ࡱ; ~  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}Root EntryF ஺Book )YCompObj^SummaryInformation(p FMicrosoft Excel 5.0 WorksheetBiff5Excel.Sheet.5;  Oh+'Oh+'0@ P08G,$n     ^ ``iݜ*o?H6A^BbzbO/(fvEE>j(9jJ2 """)))000___UUUMMMBBB999 ```( +9%I;/]E:IS+!!lYQGj2ga1aS {Cg..&YQFh.IRj#SjJu3lJA7e ,N, QdoV YC6r3w_GqC-}zn#&s[ Rp$L 6n{DuqVJ4H2Aph6B"}3%6Z\H"BMXR ӕ$sVo<gX 6END.PIC FEIGEN31.PIC FEIGEN25.PICM% FEIGEN24.PIC. FEIGEN23.PIC 8 FEIGEN22.PICC FEIGEN21.PICM ENLAR225.PIC   41}&1;cGPXY/fUm  dMbP?_*+%d LB"F??LU} F} F} F} @F}  F} `F} @F} F} F} F} F} F} (F 1A,@0.t O OO )L  O   OZLOO 1&OOOrhCurve fitting 2.NRE 545 S. Arlinghaus\T5. Bounded curve fitting--using polynomial segments to link a finite set of points.ZR The following strategy enables one to link a finite set of points with pieces\Tof a cubic curve (polynomial of de  \ sandy B CRCSPLN2/ __123Graph_A;.;__123Graph_AENLAR225.PIC;_.;__123Graph_AFEIGEN21.PIC;.0;__123Graph_AFEIGEN22.PIC;.0;__123Graph_AFEIGEN23.PIC;#.0;__123Graph_AFEIGEN24.PIC;/S.0;__123Graph_AFEIGEN25.PIC;I.0;__123Graph_AFEIGEN31.PIC;MC.0;__123Graph_AFIG26END.PIC;/A.7__123Graph_AGR-2.PIC;YI.;__123Graph_AORBIT215.PIC;G.;__123Graph_AORBIT225.PIC;=.1;__123Graph_AORBIT235.PIC; .;__123Graph_AORBIT245.PIC;O.;__123Graph_AORBIT255.PIC;.:__123Graph_AORBIT28.PIC;.1;__123Graph_APOS-GRF2.PIC;.;__123Graph_APOS-GRF3.PIC;e.;__123Graph_ASPLINE-2.PIC;.9__123Graph_AZOOM-2.PIC;./ __123Graph_X;).;__123Graph_XENLAR225.PIC;o.;__123Graph_XFEIGEN21.PIC;.;__123Graph_XFEIGEN22.PIC;.;__123Graph_XFEIGEN23.PIC;A.;__123Graph_XFEIGEN24.PIC;.;__123Graph_XFEIGEN25.PIC;.;__123Graph_XFEIGEN31.PIC;.;__123Graph_XFIG26END.PIC;Y.7__123Graph_XGR-2.PIC;.;__123Graph_XORBIT215.PIC;.;__123Graph_XORBIT225.PIC;'.1;__123Graph_XORBIT235.PIC;m.;__123Graph_XORBIT245.PIC;.;__123Graph_XORBIT255.PIC;.:__123Graph_XORBIT28.PIC;=.1;__123Graph_XPOS-GRF2.PIC;.;__123Graph_XPOS-GRF3.PIC;.;__123Graph_XSPLINE-2.PIC;.9__123Graph_XZOOM-2.PIC; _Regression_Int= 8X@"1gCourier1gArial1gArial1gArial1gArial1gArial"$"#,##0_);\("$"#,##0\)"$"#,##0_);[Red]\("$"#,##0\) "$"#,##0.00_);\("$"#,##0.00\)%""$"#,##0.00_);[Red]\("$"#,##0.00\)5*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_),))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)=,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)4+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_) General_)                + ) , *  !  #  !  ! CRCSPLN2WCurrent ZOOM-2.PICh SPLINE-2.PIC POS-GRF3.PIC POS-GRF2.PIC> ORBIT28.PIC ORBIT255.PICW ORBIT245.PIC ORBIT235.PIC ORBIT225.PIC ORBIT215.PICGR-2.PIC1  FIG2gree 3). Different cubics are spliced together to[Sform a single smooth curve composed of segments of different cubics. The splice is` Xat the data point; at the splice, the slope of the tangent line to the cubic on the leftb Zshould be the same as the slope of the tangent line to the cubic on the right. The smooth` X curve is often called an interpolating curve; estimates of unknown y-values may be madef ^between known x-values; unlike the curve1.wk1 there is no logical way to extrapolate, however.^ V The technique for making all the equations work out rests on elementary calculus,bZto ensure continuous curvature (that the second derivatives be continuous), and on linear JBalgebra. [Interested students might look at the handout in 2044.]WOThe following example shows how to perform the interpolation in black-box mode.ZRThe idea is to find five cubics which, when linked will form a single smooth curve^Vlinking the six points. The reason to use cubics is that from linear beam theory, "the`Xfourth derivative of the displacement of a beam is zero along any interval of the x-axisf^that contains no external forces acting on the beam" (Applications of Linear Algebra, p. 677).aY The idea is to find, simultaneously, values for a, b, c, d, in each interval that will `Xcreate a cubic piece which when linked to the next piece will be smooth; generally, each:2cubic segment is of the form ax^3 + bx^2 + cx + d."CUBIC SPLINE INTERPOLATION"EXAMPLE--CHOOSE SIX POINTS(x1, y1)=(1,1.25)(x2, y2)=(2,1.75)(x3, y3)=(3,3)(x4, y4)=(4,2.5)(x5, y5)=(5,2)>} 0`^`_dfdjbfN[^bdjed>&& !A,@"0.#t $O% &OO')L()O* +O,Z-L./O0O1 2345617&O89:O;O<rh= >?