForm, > FontFamily->"Arial", > FontSize->12]], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(Length[Elev]\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \( (*\(\(*\)\(\ \)\(To\)\(\ \)\(compare\)\(\ \)\(our\)\(\ \ > \)\(solution\)\(\ \)\(model\)\(\ \)\(to\)\(\ \)\(the\)\(\ > \)\(elevation\)\(\ \ > \)\(record\)\); \ > it\ is\ necessary\ to\ magnify\ the\ elevation\ record\ by\ a\ > factor\ \ > of\ 50. \ \ There\ is\ also\ a\ time\ lag\ between\ the\ solution\ > model\ and\ > \ the\ elevation\ record\ of\ 6\ time\ \(\(units\)\(:\)\)\ **) \)], > "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(\(M = 50;\)\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \( (*\(*\)\(\ \)\(Lag\)\(\ \)\(\(Factor\)\(:\)\)\ **) \)], > "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(\(Lag = 6;\)\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \( (*\(*\)\(\ \)\(The\)\(\ \)\(abbreviation\)\(\ > \)\("\<rsElev\>"\)\(\ \)\ > \(stands\)\(\ \)\(for\)\(\ \)\("\<re-scaled Elevation\>"\)\ **) \)], > "Input"], > > Cell[BoxData[ > \(\(rsElev = > Table[{i - Lag, M*\(Elev[\([i]\)]\)[\([2]\)]}, {i, 1, > L}];\)\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \(plt4 = > ListPlot[rsElev, PlotJoined \[Rule] True, > PlotStyle \[Rule] RGBColor[0, 0, 1]]\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \( (*\(\(*\)\(\ \)\(Here\)\(\ \)\(we\)\(\ \)\(compare\)\(\ > \)\(our\)\(\ \ > \)\(solution\)\(\ \)\(model\)\(\ \)\(with\)\(\ \)\(our\)\(\ > \)\(well\)\(\ \ > \)\(elevation\)\(\ \)\(data . \ \ "\<Not close, and no cigar\>"\)\); \ > because\ we\ are\ attempting\ two\ arbitrary\ values\ for\ p\ > and\ \ > \(\(q\)\(:\)\)\ \ **) \)], "Input"], > > Cell[BoxData[ > \(Show[{plt3, plt4}]\)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > \( (*\(\(*\)\(\ \)\(Remark : \ \ One\ can\ see\ a\ downward\ > trend\ in\ \ > the\ elevation\ record . \ \ I\ don' > t\ know\ what\ physical\ factor\ "\<out there\>"\ causes\ > this\ \ > trend . \ \ If\ we\ could\ identify\ it\)\), \ > and\ incorporate\ it\ into\ the\ Differential\ Equation; > perhaps\ our\ "\<Elevation DE Model\>"\ could\ be\ more\ > accurate\ in\ \ > accounting\ for\ "\<the behavior\>"\ of\ the\ elevation\ > \(\(record\)\(.\)\)\ \ > **) \)], "Input", > FontFamily->"Arial"], > > Cell[BoxData[ > StyleBox[\(\( (*\(\(*\)\(\ \)\(Again\)\); \ > what\ I' m\ looking\ for\ is\ to\ find\ a\ way\ to\ estimate\n > the\ values\ p\ and\ q, \ > such\ that\ the\ solution\ y \((t)\)\ to\ the\ above\ DE, \ > \(becomes\ > \ a\ model\ that\ fits\ my\)\n\(water\ elevation\ data; \ > i . e . \ a\ model\ in\ the\ least\ square\ sense\), \ > showing\ a\ correlation\ \((of\ say)\)\ 0.95, \nor\ above, \ > without\ having\ to\ engage\ into\ trial\ and\ > \(\(error\)\(.\)\)\ \ \ > **) \)\(\ \)\), > FontFamily->"Arial"]], "Input"] > }, > FrontEndVersion->"5.0 for Microsoft Windows", > ScreenRectangle->{{0, 1024}, {0, 680}}, > WindowSize->{1016, 648}, > WindowMargins->{{0, Automatic}, {Automatic, 0}}, > PrintingCopies->1, > PrintingPageRange->{Automatic, Automatic} > ] > > (******************************************************************* > Cached data follows. If you edit this Notebook file directly, not > using Mathematica, you must remove the line containing CacheID at > the top of the file. The cache data will then be recreated when > you save this file from within Mathematica. > *******************************************************************) > > (*CellTagsOutline > CellTagsIndex->{} > *) > > (*CellTagsIndex > CellTagsIndex->{} > *) > > (*NotebookFileOutline > Notebook[{ > Cell[1754, 51, 1536, 25, 200, "Input"], > Cell[3293, 78, 5062, 69, 542, "Input"], > Cell[8358, 149, 72, 2, 29, "Input"], > Cell[8433, 153, 155, 4, 29, "Input"], > Cell[8591, 159, 652, 11, 67, "Input"], > Cell[9246, 172, 181, 4, 29, "Input"], > Cell[9430, 178, 83, 2, 29, "Input"], > Cell[9516, 182, 378, 9, 29, "Input"], > Cell[9897, 193, 128, 3, 29, "Input"], > Cell[10028, 198, 101, 2, 29, "Input"], > Cell[10132, 202, 155, 3, 31, "Input"], > Cell[10290, 207, 271, 5, 29, "Input"], > Cell[10564, 214, 631, 13, 48, "Input"], > Cell[11198, 229, 994, 24, 29, "Input"], > Cell[12195, 255, 158, 4, 29, "Input"], > Cell[12356, 261, 324, 6, 29, "Input"], > Cell[12683, 269, 5762, 79, 618, "Input"], > Cell[18448, 350, 68, 2, 29, "Input"], > Cell[18519, 354, 433, 7, 67, "Input"], > Cell[18955, 363, 67, 2, 29, "Input"], > Cell[19025, 367, 108, 2, 29, "Input"], > Cell[19136, 371, 68, 2, 29, "Input"], > Cell[19207, 375, 170, 2, 30, "Input"], > Cell[19380, 379, 149, 4, 29, "Input"], > Cell[19532, 385, 157, 4, 29, "Input"], > Cell[19692, 391, 350, 5, 50, "Input"], > Cell[20045, 398, 74, 2, 29, "Input"], > Cell[20122, 402, 519, 9, 67, "Input"], > Cell[20644, 413, 577, 10, 86, "Input"] > } > ] > *) > > > > (******************************************************************* > End of Mathematica Notebook file. > *******************************************************************)