Geomathematics GEOL 351
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This 38 page Class Notes was uploaded by Jessica Braun MD on Saturday September 12, 2015. The Class Notes belongs to GEOL 351 at West Virginia University taught by Thomas Wilson in Fall. Since its upload, it has received 37 views. For similar materials see /class/202715/geol-351-west-virginia-university in Geology at West Virginia University.
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Date Created: 09/12/15
Name Gaol 351 Gaomathamatlcs Trigonometry Inclass problems lesh up on Thursday 1 Anhe base nfa clefyuu are standing ah mp 52 geulugxca Umt A The my face Ts The mp 52 mhe my face Yuu walk a distance x e 200 feet away rum the ram scary Lnnkmg Ts 23 abuve the hunznn Vl hat s the uffset dung thxs reum Top of Unit A h cw Face Top ofUnitA gt maps They and hhhg thexrradm transmmerthuugh alumeterbarumeter and axepuxpese Bmhwh campass Continued nextpage Note what variables you can measure directly or calculate Note what the rescue team needs to know in order to nd you and provide that information Assume that the summit of Mt Everest is located along the bearing N25E from your location 3 In the example illustrated below a stream erodes less resistant fault gauge leaving an exposed fault scarp on the distant bank You are unable to traverse the stream or make your way to the top of the exposure Using your Brunton compass you stand on the left edge of the stream and measure the angle a formed by the top of the cliff and the horizontal You walk to the left 175 feet and measure angle b Angle a measure 310 and angle b 19 How can you determine the cliff height What is the width of the stream Fault Scarp d gt Stream gt 4 The three point problem uses elevations measured at three points on a stratigraphic surface to determine the strike and dip of that surface The elevations and locations of these points can be measured at the surface or more likely in the borehole In the following problem you have data from three boreholes located in the map below indicating subsea depths to the top of the Oriskany Sandstone as shown 2 Scale 110000 Geol 351 Geomathematics Derivative Concepts amp Basic Calculations The derivative is essentially the slope of a function at a given point We ve talked about slopes on numerous occasions throughout the semester We estimated agedepth relationships and sedimentation rates for different periods of time In most of the cases we ve worked with up to this point those slopes have been constant over a certain time period Sedimentation rates in the North Sea for example were roughly linear during the 0 to 10 000 period before present and the 10000 to 15000 year before present periods We noted that in many instances we would eXpect the agedepth depthporosity relationships to vary with depth because of the in uence of compaction As time passes the increased weight of sediment bearing down on a layer deposited at an earlier time will push grains closer together and reduce the open space or pore space within the layer The layer will get thinner and the porosity will decrease with increased depth of burial The porosity depth relationship in unconsolidated sediments is often written as 1508 In this equation q is the initial porosity at a depth 2 0 and c is a constant a compaction factorThe porosity depth relationship is shown below over the 1 to 5 km range of depths z for an initial porosity of 05 and compaction factor of 05 km39l The rate of change of porosity A between 1 and 2 kilometers could be estimated as g where A is the difference in porosities measured at 1 and 2 kilometers 2 1 and Azis the difference in the corresponding depths 22 7 21 PorosityDepth Relationship Slope 2 3 2 km 2 amp 1 are 0184 and 0304 at depths 2 of2 and 1 km respectively This gives a slope or M gradient of 0 12 per kilometer forE Spatial variations are usually referred to as gradients 115 A while temporal changes are referred to as rates You can see from the graph that this El is an average estimate of the porositydepth gradient between 1 and 2 kilometers since the line representing this gradient or slope intersects the line at these two points This measure provides a useful general reference to how porosity is changing between depths of