/O (x6, y6)=(6,1.75)!F">Basic Theorem--see notes in 2044 for derivation--from textbookQ#IGiven n points (x_1,y_1),...,(x_n,y_n) with x_(i+1)-x_i=h, i=1,2,...,n-1,$the cubic spline%&@&8a_1(x-x_1)^3+b_1(x-x_1)^2+c_1(x-x_1)+d_1, x_1<= x <= x_2'@'8a_2(x-x_2)^3+b_2(x-x_2)^2+c_2(x-x_2)+d_2, x_2<= x <= x_3(S(x) =@(8........................................................)a)Ya_(n-1)(x-x_(n-1))^3+b_(n-1)(x-x_(n-1))^2+c_(n-1)(x-x_(n-1))+d_(n-1), x_(n-1) <= x <= x_n*@+8that interpolates these points has coefficients given by,-a_i = (M_(i+1)-M_i)/6h. b_i = (M_i)/21/)c_i = (y_(i+1)-y_i)/h-((M_(i+1)+2M_i)h/6)0 d_i = y_i1=25for i = 1,2,...,n-1, where M_i=S''(x_i), i=1,2,...,n.3K4CSome substitutions and algebraic manipulations make it possible to P5Hrewrite this system of equations as a single matrix equation. When thisO6Gsingle matrix equation is coupled with the physical assumption that theS7Knatural spline is allowed to extend freely beyond the interpolated region--P8Hthat is, when the ends shoot off on a straight line so that S''(x)=0 and?97M_1=M_n=0--the following matrix equation is the result.: ;1 0 0 0 ... 0 0 0 ;M_1~ ; <1 4 1 0 ... 0 0 0 <M_2< y_1-2y_2+y_3 =0 1 4 1 ... 0 0 0 =M_3= y_2-2y_3+y_4> ?........................ ? * ?... ?=? ............D6 l JU NNVo D "5 A OTSWTC AKK @AA,@B0.Ct EOF GOOH)LIJOK LOOZQ LR S OT OU V X Y Z [ 1\ &O] _ @ A0 0 0 0 ... 1 4 1AM_(n-1)Ay_(n-2)-2y_(n-1)+y_n B0 0 0 0 ... 0 0 1 BM_n~ BCdE\Use this latter equation to find coefficients for the cubic spline to fit the sample points. FEquation from Black Box:*G? GM1~ G G (x1, y1)=(1,1.25)*H?@? HM21H ???H (x2, y2)=(2,1.75)*I?@?I times IM3 I=I 6 times-I H ?@I (x3, y3)=(3,3)*J?@? JM4+J I @J (x4, y4)=(4,2.5)*K?@? KM51K ?J @?K (x5, y5)=(5,2)*L? LM6~ L L (x6, y6)=(6,1.75).O&Solve the matrix equation for M1...M6: QM1*Q?~ Q  RM2Rwo%ѿRwo%?RmVi_Rc,?Rc,sRc,s?~ R ? SM3 S=S6 timesSmVi_?SmVi_SmVi_?Sc,Sc,?Sc, S times~ S  TM4Tc,Tc,?Tc,TmVi_?TmVi_TmVi_?~ T  UM5Uc,s?Uc,sUc,?UmVi_Uwo%?Uwo%ѿ~ U ? VM6*V?~ V ~ X~ XYZ?Y GRRRRA,@0.t O OO)LO OZLOO 1&OO c1 c2 c3 c4 c5~ ?~  M1=LK"Z?'D D DD~ ? G=?'D D DD~ @JS?D[ M4=u*M'D D DD~ @Im?D\ M5=Ȝӿ'D D DD~ ?D M6 d1 d2 d3 d4 d5~  M1~ ? G=?~ @:fT?+H3M~ @u_ ]?O@aӿ~ @>6Equation of cubic spline to fit the six sample points.IAFive equations to fit curve pieces between the six sample points.g|_\?(x-1)^3 +0(x-1)^2 +0.174641 (x-1) +1.25(x-2)^3+0.97607(x-2)^2+1.15071 (x-2) +1.75 S(x)=B_zs?(x-3)^3-1.6543(x-3)^2+0.47248 (x-3) +3:f(x-4)^3+0.39114(x-4)^2-0.79066 (x-4) +2.5u(x-5)^3+0.08971(x-5)^2-0.3098 (x-5) +2:G Tk0F*>>>>0<@DDDDBM}A,@0.t O OO)LO OZLOO 1&OOD<Plot point by point; tenths of a unit between sample points.x value spline fit~ ??Q Gg|_\?LL6Z?L?