.MCAD 306000000 Z  docDocument<mcObjectIee (d2_graph_format graphData axisFormatLLtrace2D      dim_formatCmasslengthtimecharge temperature luminosity substanceew RomanNumericalFormat@dii shpRectEfmcDocumentObjectStateJ mcPageModel:????mcHeaderFooter99 ComputeEngine=BuiltInsBO SerialAnyvalH@P@H @Q@HR@HMbP?S@H units_classA TextState0 TextStyle/@ Arial Normalfont_style_list> font_style?  VariablesTimes New Roman?  ConstantsTimes New Roman? TextArial? Greek VariablesSymbol? User^1Arial? User^2 Courier New? User^3System? User^4Script? User^5Roman? User^6Modern? User^7Times New Roman? SymbolsSymbol? Current Selection FontArial? Undefined Font? HeaderArial? FooterArial? Rotated Math FontTimes New Roman/ TextRegion docRegion7shpBoxD 1C )):): CharacterMapRangeMap,7ENVIRONMENTAL PHYSICS COMPUTER LAB #2 The Hubbert Model ChrPropMap(7 RangeElem-7  ChrPropData) RangeData.lArial0,0,128 ParPropMap*7 -7  ParPropData+EmbedMap" -LinkMap 7 -7LinkData!@NormalArial @Da@vp888D8 We continue the work done in Lab #1, where we modeled the annual consumption as growing at a constant rate. If it is perceived that the resources are approaching depletion, the rate of growth is decreased as other resources are used instead. We will denote nj the yearly consumption during year j, and qj the cumulative consumption until year j (not included). In this program we use the "step function" also known as the Phi function or Heaviside function: F(x) = 1 if x > 0 and if x = 0, F(x) = 0 if x < 0. We use it in the model to insure that the annual consumption is zero when the cumulative consumption reaches the value of the total resources available qtot. The continuum version of this model was invented by the American geophysicist Hubbert (see Ch.3 of McFadden, Hunt and Campbell), and predicted correctly a maximum in the US oil production. We are going to use the discreet version of the Hubbert model to simulate the world oil consumption. In 1992 the annual world oil consumption was 125 quads/yr. The oil resources remaining that year were 7000 quads.(`8-)@ Arial0,64,128-)n Arial0,64,128-)N Arial0,0,128-,)n Arial0,64,128-)N Arial0,0,128-)n Arial0,64,128-)@ Symbol0,64,128-)n Arial0,0,128 -!)@ Arial0,0,128"-#)@ Arial0,0,128 $-%)@ Arial0,0,128"&-')@ Arial0,0,128$(-))@ Arial0,0,128&*-+)@ Arial0,0,128(,--)@ Arial0,0,128*.-/)@ Arial0,0,128,0-1)@ Arial0,0,128.2-3)@ Arial0,0,12804-5)@ Arial0,0,12826-7)@ Arial0,0,12848-9)@ Arial0,0,1286:-;)n Arial0,0,1288<-=)@ Symbol0,0,128:>-?)n Arial0,0,128<@@-@A)n Arial0,64,128>@B-@C)N Arial0,0,128@@@D-@E)n Arial0,64,128@B@F-@G)@ Arial0,64,128@D@H-@I)@ Arial0,64,128@F@J-@K)n Arial0,64,128@H@L- @M)j Arial0,64,128@J@N-@O)n Arial0,64,128@L@P-@Q)@ Arial0,64,128@N@R-@S)@ Arial0,64,128@P@T-@U)n Arial0,64,128@R@T*8@V-8@W+"@X- 8@Y-8@Z!