l and 2 kilometers in this area but does not tell us how the porosity changes at any one point in particular We could ask a similar question but re ne our depth range down to 100 meters instead of l kilometer For example in the plot below we compare the gradient between 1 and 11 km to the gradient between 1 to 2 kmfz in this case is w or 014per kilometer The gradient is a little steeper in the shallower part of the curve PorosityDepth Relationship 034 032 030 Gradlent 028 to 2 km 1 03926 Gradient 024 10t011km 022 020 018 016 08 10 12 14 16 18 20 22 Zkm As Waltham notes the derivative is a measure of the slope of a curve in the limit that the AZ or whatever the independent variable happens to be goes to zero As AZ goes to zero we are looking at the slope of a line that intersects the curve at only one point This makes that line a tangent The derivative is the tangent of the line intersecting a curve at a single speci ed point x de In class we note that the der1vat1ve of the eXponent1al functlon d x is just ex The function is d2 1ndestruct1ble It does not change w1th success1ve d1fferent1atlons 1e d i ex and so on no x matter how many times you differentiate ex the result is the same ex However if X is multiplied 116 by a constant the derivative does change 1fwe differentiate the function 2 with respect to 2 d2 YL H L ie 39 Firstyou evaluate the derivative of ex with respect to x where x cz then we multiply that Stumpy A Lg a a a a L r Ligde dgxissiml dY 4239 d W ex and dY d7cz dz dz d In this twostep process the 65 cancel out and we get 2 ygte1 is a composite functi on or a function of another function We consider the term in the exponent as a separate function in this case xz 42 L ab ExerciseHomework a a Ay Ax Ay 0 11 a u y A2 d case fory E 752 Liz PM I do 1 c c 3 f0 dz simplici 1 Open Excel In cell A1 enter Z then generate a Column ofnumbers from 0 to 10 with increments 0f0 1 running from cell A2 to A102 2 In ce11 B1 39 39 r In ce11 B2 enter an formula into cells 133 through 13102 3 d Title column c 7 dPhidz ie Leave cell c2 blank and enter bitb2 in cell 3 Copy the formula into cells c4 through 3102 L 39 39 39 39 39 in a n t nnar Our do computation of Educ not equal of figure below 03 v re airaz a Puvuslyz Mamadz l at uamaim unaslbi u 2 u Elmms u uablm u 3 u unmazz u meta u A u 5 w n E7EI32EIEI5 n mmaa n bubsanbb n mama 5 Construct a plot of the calculated quantities in columns B and C 6 Why don t the values in column C equal those in column B There are two reasons 1 our delta 2 is not in nitesimally small and 2 we left out our AZ Consider the basic de nition the denvat1ve 1s a 111 the The denvat1ve is not s1mply a difference 1t 61 is a slope Speci cally it becomes the slope of the tangent line when written asd Z What have you actually calculated in column C Using the computer as an aid we can approximate the derivative just by making our AZ suf ciently small 7 An intentional mistake on my part made to emphasize the point above The derivative is more than a difference it is a slope What do you need to do to improve the agreement between the numbers in columns C and B Part 2 In this exercise we ll use the computer to compute the derivative of the cos6 and test out the d cos 9 idea that Sin 9 1 In Sheet 2 in your Excel workbook enter Degrees in cell Al 0 and l in cells A2 and A3 Highlight these two cells and drag the lower right comer down to cell 362 generating a sequential set of numbers running from 1 to 360 2 In Column B convert these numbers into radians Remember the Excel function pi generates the number n amp radians 7239 times angle 180 3 Next generate the function cosr9 in Column C where 6 is in radians 4 In Column D compute the difference in cosr9 terms ie C3C2 d cos 9 d6 6 Graph these columns D and E the difference of cosines and sinr9 7 Plot and compare columns D and E 8 Why do you end up with a discrepancy in the expected relationship What have we omitted in the above computation 5 In Column E compute the sine ie Parts 1 and 2 are due this Thursday for 10 points extra credit A Prepare a plot of the results obtained in parts 1 and 2 Hand that in with your name on it B On each plot note the error that leads to the discrepancy noted above and indicate how it should be corrected The basic