~ [@S-EH?~ ^@lG,?~ @`@ò?~ a@kw&ls?~ ?QO? ?~ d@)x ?~ @e@i:@ս?~ f@Nf3?~ g@$tOO?~ @?X N?L"M=?L7/same value using equations from above and below~ `s@y&H@@~ t@-.E@~ t@sÖ@~ @u@ma0Y@:nH)---------------------A,@0.t O OO)LO OZLOO 1&O~  @I_C@~ v@@~  w@10@~ w@XsX@~ `x@M5K@~ @@X NJ5o?LT?L+H3M?L@7/same value using equations from above and below~ y@v?e@~ @z@%q@~ z@/\@~ {@>L@~ @XV@~ |@Pn>"d(@~ `}@ȗz@~ ~@@~ ~@&$A@~ @@R Hu?L_ ]?LO@a?L7/same value using equations from above and below~ @oU+v?~ @@z4?~ @O?~ @ Ca7?~ @?~ @Fwp?~ Ё@c'?~  @U0Q?~ p@̳PZ?~ @9 ?8-----------------------A,@~ ??70.325358*(C103-1)^3+0*(C103-1)^2+0.174641*(C103-1)+1.25~ [@?70.325358*(C104-1)^3+0*(C104-1)^2+0.174641*(C104-1)+1.25QA,@0.t O OO)L O   O Z LOO 1&OOOrh /O~ ^@?70.325358*(C105-1)^3+0*(C105-1)^2+0.174641*(C105-1)+1.25~ @`@?70.325358*(C106-1)^3+0*(C106-1)^2+0.174641*(C106-1)+1.25~ a@?70.325358*(C107-1)^3+0*(C107-1)^2+0.174641*(C107-1)+1.25~ ??70.325358*(C108-1)^3+0*(C108-1)^2+0.174641*(C108-1)+1.25~ d@?70.325358*(C109-1)^3+0*(C109-1)^2+0.174641*(C109-1)+1.25~ @e@?70.325358*(C110-1)^3+0*(C110-1)^2+0.174641*(C110-1)+1.25~ f@?70.325358*(C111-1)^3+0*(C111-1)^2+0.174641*(C111-1)+1.25~ g@?70.325358*(C112-1)^3+0*(C112-1)^2+0.174641*(C112-1)+1.25~ @F>-0.87679*(C113-2)^3+0.976076*(C113-2)^2+1.150717*(C113-2)+1.75~ @j@F >-0.87679*(C114-2)^3+0.976076*(C114-2)^2+1.150717*(C114-2)+1.75~ k@F >-0.87679*(C115-2)^3+0.976076*(C115-2)^2+1.150717*(C115-2)+1.75~ l@F >-0.87679*(C116-2)^3+0.976076*(C116-2)^2+1.150717*(C116-2)+1.75~ n@F >-0.87679*(C117-2)^3+0.976076*(C117-2)^2+1.150717*(C117-2)+1.75~ @F >-0.87679*(C118-2)^3+0.976076*(C118-2)^2+1.150717*(C118-2)+1.75~ @p@F>-0.87679*(C119-2)^3+0.976076*(C119-2)^2+1.150717*(C119-2)+1.75~ p@F>-0.87679*(C120-2)^3+0.976076*(C120-2)^2+1.150717*(C120-2)+1.75~ q@F>-0.87679*(C121-2)^3+0.976076*(C121-2)^2+1.150717*(C121-2)+1.75~  r@F>-0.87679*(C122-2)^3+0.976076*(C122-2)^2+1.150717*(C122-2)+1.75~ @A90.681818*(C123-3)^3-1.6543*(C123-3)^2+0.472488*(C123-3)+3~ `s@A90.681818*(C124-3)^3-1.6543*(C124-3)^2+0.472488*(C124-3)+3~ t@A90.681818*(C125-3)^3-1.6543*(C125-3)^2+0.472488*(C125-3)+3~ t@A90.681818*(C126-3)^3-1.6543*(C126-3)^2+0.472488*(C126-3)+3~ @u@A90.681818*(C127-3)^3-1.6543*(C127-3)^2+0.472488*(C127-3)+3~  @A90.681818*(C128-3)^3-1.6543*(C128-3)^2+0.472488*(C128-3)+3~ v@A90.681818*(C129-3)^3-1.6543*(C129-3)^2+0.472488*(C129-3)+3~  w@A90.681818*(C130-3)^3-1.6543*(C130-3)^2+0.472488*(C130-3)+3~ w@A90.681818*(C131-3)^3-1.6543*(C131-3)^2+0.472488*(C131-3)+3~ `x@A90.681818*(C132-3)^3-1.6543*(C132-3)^2+0.472488*(C132-3)+3~ @D<-0.10044*(C133-4)^3+0.391148*(C133-4)^2-0.79066*(C133-4)+2.5~ y@D<-0.10044*(C134-4)^3+0.391148*(C134-4)^2-0.79066*(C134-4)+2.5~ @z@D<-0.10044*(C135-4)^3+0.391148*(C135-4)^2-0.79066*(C135-4)+2.5~ z@D<-0.10044*(C136-4)^3+0.391148*(C136-4)^2-0.79066*(C136-4)+2.5D lQQQQQQQQXXXXXXXXXXSSSSSSSSSSVVV !A,@"0.#t $O% &OO')L()O* +O,Z-L./O0O~ {@D <-0.10044*(C137-4)^3+0.391148*(C137-4)^2-0.79066*(C137-4)+2.5~ !@D!<-0.10044*(C138-4)^3+0.391148*(C138-4)^2-0.79066*(C138-4)+2.5~ "|@D"<-0.10044*(C139-4)^3+0.391148*(C139-4)^2-0.79066*(C139-4)+2.5~ #`}@D#<-0.10044*(C140-4)^3+0.391148*(C140-4)^2-0.79066*(C140-4)+2.5~ $~@D$<-0.10044*(C141-4)^3+0.391148*(C141-4)^2-0.79066*(C141-4)+2.5~ %~@D%<-0.10044*(C142-4)^3+0.391148*(C142-4)^2-0.79066*(C142-4)+2.5~ &@@&8-0.0299*(C143-5)^3+0.089712*(C143-5)^2-0.3098*(C143-5)+2~ '@@'8-0.0299*(C144-5)^3+0.089712*(C144-5)^2-0.3098*(C144-5)+2~ (@@@(8-0.0299*(C145-5)^3+0.089712*(C145-5)^2-0.3098*(C145-5)+2~ )@@)8-0.0299*(C146-5)^3+0.089712*(C146-5)^2-0.3098*(C146-5)+2~ *@@*8-0.0299*(C147-5)^3+0.089712*(C147-5)^2-0.3098*(C147-5)+2~ +@@+8-0.0299*(C148-5)^3+0.089712*(C148-5)^2-0.3098*(C148-5)+2~ ,@@,8-0.0299*(C149-5)^3+0.089712*(C149-5)^2-0.3098*(C149-5)+2~ -Ё@@-8-0.0299*(C150-5)^3+0.089712*(C150-5)^2-0.3098*(C150-5)+2~ . @@.8-0.0299*(C151-5)^3+0.089712*(C151-5)^2-0.3098*(C151-5)+2~ /p@@/8-0.0299*(C152-5)^3+0.089712*(C152-5)^2-0.3098*(C152-5)+2~ 0@@08-0.0299*(C153-5)^3+0.089712*(C153-5)^2-0.3098*(C153-5)+2&@VVVVVVRRRRRRRRRR]><`@@A@:.[W5.  &APage &P"<??3c?3333QQ;ocQE4333QQ;QE4D FAb 3OV{B 3 43*4523  43"  3O %3OQ44$%3O&Q43      4444 f.3?3?4?4S-EH?5333333?5lG,?6?6ò?7ffffff?7kw&ls?8?8QO? ?9?9)x ?:333333?:i:@ս?;?;Nf3?<ffffff?<$tOO?=@=?>@>=??@?{J@@ffffff@@npmF@A333333@ATCؕ{@B@B*5{@C@CY[M@D@D+o&*D@Effffff@ESwP@F333333@F-^q'@G@G@H@Hy&H@@I @I-.E@Jffffff @JsÖ@K333333 @Kma0Y@L @LI_C@M @M@N @N10@Offffff@OXsX@P333333@PM5K@Q@Q@Rffffff@Rv?e@S@S%q@T333333@T/\@U@U>L@V@VXV@Wffffff@WPn>"d(@X@Xȗz@Y333333@Y@Z@Z&$A@[@[@\ffffff@\oU+v?]