@NormalArial @[eqRegion3@D@#@\tree1 p@]1 @\@^1@@]@_1d@^n@`1@^0@a1@]125@b3@D@c1 p@d1 @c@e1@@d@f1d@eq@g1@e0@h1@d0@i3@DW4@j1 p@k1 @j@l1@@k@m1d@lYear@n1@l0@o1@k1992@p3@D@q1 p@r1 @q@s1d@rj@t1@r@u1t@t0@v1@t99@w3@D@x1 p@y1 @x@z1d@yk.0@{1@y0.05@|3@DL&@}1 p@~1 @}@1d@~k.1@1@~.0032@3@D@1 p@1 @@1d@q.tot@1@7000@3@DPW:m@1 p@1 @@1p@@@10@@10A@@10A@@1@@@1@@1e@Vq@1@@1d@j@1@1@1@@1e@Un@1@@1d@j@1@1@1@@1e@WYear@1@@1d@j@1@1@1p@@10@@10A@@10A@@1@@@1@@1@@@1d@q@1@j@1@@1d@n@1@j@1@@1@@@1@@@1d@n@1@j@1p@@1@@1@@@1t@1@1@k.0@1@@1d@k.1@1@j@1@@1d@\F@1p@@1@@1d@q.tot@1@@1d@q@1@j@1@@1@@@1d@Year@1@j@1@1@3@DWDX@1 p@1@@1@@@1@@@1@@@1f@ 188.169601@1@0@1@@1d@ _n_u_l_l_@1@ _n_u_l_l_@1@@1d@n@1@j@1@@1@@@1@@@1B@@1d@2.091@1@@1d@10@1@3@1@1990@1@@1@@@1t@1992@1@@1d@k.0@1@k.1@1@ _n_u_l_l_@1@@1d@Year@1@j@< ^N     @@D&300 Copyright @ Miron Kaufman 1998(@-@)@nArial*@-@+"@- @-@!@NormalArial @3@DG&"Hd@1 p@1@@1@@@1@@@1@@@1t@10000@1@0@1@@1d@ _n_u_l_l_@1@q.tot@1@@1d@q@1@j@1@@1@@@1@@@1B@@1d@2.091A1@A1dA10A1A3A1@A1dA1.992A1AA1dA10A1A3A1@A 1dA _n_u_l_l_A 1A _n_u_l_l_A 1@A 1dA YearA 1A jA; N^     A@DY<he44^4^AwNote a maximum in the annual rate nj. It occurs at j = k0/k1. Can you explain this? By experimenting with the values of k0 and k1 you can get qualitatively different outcomes. Run the simulation for: (a) k0 = 0.05 and k1 = 0.002; (b) k0 = 0.05 and k1 = 0.004. Describe qualitatively what happens in each case by looking at the plots of n versus time and q versus time.(#wA-#A)A~ 0,0,128A-A)A^ 0,0,128AA-A)A~ 0,0,128AA-A)A^ 0,0,128AA-A)A~ 0,0,128AA-A)A^ 0,0,128AA-@A)A~ 0,0,128AA-A)A^ 0,0,128AA -A!)A~ 0,0,128AA"-A#)A^ 0,0,128A A$-NA%)A~ 0,0,128A"A&-A')A^ 0,0,128A$A(- A))A~ 0,0,128A&A*-A+)A^ 0,0,128A(A,-A-)A~ 0,0,128A*A.-A/)An Arial0,0,128A,A0-A1)A^ 0,0,128A.A2- A3)A~ 0,0,128A0A4-A5)A^ 0,0,128A2A6-wA7)A~ 0,0,128A4A6A*wA8-wA9+"A:- wA;-wA-A?)A=nArial*A@-AA+"AB- AC-AD!@NormalArial AE@D44,7,7@Next we are going to fit the Hubbert formula: N(t)= Nmaxexp[-(t-Tmax)2/2s2] to the US annual oil consumption expressed in quads/year versus time. The data from 1980 to 1996 was obtained from the DOE web site: http://www.eia.doe.gov/(5AF-5AG)AEn Arial0,0,128AH-AI)AEN Arial0,0,128AFAJ-AK)AEn Arial0,0,128AHAL-AM)AE@ Arial0,0,128AJAN-AO)AE@ Arial0,0,128ALAP-AQ)AE@ Arial0,0,128ANAR-AS)AEN Arial0,0,128APAT-AU)AEN Arial0,0,128ARAV-AW)AEN Arial0,0,128ATAX-AY)AEN Arial0,0,128AVAZ-A[)AEN Arial0,0,128AXA\-A])AE@ Symbol0,0,128AZA^-A_)AE@ Arial0,0,128A\A`-Aa)AE@ Arial0,0,128A^Ab-Ac)AE@ Arial0,0,128A`Ad-Ae)AE@ Arial0,0,128AbAf-EAg)AEn Arial0,0,128AdAh-Ai)AE@ Arial0,0,128AfAj-Ak)AEn Arial0,0,128AhAl-Am)AE@ Arial0,0,128AjAn-Ao)AE@ Arial0,0,128AlAp-Aq)AE@ Arial0,0,128AnAr-As)AE@ Arial0,0,128ApAt-Au)AE@ Arial0,0,128ArAv-Aw)AE@ Arial0,0,128AtAx-Ay)AE@ Arial0,0,128AvAz-A{)AE@ Arial0,0,128AxA|-A})AE@ Arial0,0,128AzA~-A)AE@ Arial0,0,128A|A-A)AE@ Arial0,0,128A~A-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AA-A)AE@ Arial0,0,128AAAF*A-A+"A- A-A!@NormalArial A3@D8YA1 pA1 AA1dAusoilA1pAA10AA10AAA10AAA10AAA10AAA10AAB10AAB10ABB10ABB10ABB10ABB10ABB10ABB10ABB10ABB 10ABB 10AB B 10AB B 10AB B 10AB B10AB B10ABB10ABB10ABB10ABB1@BB1B35.864B1B34.663B1B34.735B1B33.841B1B33.527B1B 32.845B1B 33.553B1B 34.211B1B 34.222B1B 32.865B1B32.196B1B30.922B 1B31.051B!1B30.054B"1B30.231B#1B31.931B$1B34.202B%1B29.0B&1B21.0B'1A18.0B(1A16.0B)1A12.5B*1A7.3B+1A5.1B,1A2.5B-3@DB.1 pB/1 B.B01dB/yearB11pB/B210B1B310AB2B410AB3B510AB4B610AB5B710AB6B810AB7B910AB8B:10AB9B;10AB:B<10AB;B=10AB10AB=B?10AB>B@10AB?BA10AB@BB10ABABC10ABBBD10ABCBE10ABDBF10ABEBG10ABFBH10ABGBI10ABHBJ10ABIBK1@BJBL1BJ1996BM1BI1995BN1BH1994BO1BG1993BP1BF1992BQ1BE1991BR1BD1990BS1BC1989BT1BB1988BU1BA1987BV1B@1986BW1B?1985BX1B>1984BY1B=1983BZ1B<1982B[1B;1981B\1B:1980B]1B91970B^1B81965B_1B71960B`1B61955Ba1B51950Bb1B41940Bc1B31930Bd1B21920Be3@DP ZBf1 pBg1 BfBh1dBgjBi1BgBj1tBi0Bk1Bi24Bl@D ? ZZ@When using a semi-logarithmic graph we see that the linear dependence predicted by the exponential model does not describe well the data from 1970 to 1996. (Bm-Bn)Bln Arial0,0,128*Bo-Bp+"Bq- Br-Bs!@NormalArial Bt3@D@ Bu1 pBv1BuBw1@BvBx1@BwBy1@BxBz1fBy35.864B{1By2.5B|1BxB}1@B|B~1B|B1BwB1dBusoilB1BjB1BvB1@BB1@BB1BBB1dB1.996B1BB1dB10B1B3B1B1910B1BB1@BB1BB1BB1dByearB1BjB= NO     B@D ?   COPYRIGHT @MIRON KAUFMAN, 1998(B-B)BnArial*B-B+"B- B-B!@NormalArial B@D @9  988B0We do the fitting by using the minerr(a,b,c) function. This returns the values of the parameters a, b, c that come closest to satisfying the equations and inequalities in a solve block. A solve block must start with the statement given. Mathcad evaluates the minerr function using the Levenberg-Marquardt method which requires a guess value for each unknown to begin the search for solutions. So the guess values must appear in the worksheet before given. The minerr function returns a vector whose first element is a, second element is b and third is c.(0B-B)Bn Arial0,0,128B-B)Bl Arial0,0,128BB-B)Bn Arial0,0,128BB-9B)Bn Arial0,0,128BB-B)B~ 0,0,128BB-B)Bl Arial0,0,128BB-dB)B~ 0,0,128BBB*0B-0B+"B- 0B-0B!@NormalArial B@DY k h %vvvGuessing values:(B-B)Bn Arial0,0,128*B-B+"B- B-B!