lessons illustrated in this exercise are 1 that computers can easily compute derivatives as long as we use a small difference in our independent variable and 2 don t forget to include the denominator 118 Can 351 r IVIaLhEmztics fur Genlngins 7 Class Exercise ammw m 5mm n quotmy mmm 005W Mm my mu m mch mm mu 4 w my 3911 9mm mang af rm W yxubahd y mm Mm 1 nrllYz F06 quot2 27m usngXcEL m ymbahd y Mmmmyma 1mm dnyhym39de Mammrmmm mm was my szmylz and mm mm My ammawnmm v mama awmg EXCILmd moMLM 3911 ms lElElmd nu mus 12m 13 Clmkunce u Andwhd mm am Hm key cm on 3914 m ceur Ce 1 Bounce wane magma Dug 3914 1mm ugh mm af rm ma Ce dwwnm m 52 onmo 3914 scrumquot 194qu was sun Lenny mi 3914 comm an my mum rm mu 9 5m x mam mm m gm m m Comma A m men number s m m mmyebble ms In Ce A5 m msmdmi damnun arm mm Hm I g 7 35MB N Chckomlelmaseyanm 1 knmsmm a 7 ugh m expmd m mam um 7 mm g 1 i away I i imean pebme mass 35H m istandavd mmquot uvpebme masses 48 7 In Ce Es emu 3914 an 35 1 ma Mtg 57 emu me ea 1 m mm ma shrdzrd demcnn m a mm mm The values in column B and rows 5 and 6 can be referred to later in our computation of pm Cell references in EXCEL can be either absolute or relative The absolute reference is useful because it allows us to refer back to a specific cell and avoids having that value change when a formula cell into which it is entered is copied The absolute reference is fixed reference Whether a reference is fixed or absolute depends only on how you refer to it The following table illustrates various absolute references Formula Result when formula is copied 2Al The column remains constant 7 fixed 2Al The row remains constant 7 fixed 2Al The row and column remain constant 7 fixed Now in our example the mean mass is stored in cell B5 and the standard deviation in cell B6 e7m7rh2 2 2 110 p 27z 2 When we go to compute the normal probability density of a mass distribution having average value 35018 and standard deviation 48 then we can make absolute reference to cells B5 and B6 Simplifying Complex Calculations Here are some suggestions that might be useful when undertaking complex computations of the sort that you might be asked to do on the job This advice applies whether you use PSIPlot or EXCEL We can break any complicated mathematical expression down into parts that can be solved for individually and then combined later in a complicated mathematical expression The computation of pm is actually not that complex and you might prefer not to take this approach in the future but the suggestions are made for you to consider When complicated expressions are broken down into parts individual parts can be checked for accuracy and there is less likelihood of error showing up in the final solution 1 In the present application we could compute the factor separately and store it in a cell for later 27z use To do this go to cell B7 and type in 12 3141593B6quot2quot05 In cell A7 enter Inverse of square root of 2pisquot2 This will identify the value 00083 that appears in cell B7 We could also calculate out the variable 2 Go to cell 12 and enter 127B5B6 This is the z value for a mass of 100 grams You should get Z 521 returned to cell J1 You may be saying to yourself that you could just as easily have entered Ize 3501sy4s But again W p r 39 h h 39 quot 39 h39 39 39 you a simple a r help you 39 rquot 4 we just dealtwith The present example is offered as an illustration of the possible use Copy this cell into cells J3 through J52 Computing the probability distribution pm Go to column K and enter the following formula in cell K2 B7 EXP 05 J2A2 1 7 is the constant V 27I m m m J2 is z for a pebble having mass 200 grams The J39s equal S EXP is the natural base 9 S hurt Cut Keys Copy cell K2 into cells K3 through K52 Ctrl C to co py Your window should look like the one below Ctrl V to past Plotting the Probability Distribution pm vs m We ll review the procedures for selecting nonadjacent columns Take Notes Click on the chart Wizard icon on the EXCEL A z I menu bar 39l see illustration at right I39D v This Will bring up the Chart Wizard Window below Select the illustrated options then click on Next Chan Wizald Step 1 of 4 Chan Type Standard Types l Custom Types ll Chart subtype ghart type catter with data points connected by mouthed