@]z4?^333333@^O?_@_ Ca7?`@`?affffff@aFwp?b@bc'?c333333@cU0Q?d@d̳PZ?e@e9 ? ]> wp@A@.[W5.  &APage &P"w??3?3333QQ;QE4D FA " 3ODz 3 43*4523  43"  3O %3OQ44$%3O&Q43      4444 f@3?4S-EH?5lG,?6ò?7kw&ls?8QO? ?9)x ?:i:@ս?;Nf3?<$tOO?=?>=??{J@@npmF@ATCؕ{@B*5{@CY[M@D+o&*D@ESwP@F-^q'@G@Hy&H@@I-.E@JsÖ@Kma0Y@LI_C@M@N10@OXsX@PM5K@Q@Rv?e@S%q@T/\@U>L@VXV@WPn>"d(@Xȗz@Y@Z&$A@[@\oU+v?]z4?^O?_ Ca7?`?aFwp?bc'?cU0Q?d̳PZ?e9 ? > ????m@@ATE OF PRODUCTION OF NEW SOIL ubic spline interpolation.Fig. 3a. oil thickness roduction of new soil qq`     }ENLAR225.PIC@@ Soil thickness Production of new soil vq}FEIGEN21.PICATE OF PRODUCTION OF NEW SOIL ubic spline interpolation, Fig. 2 Column A--x values Column C--first Feigenbaum iterates qq}FEIGEN22.PICATE OF PRODUCTION OF NEW SOIL ubic spline interpolation, Fig. 2 Column A--x values Column D--second Feigenbaum iterates qq}FEIGEN23.PICATE OF PRODUCTION OF NEW SOIL ubic spline interpolation, Fig. 2 Column A--x values Column E--third Feigenbaum iterates qq}FEIGEN24.PICATE OF PRODUCTION OF NEW SOIL ubic spline interpolation, Fig. 2 Column A--x values Column F--fourth Feigenbaum iterates qq}FEIGEN25.PICATE OF PRODUCTION OF NEW SOIL ubic spline interpolation, Fig. 2 Column A--x values Column G--fifth Feigenbaum iterates qq}FEIGEN31.PICATE OF PRODUCTION OF NEW SOIL ubic spline interpolation, Fig. 2 Column A--x values Column E--third Feigenbaum iterates qq}FIG26END.PICATE OF PRODUCTION OF NEW SOIL ubic spline interpolation, Fig. 2 Column A--x values Column Z--26th Feigenbaum iterates qq}GR-2.PIC@@ Soil thickness Production of new soil vq}ORBIT215.PICORBIT OF 1.5 Under transformation of Figure 2. Columns A through Y Columns B through Z qq}ORBIT225.PIC@@ORBIT OF SEED VALUE, X=2.5 Graphical Analysis Soil thickness roduction of new soil qq}ORBIT235.PICORBIT OF 3.5 Under transformation of Figure 2. Columns A through Y Columns B through Z qq}ORBIT245.PICORBIT OF 4.5 Under transformation of Figure 2. Columns A through Y Columns B through Z qq}ORBIT255.PICORBIT OF 5.5 Under transformation of Figure 2. Columns A through Y Columns B through Z qq}ORBIT28.PIC?