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Temperoture depthduringsummer Plotte versus for Loke, Michigon 23 228 49 228 9 I 2A6 137 t3I 1 83 117 229 tlt 272 ilt
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" , :c o n t i n u e d Solution As just described, want to usenaturalsplineend conditions performthis we to analysisUnfortunately,because usesnot-a-knotend conditions,the built-in MATLAB it spl ine function doesnot meetour needs Further,the sp1 ine functiondoesnot returnthe first and secondderivatives we require for our analysis However, it is not difficult to develop our own M-file to implementa natural spline and retum the derivatives Sucha codeis shownin Fig 1612 After somepreliminaryerror trapping,we setup and solveEq (1627)for the second-order (c) Notice how coefficients
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SPLINES AND PIECEWISE INTERPOLATION
coniinued
I - 1:m %perforn interpolaLions at desired values - Splinelnrerp(x, lyt'(r),dy{r),d2(i)l n, a, b, c, d, xx(:)); end enC frrrlctfon hti = 11*, i) hl'r = x(r + i) - x(i); for end iun -l n iC'J fdri, , x , , 1) - y(j)) fcld - (y(i) r' (r(i) x(j)); enC function lyyy, dyy, d2yl -gp] inelnterp
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, 000U001 & xr <_,- (ii xi >= x(ri) x + i) + 0000001 yyy-a (1i) +b ( ii) * (xi-x (ri) ) +c (ii) * (x!,x (ir ) ) ^2+o( ii) *(xi x(ri))^3; * d y i , = 5 1i I ) + 1 c ( i , i ) * ( x l - x ( i i ) ) + I * r l ( i i ) * ( x i - x { , i ) ) ^ 2 ; d2y-2*c (ii ) +5*d (ii) * (zi-x (ii ) ) ;
br eak end end end
FIGURE t6t2 lContinued)
we use two subfunctions, h and fd, to compute the required finite differencesOnce Eq (1627) is set up, we solve for the c's with back division A loop is thenemployed t0 generate other coefficients(a, b, and d) the At this point, we have all we need to generate intermediate values with the cubic equation: f (x) : ai * bt(x - rr) + ct(x * x)2 * di@ - xi)3 We can also determine the lirst and second derivatives by differentiating this equation twice to give '(x) : bi * ci(x - x;) * 3diQ - x;)z f"{x1 :2ci + 6di, - xi) f As in Fig 1612, these equations can then be implemented in another subfunctign, splineTnterp, to determinethe values and the derivativesat the desiredintermediate values Here is a script file that usesthe naLspl ine function to generatethe splineandcreats plots of the results: z = i0 23 49 91 137 IB3 22t) 2721; r=1228 228 228 206 139 177 I1 l 17L1;
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I67 CASE STUDY
continued
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As in Fig \613, the thermoc\ine appearsto be \ocated at a depth of about 115 m We can use root \oeation (zero second denrative) or optimization methods (minimum fust denvative) to refine this estimateThe result is that the thermocline is located at 1135 m where the gradientis - 161'C/m
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