Llnes Press and hold to Elew sample ltBaln I Nextgt I nish l Cancel El The next Window to appear requests information about the series of cells that you want to plot Click on the Series folder see below hark w ar Sign 2 039 4 Mmm gala vanqe Seuss m m r caLumns cm W m Ifyuuwantedtu r u m u M H vanuuspebblemasses Make S ome Clitk 0K tes changeme plulmleandaxlslabelstu sulth pmmem see beluw he re man man 39mee Wee emheee Legend Deeeeebe q heme r vvahahmtyD smhutmnarveh vaJueOOaxxs Pebb e Mass qvams lame V axxs thamhty Semnd ateqmv X1 axx 52mm va ue m axxs We em 5mer mm Mew Ehavl let I an hall Local H were rhanr E r 5 new weez ham E a meme m We eeee New Your Worksheet should look something like that shown below Probability Distribution of Pebble Masses Plubablllly Euu Pehhle Mass grams Changing Absolute Cell Values One Very powerful aspect of the absolute cell referencing features in EXCEL is that this Will allow you to reproduce the entire series of computations for a different mean and different standard deviation simply by changing the Values you entered in cells B5 and B6 Go ahead and change the mean Value to 400 Note What happens to your plot The change is automatically made to your plot Now change the standard deviation to 00 Note these various changes Experiment Plot Esthetics or 39 39 n quotIn 39 rr 39 L 439 ckgrounds text colors etc Several cosmetic features will be illustrated in the class piesentation so please take notes 39 39 39 To dn tbi 39 39 panel The menu shown below will drop down Click on Drawing Stan eld Favmattl q halt ll ooolo antlal Yaalhax Vawl q Extemal Date F l Pltluve pwotloole aeoowoo Probability Distribution ofPebble Mas vlsualm Web Wavdmt Plubabllly Pebble Mass lnramsl in appea AgtoShapesv Do l 39 39 IJDLaW39 G Add Text Box Format text in text box Format background in text box Format Cells Form at Chart 92 ut might end up e mi Mew insert ngmat Ian s gata Mnduw ea 5 mmssaeawm g zr l g li hv a 3511 49 mm Pmbammy we Statement ofproblem Compute and plot the normal probability densities ofpebble masses tabulated in Table 71 of Waltham39s text Clo 4mmv n Read Y gmquot U a a m U pyaaoensvaaemnasart maaensxus Generating Histograms with Excel I 39 E 1 histograms similar r r 39 l quot U eielii ie Iiiiilailyfm quot 39 l 39 39 H39 provided to increase your familiarity wiai Excel functionality Emel will generate data to hose we develoPed using PsiPlot First click on tools and examine clie amp down window see below 39 To install d1 Data A AnzlysisTool val r I n i cliek 0K and clie add in should take place Cancel Analysis TuoLDak Provides func ons and interfaces far Financial and scientific data analysis Nawxemmm m Tunis mzmutem andymlshmlld mwsee aDmaAmJyss aptmnl m 1m 521mm Dam Analysis aplmn an msnnzm who 5 542 m Mew lnsevt ngmat DD 5 Data Mndaw M J 532 an F7 Shavewmkhaak 1 MA S dEM Evatech n ass une asses 2 374 224 W39 i 389 242 i 358 255 i 395 255 i 371 255 L 334 259 i 224 277 i 335 283 4mm 17B 255 283 425 thn yml click an Data Analysis m m shave kg m fallnwmg wmdnw wm apn up Highligh m Hlsmgnm Analysis Tan and chck OK 15 m3 mm A W s w y Arum Wuhan MthVwkatnn arme u39nhtnn Mar Fewer mm 39 yourcursoi39 i ii 1 LL should look like that shown below JE Elle 5am Mew Insevt Foimez owls gaze Mndaw help I s L e A l mnmaa Mggeaglnv g zl gwiihl vl v Bl A2 L l AlallDlElFlGlHlllJ L Sorted Masses Ems i l 224 225 23 A 242 25a man i i 255 275 Input Range 5 i 25h 73m Cancel i l 2E5 325 WWW 4 L l 259 35 rtehels e vl i 277 375 5 i 283 4 LB I 253 425 r om z i L I 283 ASE 0 New kasheet Plv 172 284 475 g l 287 Sun r Ne whim k l 5 2am I39 Paveta Salted hlsl gvam 2 g r imitative eeeeneie i I havtout oz 1 3m r39 F la am Click on the 39 39 39 ou in the Finall click on the i i i L c 1 i ii L outprut to Your window should look something like the following 1m In ut mp9 Range 31 En Range cancel 39 Lahels lielD pi F guruutRange Mama a P New Worksheet Ely P New urkbnnk l Pareto sorted histugram I unwatlve Percentage I gain Output Also check the Chart Output option box Then click OK Your woiksheet should look like the following 3512 am Mew insevt ngmat m1 gate mm H An JDD EHQBLIVMEEEQ nv gi i D2 j 1Em A a 1 c 1W1 m 1n17 1 1 n E F G 1 H 1 1 1 J 1 K 1 L 374 224 419710 339 242 Histogram 358 2513 a 395 2513 g u 334 2139 