@@RBITS OF 2.5 AND 2.8, FIGURE 5a raphical Analysis oil thickness roduction of new soil qq}POS-GRF2.PIC@@ATE OF PRODUCTION OF NEW SOIL ubic spline interpolation.Fig. 3a. oil thickness roduction of new soil qq}POS-GRF3.PIC@@ATE OF PRODUCTION OF NEW SOIL ubic spline interpolation.Fig. 3a. oil thickness roduction of new soil qq`}SPLINE-2.PIC@@ Soil thickness Production of new soil vq}X=Y.PIC?@@RBITS OF 2.5 AND 2.8, FIGURE 5a raphical Analysis oil thickness roduction of new soil qq}XY.PIC?@@RBITS OF 2.5 AND 2.8, FIGURE 5a raphical Analysis oil thickness roduction of new soil qq}ZOOM-2.PIC?@?@ Soil thickness vv"   &APage &P"pv??3 CRCSPLN2K3333QQ;cQ;3     4E4DFA 3O4n93*@43*@#4523   43" 3      4444 fN??333333??ffffff???333333?? ffffff? @ @ @ ffffff@333333@@@@ffffff@333333@@@ @ffffff @333333 @ @ @ @ffffff@333333@@ffffff@ @!333333@"@#@$ffffff@%@&333333@'@(@)ffffff@*@+333333@,@-@.ffffff@/@0333333@1@2@3?4S-EH?5lG,?6ò?7kw&ls?8QO? ?9)x ?:i:@ս?;Nf3?<$tOO?=?>=??{J@@npmF@ATCؕ{@B*5{@CY[M@D+o&*D@ESwP@F-^q'@G@Hy&H@@I-.E@JsÖ@Kma0Y@LI_C@M@N10@OXsX@PM5K@Q@Rv?e@S%q@T/\@U>L@VXV@WPn>"d(@Xȗz@Y@Z&$A@[@\oU+v?]z4?^O?_ Ca7?`?aFwp?bc'?cU0Q?d̳PZ?e9 ?> e9  &APage &P"???3 CRCSPLN2K3333QQ;cQ;3    4E4DFAe 3O4n9 3*?@43*?@#4%  3OVQ Soil thickness'4523   43" 3      4444 f\??333333??ffffff???333333?? ffffff? @ @ @ ffffff@333333@@@@ffffff@333333@@@ @ffffff @333333 @ @ @ @ffffff@333333@@ffffff@ @!333333@"@#@$ffffff@%@&333333@'@(@)ffffff@*@+333333@,@-@.ffffff@/@0333333@1@2@3?4S-EH?5lG,?6ò?7kw&ls?8QO? ?9)x ?:i:@ս?;Nf3?<$tOO?=?>=??{J@@npmF@ATCؕ{@B*5{@CY[M@D+o&*D@ESwP@F-^q'@G@Hy&H@@I-.E@JsÖ@Kma0Y@LI_C@M@N10@OXsX@PM5K@Q@Rv?e@S%q@T/\@U>L@VXV@WPn>"d(@Xȗz@Y@Z&$A@[@\oU+v?]z4?^O?_ Ca7?`?aFwp?bc'?cU0Q?d̳PZ?e9 ?> e9  &APage &P"?es??3 CRCSPLN2K3333QQ;3dQ;3     4E4DFA', 3On 3*@43*@#4%^ 3OVQ Soil thickness'4%Nw3OQ Production of new soil'4523   43" 3      4444 f.k??333333??ffffff???333333?? ffffff? @ @ @ ffffff@333333@@@@ffffff@333333@@@ @ffffff @333333 @ @ @ @ffffff@333333@@ffffff@ @!333333@"@#@$ffffff@%@&333333@'@(@)ffffff@*@+333333@,@-@.ffffff@/@0333333@1@2@3?4S-EH?5lG,?6ò?7kw&ls?8QO? ?9)x ?:i:@ս?;Nf3?<$tOO?=?>=??{J@@npmF@ATCؕ{@B*5{@CY[M@D+o&*D@ESwP@F-^q'@G@Hy&H@@I-.E@JsÖ@Kma0Y@LI_C@M@N10@OXsX@PM5K@Q@Rv?e@S%q@T/\@U>L@VXV@WPn>"d(@Xȗz@Y@Z&$A@[@\oU+v?]z4?^O?_ Ca7?`?aFwp?bc'?cU0Q?d̳PZ?e9 ?