@NormalArial B3@D[ q h )B1 pB1 BB1dBN.maxB1B90B3@D[ 1q h *B1 pB1 BB1dBT.maxB1B1975B3@DHZ nl Wh +B1 pB1 BB1dB\sB1B20B@D   c(-This is the function that we fit to the data:(-B--B)B~ 0,0,128*-B--B+"B- -B--B!@NormalArial B3@D( >  bB1 pB1 BB1@BB1dBhubbertB1pBB1 BB1 @BB1 @BB1dBTB1BN.maxB1BT.maxB1B\sB1BB1dBN.maxB1BB1dBeB1BB1K@BB1BB1p@BB1BB1dBTB1BT.maxB1B2B1BB1tB2B1BB1dB\sB1B2B@D  <(We start the solve block with:(B-B)B~ 0,0,128*B-B+"B- B-B!@NormalArial B3@D   >B1 pB1BgivenB3@D /; " ;B1 pB1,BB1@@BB1@BB1tB0B1B24B1BB1dBjB1BC1p@BC1CC1@CC1dCusoilC1CjC1CC1dChubbertC1pCC1 CC 1 @CC 1 @C C 1@C C 1dC yearC 1C jC1C N.maxC1C T.maxC1C\sC1B2C1B0C3@Dx $  C1 pC1,CC1tC1C1C1C3@D  7$ * C1 pC1,CC1tC2C1C2C@DQ - ` B0%n%nAThere are three constraints above. They use the boolean equals which is obtained from the evaluation palette or by simultaneously hitting control and = keys. The first constraint is the least squares criterion: the sum of squared distances from the data to the curve is as small as possible. The other two constraints are required by the minerr function which needs the same number of constraints as arguments.(m1C-1C)C~ 0,0,128C -C!)Cl Arial0,0,128CC"-C#)Cl Arial0,0,128C C$-C%)Ch Arial0,0,128C"C&-0C')C~ 0,0,128C$C&C*C(-C)+"C*- C+-C,!@NormalArial C-3@D h AC.1 pC/1 C.C01p@C/C110C0C210AC1C310AC2C41@C3C51C3\sC61C2T.maxC71C1N.maxC81C/C91dC8minerrC:1pC8C;1 C:C<1 @C;C=1dC<N.maxC>1C<T.maxC?1C;\sC@@D) @ 8 O87Those are the values of the 3 parameters max, TM and s:(6/7CA-/CB)C@~ 0,0,128CC-CD)C@^ 0,0,128CACE-CF)C@~ 0,0,128CCCG-CH)C@@ Symbol0,0,128CECI-CJ)C@@ Arial0,0,128CGCICA*7CK-7CL+"CM- 7CN-7CO!@NormalArial CP3@D+ A 8 UCQ1 pCR1CQCS1dCRN.maxCT1CRCU1+@CTSerial_DisplayNodeFCV1CTCW3@DD $a X TCX1 pCY1CXCZ1dCYT.maxC[1CYC\1+@C[@FC]1C[C^3@Dj | x SC_1 pC`1C_Ca1dC`\sCb1C`Cc1+@Cb@FCd1CbCe@D ?  \ COPYRIGHT @MIRON KAUFMAN, 1998(Cf-Cg)CenArial*Ch-Ci+"Cj- Ck-Cl!@NormalArial Cm3@D 0 WCn1 pCo1 CnCp1dCoYEARCq1CoCr1 @CqCs1tCr1900Ct1Cr1905Cu1Cq2100Cv3@D' "B( Cw1 pCx1CwCy1@CxCz1@CyC{1@CzC|1fC{35.864C}1C{ 0.119509C~1CzC1dC~N.maxC1C~C1 CyC1@CC1dCusoilC1CjC1CC1dChubbertC1pCC1 CC1 @CC1 @CC1dCYEARC1CN.maxC1CT.maxC1C\sC1CxC1@CC1@CC1BCC1dC2.1C1CC1dC10C1C3C1C1900C1CC1dCT.maxC1CC1 CC1@CC1dCyearC1CjC1CYEARC, ^^     C@DE^0-$-$@]The model predictions for the US oil consumption (quads/year) in the years 2000 and 2050 are:(]C-]C)C~ 0,0,128*]C-]C+"C- ]C-]C!@NormalArial C3@Dt=aC1 pC1CC1@CC1dChubbertC1pCC1 CC1 @CC1 @CC1tC2000C1CN.maxC1CT.maxC1C\sC1CC1+@C@FC1CC3@Dn=`C1 pC1CC1@CC1dChubbertC1pCC1 CC1 @CC1 @CC1tC2050C1CN.maxC1CT.maxC1C\sC1CC1+@C@FC1CC@D)V Copyright @ Miron Kaufman 1998(C-C)CnArial*C-C+"C- C-C!@NormalArial