1 E 335 283 2 2513 283 1 Bin 34m 283 1 374 284 423 287 338 29m 373 294 342 311 242 311 318 312 454 313 341 3117 mnandprennenc appendquot quot 39 quot 39 quot You can edit your histogram plot to increase plot size change label names etc Histogram 25 gt20 0 515 3 requency 510quot u 5quot H V o quot1n1n1 1 1 1 1 1 1391ng v39393939393 ewraewrsezveag uh Fnrmat Data 52 g1 pawl m NewDataeheelsgmome em 9mm m r5 Gavwdth u 3 r quotmm 25 M g1 5 n T nmnt Hand m a p ot of me probabmty densuy funcuon generated m we rst part of ms exercwse for me pebb e mass samp e Make sure u 5 correcuy abe ed and thatyourname 5 on n Elli Tomhwilson romwisonmaiwvuedu Dep l Geology and Geography West Virginia University v XQuVirgimuLiniu mn mi The cross sectional area tqry ideposi l39 see handon 7 The whickness M n sedhnneizry body varies as shown helow Olles on 9 Distance m Thickness m H mm 1 2mm 2 ADD 3 mm 2 5 15mm 2 1 2n n n Sediment Thickness Variations Thickness m F39uiy Thickness m 1 Thickness m n sun isnn ZUDEI mun Distance hm v XQququmwmq x ax3 bxz cx x Iax2bxcdx n 3 2 1 D For The 539 order polynomial you derive you39ll have 6 Terms including The consTunT J K L o Cuemmemslmagyanuw Scaled x Pvuducl anms lmegvalmnlacluvs laws as 15 34 15 a3 14 a2 15 al 12 mm mm o mm 39x T 2857E12x4 1303E08x3 2173E05x2 1423E02x 7784E02 l J l K l L l M l N l o Cuef cienls Inlegrallan c led x Fmdum actors lntegrahnn actors terms all 2EEE 12 115 5 72E13 3 2UE1B 1E3ECld 33 1 SEE DB 114 3 25EUQ l EUE13 62EIE04 52 gt2 TIEUS 13 7723E JE E DUENJEJ 7579304 31 142502 10 7 10503 ADUEHJB 2545434 30 V7 7BEVD239 1 v7 TBEVDQ 2 DUE03 r1 SBEHIQ sum D73 73 quotWuz39vllgvmilmvmxy nuTes To seT Thisupg l x 7 7 wt3 bxz cx x IULDC X Cdx7 3 2 1 I For The 539 order polynomial you derive you39ll have 6 Terms including The consTanT J o Pvuducl K L Cuemmems imagyanum Sealed x acims lnlegvaliunlacluvs levms as 15 a4 15 a3 14 a2 15 al in If you39ve followed along on This now open anoTher spreadsheeT and proceed wiTh The evaluaTions for The 5M order polynomial X Unvwm39 fuI calculus links hTTp archivesmaThuTkeduvisuolcalculus hTTparch ivesmoTh uTkeduuTkcalculus1 42TochTml hTTparch ivesmoTh uTkeduvisualcalculu s4subsTiTuTions3indexhTm I Wsz39vilginnUmwaly aT is The volume of MT 7Fuji The volume of each of The liTTle disks see below represenTs a AV v MalVirgmuleu mI 2 is The volume of a 7quot dZ disk having radius r and Thickness dz Zmax VIZ nde ToTal volume The sum of all disks wiTh Thickness dz 39Vmamuuumwmg fhaf for39 MT Fuji r2 can bet 13y The following polynomial r2 gigwoom To find The volume we evalua re The defini re in regr al 4002 7 sooJZ 3 V 400km2 l0 3 J5 V jg dzi 3Mdz7rj3400dz 0 3 0 Jg 0 V me wwummsu Eldefinifequot solufion V J 7271de 3 2 15 4002 8002 4002 6 15x5 0 7r600 16001200 2007 628km3 VwugiiwLiiuLi relief in about 11 kilometer J cover The base of This volcpno n u I 398 y Wz mnuLumnlg V infegr a l e only fnbm ya139ion of M11 Fuji is 339 7 2 wM40W 3 J3 Consider who happens when z3km Ramusm M m mmin 39v mmmmuaim The ToTaI naTuraI sTr39ain s is The sum of an infiniTe number of infiniTer small exTensions In our example This gives us The definiTe L inTegrul I 1dL l lll 1 L lnLgt1nLgt In 5 lnS muLmu9 v 2 can simpiifyihe prubiem and siiii ubmm ausefui resuii Appruximmeihe average densquot 25 11000 N 2 R 2 MZ47quot PAY MI 47W pdr F y WWW can smpwy we prub em and SM ubmm ausefu resuquot Appmmmme average densquot as 1 1000 mu m I 4nr211000drJ39 4Irrz4500dr n ma 4 m 4 m 37 A 3m311L The result 602 X m w m m 10 Kg Is clos to me 44000 18000 y 3 m3 3 may genem y accepted quot 3 8quot value of 597 X 104 kg V0 unwath 2mg 82 is The disTance in km f bm The base of The EarTh39s crusT i DeTermine The heaT generaTion raTe GT 0 10 20 and 30 km from The base of The crusT 0 kW kW 10 kW kW EH EW 39 km ii WhaT is The heaT generaTed in a in a small boxshaped volume Az Thick and 1km x 1km surface Since AVAAz AV Azkm3 Vmwmm 139 generated will be QAV QAzkm3 This is a differential quantity so there is no need to integrate iii amp iv Heat generated in the vertical column In this case the sum extends n over a large range of AZ so q Q AZ However integration is qJdz the way to go 1de EU v quotguiiummsm the flow rate at the a require evaluation of the definite integral qjde 30 q 225kW 40O