> e9  &APage &P"?of??3 CRCSPLN2K3333QQ;kdQ;3     4E4DFA 3O4n93*@43*@#4523   43" 3      4444 fy??333333??ffffff???333333?? ffffff? @ @ @ ffffff@333333@@@@ffffff@333333@@@ @ffffff @333333 @ @ @ @ffffff@333333@@ffffff@ @!333333@"@#@$ffffff@%@&333333@'@(@)ffffff@*@+333333@,@-@.ffffff@/@0333333@1@2@3?4S-EH?5lG,?6ò?7kw&ls?8QO? ?9)x ?:i:@ս?;Nf3?<$tOO?=?>=??{J@@npmF@ATCؕ{@B*5{@CY[M@D+o&*D@ESwP@F-^q'@G@Hy&H@@I-.E@JsÖ@Kma0Y@LI_C@M@N10@OXsX@PM5K@Q@Rv?e@S%q@T/\@U>L@VXV@WPn>"d(@Xȗz@Y@Z&$A@[@\oU+v?]z4?^O?_ Ca7?`?aFwp?bc'?cU0Q?d̳PZ?e9 ?> e9  &APage &P"???3 CRCSPLN2K3333QQ;dQ;3     4E4DFA 3O4n93*@43*@#4523   43" 3      4444 f·??333333??ffffff???333333?? ffffff? @ @ @ ffffff@333333@@@@ffffff@333333@@@ @ffffff @333333 @ @ @ @ffffff@333333@@ffffff@ @!333333@"@#@$ffffff@%@&333333@'@(@)ffffff@*@+333333@,@-@.ffffff@/@0333333@1@2@3?4S-EH?5lG,?6ò?7kw&ls?8QO? ?9)x ?:i:@ս?;Nf3?<$tOO?=?>=??{J@@npmF@ATCؕ{@B*5{@CY[M@D+o&*D@ESwP@F-^q'@G@Hy&H@@I-.E@JsÖ@Kma0Y@LI_C@M@N10@OXsX@PM5K@Q@Rv?e@S%q@T/\@U>L@VXV@WPn>"d(@Xȗz@Y@Z&$A@[@\oU+v?]z4?^O?_ Ca7?`?aFwp?bc'?cU0Q?d̳PZ?e9 ?> e9  &APage &P"???3 CRCSPLN2K3223QQ;d1Q;13    4E4DFA 3O4n93*?@43*@#4523   43" 444 d       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abc> c;  &APage &P"F??3 CRCSPLN2K33QQ;eQ;3    4E4DFA_C g 3O7 3*43*#4%  3OQ Columns A through Y'4%N=wi3O~Q Columns B through Z'4523   43" 44%M3O(Q 1.ORBIT OF 5.5 Under transformation of Figure 2.'44 26@?     ? !"#$%&'()*+,-./01> 1  &APage &P"?5 ??ti3 CRCSPLN2K33QQ;KeQ;3    4E4DFA_C g 3O7 3*43*#4% 3OQ Columns A through Y'4%N=wi3O~Q Columns B through Z'4523   43" 44%M3O(Q 1.ORBIT OF 4.5 Under transformation of Figure 2.'44 2.@XV@     XV@ !"#$%&'()*+,-./01> 1XV  &APage &P"F@5 ??ti3 CRCSPLN2K33QQ;eQ;3    4E4DFA_C g 3O7 3*43*#4% 3OQ Columns A through Y'4%N=wi3O~Q Columns B through Z'4523   43" 44%M3O(Q 1.ORBIT OF 3.5 Under transformation of Figure 2.'44 2& @I_C@     I_C@ !"#$%&'()*+,-./01> 1I  &APage &P"F@5 ??ti3 CRCSPLN2K3223QQ;e1Q;13    4E4DFACg 3O497 3*@43*@#4% 3OVQ Soil thickness'4523   43" 44%M3O(Q 0-ORBIT OF SEED VALUE, X=2.5 Graphical Analysis'44 d      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abc> cIT O  &APage &P"FSEED??ap3 CRCSPLN2K33QQ;eQ;3    4E4DFA_C g 3O7 3*43*#4% 3OQ Columns A through Y'4%N=wi3O~Q Columns B through Z'4523   43" 44%M3O(Q 1.ORBIT OF 1.5 Under transformation of Figure 2.'44 2?QO? ?     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