vi To generate 100MW of power 100 OOOkW 4444ka 225kVmZ vgtlnmelmubm Volume of The ear39Th an obIaTe spher39oid 4le In This equaTion r39 varies from r39e aT The equaTor39 To r390 aT The poles z r39epr39esenTs disTance along The ear39Th39s r39oTaTion axis and varies from r39P To r39P The equaTor39iaI radius is given as 6378km and The polar39 r39adius as 6457km v muunmism i WhaT is The volume of The disc in Terms of r and AZ ii WriTe an expression for The volume of The EarTh assuming iT is filled by a sTack of N disks wiTh Thickness AZ iii WhaT happens To This expression when AZ goes To 0 vgtlugmiUmLmn iv Given re 6378km and rP 6357km whaT is The volume of The earTh v How does This compare To The volume of a sphere wiTh average radius ra rP2 ie 63675km v Vasmwmummsuy 29 t t0 eXp xX i CalculaTe The area in discreTe form using recTangles wiTh widTh AX and heighT Ti LeT39s bring up Excel and Tackle This one 39Vwmngmummsg 1 Hand integral worksheets in before leaving 2 Finish up problem 97 today 3 We39ll talk more about problem 99 and 910 on Thursday and have a final review V mmiinmmn Geol 351 Mathematics for Geologists Extra Credit Intro Statistics Exercise The data used in this problem are taken from an exercise prepared by Stephen Nelson Tulane University for his Geology 204 Natural Disasters class see httpwwwtulane eduNsanelsonge01204hmwk5700htm You can visit Dr Nelson s site for more details on the problem In this problem we will analyze maximum discharge data from Dry Creek 7 a stream in Louisiana 7 to estimate ood frequency and the annual probability of occurrence for oods of a given discharge Our interest here is to introduce you to the basic idea and also to give you some additional experience using PsiPot and Excel to manipulate plot and analyze data The data are contained in an Excel le Dischargexls placed on the class HDrive In this exercise you will learn how to use Excel and PsiPlot to 1 rank a list of data peak discharge 2 compute the recurrence interval for observed discharges 3 generate a lognormal plot of the recurrence intervals vs discharge 4 predict recurrence intervals for discharges larger than those observed in the time period covered by the data 50 years and 100 years 5 and finally calculate the exceedence probability 1 Compute the rank EXCEL Using Excel gt gt Help we nd that Excel has a spreadsheet function RANK that will allow us to rank a column or specified range of data in a worksheet Rank Returns the rank of a number in a list of numbers The rank of a number is its size relative to other values in a list If you were to sort the list the rank of the number would be its position Excel Smtax RANKnumberref0rder Number is the number whose rank you want to find Ref is an array of or a reference to a list of numbers Nonnumeric values in ref are ignored Order is a number specifying how to rank number I If order is 0 zero or omitted Microsoft Excel ranks number as if ref were a list sorted in descending order I If order is any nonzero value Microsoft Excel ranks number as if ref were a list sorted in ascending order RANK gives duplicate numbers the same rank However the presence of duplicate numbers affects the ranks of subsequent numbers For example in a list of integers if the number 10 appears twice and has a rank of 5 then 11 would have a rank of 7 no number would have a rank of 6 Examples IfAl A5 contain the numbers 7 35 35 l and 2 respectively then RANKA2A1A51 equals 3 RANKA1A1A51 equals 5 Excel Calculation In cell D5 of dischargexls type in the following RANKC5C5C250 In this Excel formula C5 is the number whose rank we wish to determine C5C25 specifies the range of numbers relative to which the value C5 is to be ranked The dollar sign before each row number indicates that the row number will remain unchanged when the formula is copied we ll talk more about this in a later exercise Copy this formula into cells D6 through D25 Examine the formula in some of the copied cells and note that the references to rows 5 through 25 remains fixed The last term in the RANK spreadsheet function is 0 and this tells Excel to rank the numbers in descending order 7 that is the largest number gets the smallest rank Column D now contains the rank for each value of discharge in descending order PsiPlot Calculation Copy Dischargepdw from the HDrive to your GDrive Open the file and note that the time is in decimal years with the beginning time set to 0 years Make a copy of the discharge rates in column3 C3 Then with the cursor in this column column3 click Column gtgt Rank and select the descending option Discharge rank will now appear ion column 3 Rename column 3 Rank 2 The recurrence interval is calculated using the following formula R n l m where R is the recurrence interval n is the number of years spanned by the data n20 and m is the rank of each discharge value in the data set Excel Calculation Thus to calculate the recurrence intervals R associated with the discharge listed in cell C5 enter the formula 2lD5 Enter and copy this formula into cells E6 through E25 Look at the numbers and note that the smallest discharge 950m3sec has the highest recurrence interval 1l This is what we would expect ie a ow rate greater than or equal to the minimum ow rate is likely to occur every year PsiPlot Calculation In PsiPlot use the MathgtgtTransformgtgtOne Line equation option Enter the equation R2lRank Discharge rank will appear in column 4 3 Generate a log normal plot of the recurrence interval versus discharge Remember that your dependant variable is the discharge and the independent variable is the recurrence interval Excel Plot To get the log normal plot in Excel double click the xaxis to bring up the Format Axis window Check the logarithmic scale check box The range should be 1 to 100 Double click the discharge axis and set the minmax values to 0 and 5000 To etthe 39 39 quot b quot onthex quot 4 39 format window Under the Axis Mode drop down list select the log decimal display option and under Range set the minimum to 1 the max will be 100 On the Discharge axis set the min to 0 and max to 5000 Leave the Axis Mode set on Linear Norm 4 Predict discharge for events with 50 and 100 year recurrence intervals In Excel solve graphically by visually tting a best t line through the data points 1 H 100 43911 1 1 0a ooyeaI recurrence intervals PsiPlot Commotion In PsiPlot we will use the MathgtgtFittinggtgtUser De ned approach Set the parameters in the User De ned Fitting window as follows INDVAR R DEPVAR Discharge PARAMS EQUATIONS DischargeAlog10RB INIT PARAMS A 000 B ENDMODEL Click on the Compile button should get a compile success il message box click OK The click Solve Take the default Save Data Options click OK The User De ned Fitting Report Window will come up Note the correlation coefficient corrl under the Goodness ofFit Statistics This should be pretty high 7 about 099 Click OK to exit Your spreadsheet will now contain three additional columns One column will contain the slope and intercept l TImE Dlschavgel Rank l Recurrencelnl l LOGRI l Pamm5 l 557 l Dep l 17 595 5555 19 555539 1552539 5 522315755575 1552539 1555 347539 27 5535 1455 5555 12 5555 15557 5 2215 1522 5551 15557 1354 5523 3 15571 1555 5555 a 5555 2 2222 5 3455 2 2222 1554 4255 r 2 5751 3155 5555 15555 25 5555 13515 25 5555 3254 3157 Now given the slope and intercept compute the peak discharge for 50 and 100 year events ie oods with 50 and 100 year occurrence intervals 5 Finally compute the J r 39 39 quotquot The J r 39 39 quotquot P6 is simply the reciprocal of the recurrence interval PelR If the recurrence interval is 5 years for example it s probability of occurrence in any given year is 15 or 20 Nelson notes that ood stage occurs when discharge reaches 2000 m3 sec What is the occurrence interval and exceedence probability for a ood with this discharge Nelson asks us to consider the following Someone has offered to sell you a 4bedroom 2story house with a 2 car garage and swimming pool on a 1 acre lot on a relatively at piece of ground on the banks of Dry Creek for what seems like a reasonable price of 50000 The last time the house was ooded it cost 40000 to repair the ood damage How often has the house been ooded in the last 20 years What is the probability that the house will be ooded in the first year that you own it Would you still consider buying the house Why or why not If you decide to do this for extra credit it will be due November 18th the last Thursday before Thanksgiving
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