I JIM/Willi HIJIWH J J \ I l WWMII'JI I W 'N H—l—t Iowa (mow ANALYSIS OF VIBRATlON-ROTATSON SPECTRA 35‘ AXIMLY smws'rmc MOLECULES. Vv'iTH JLF'FUCATI'CEN TO C03i Thesis Em fho 13¢»ng a! ['41. D. pthHfiAN STATE UNIVERSITY ¢firahn WiHéam Eoyd 19:53 IIIIIIIIIIIIIIIIIIIIIIIIIIII ' NIHIUIIIHHIIIHIWIHIJIIIHIIHIIIHlllll;HlHHlllllll 31293 01713 6619 This is to certify that the thesis entitled ANALISIS OF VIBRAIION-ROTAIION SPECTRA OF AXIALLY SYMMETRIO MOLECULES, WITH APPLICATION TO 0D3I presented by John William Boyd has been accepted towards fulfillment of the requirements for A degree in M 1M Major professor December 14, 1962 Date 0-169 LIBRARY Hips 5cm “ . . pmsié us ‘ ABSTRACT ANALYSIS OF VIBRATION-ROTATION SPECTRA OF AXIALLY SYMMETRIC MOLECULES, WITH APPLICATION TO CD31 by John William Boyd We have studied the general problem of the analysis of vibration-rotation spectra of axially symmetric mole- cules. In order to be able to express the frequency for all possible transitions (as given by the selection rules), we have written a single analytical expression for the frequency of the radiation as a function of the quantum numbers and their changes, in which we treated the changes in the quantum numbers as full variables on equal footing with the quantum numbers. Previous treatments . have given sets of apparently irreconcilably different expressions for lines in the various portions of bands; whereas, we have obtained a single expression that repre- sents all possible transitions. This, in turn, enables us to analyze a complete band or even several complete bands simultaneously. The statistical problem of estimating nmny'parameters and their reliability from a large number of data points was examined. The concept of simultaneous confidence intervals was applied to this problem. The results of the study were applied to a near John William Boyd infrared absorption spectrum of CD31 near 4600 cm’l. This spectrum consisted of both a perpendicular (45K =:t1) and a parallel (13K : 0) band. These two bands were assigned to 21%, and were analyzed simultaneously. ANALYSIS OF VIBRATION-ROTATION SPECTRA OF AXIALLY SYMMETRIC MOLECULES, WITH APPLICATION TO CD31 By John William Boyd A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1963 ACKNOWLEDGEMENT This work was done under the direction of Dr. T.H. Edwards and was supported, in part, by a National Science Foundation grant and (Ea-operative Graduate Fellowship. I also wish to acknowledge the many valuable discussions which I had with a fellow graduate student, Mr. William B. Bless. TABLE OF CONTENTS Introduction Statistical Study Assumptions Matrix Formulation Estimator of,fi Lack of bias and Minimum Variance of Estimator Unbiased Estimator of the Variance of g Unequal Variances (Weighting) Summary of Results with Minimal Assumptions Confidence Intervals Linear Independence of x Experimental Analysis of data Summary Bibliography Ul-h‘UfD-‘t LIST OF FIGURES Symmetric Top Energy To Third Order Line Positions For 1st Overtone of My Matrix Formulation Determination of'zgh Spectrum “—5 UMWUIU LIST OF APPENDICES I Introductory Statistics II Commonly Used Distribution Functions And Their Interrelations III Tabulation of Observed And Calculated Spectrum 49 52 55 INTRODUCTION With the advances in the art of high resolution infrared spectroscopy of gases (i.e., higher resolution and higher precision), we have a myriad of data with which to deal. Not only is the prdblem complicated by a large increase in the quantity of data; but in order to reproduce the observed spectra with precision on the order of the precision of the measurements, smaller terms in the theoretical expression used must be taken into account. These terms were unknown to early workers in the field and later were usually ignored. With the advent of high speed computers, the physical problems of handling large quantities of data have been reduced. However. the introduction of the smaller terms into the calculations has necessitated an extensive study into preper techniques for handling and evaluating experimental data so as to extract the maximum amount of information. In order to test and demonstrate the technique chosen, a representative vibration-rotation absorption band was sought in the near infrared. An axially symmetric molecule was chdsen as a compromise between simplicitv of the spectrum and analyticity of theoretical expressions for The band of CD31 near 4600 cm" was found the energy. to be free of obvious perturbation and well enough resolved 80 that approximately 200 transitions could be identified and measured. This band was assigned to the vibrational 1 transition 2 >14 . The explicit expression for the vibration-rotation energy of axially symmetric polyatomic molecules, through the third-order perturbation calculation with soaaiooo cm’1 and an expansion coefficient ~1/30, is given in Figure 1 (1,2) where; a),x Harmonic and anharmonic potential constants, B Inverse of principal moments of inertia, :1 Change in B with vibration, D Centrifugal stretching constants, 3 Coriolis coupling coefficient, Third-order Coriolis corrections, Total angular momentum quantum number, Component of J along unique axis, Vibrational angular momentum quantum number, Abyss-.4; Vibrational quantum number.5* The general expression for the frequency of the radiation absorbed (or emitted) when a molecule undergoes a transition from state 1 to 2 is given by the Bohr E ~E} condition. flCmmJL= 25¢ . Here each transition will be specified by the quantum numbers designating the initial state and the changes in these quantum numbers necessary to specify the final state. At this point, let us particularize to the first overtone of 13 , i.e., Ayvs==C:).S¢W$ ; thy; =22. *{3 Vibrational degeneracy SYMMETRIC TOP ENERGY TO THIRD ORDER 3%: = 2 w. (v. +i9s) + B:[I(J'+z)—K2] + 3: K2 —-2 of; :51, K ‘ + g‘é‘ X 55’ ( Vs *ifissz’+r39’) +§$¥ X41; 1"}: - g «2, (Vs+i3s)[T(T+I)-K‘] —§a<§ (vs +5 3.)}{1 —- D,[ WWI «D,K [:rcrwJK‘ - DK K‘ +2; WHAKUUHJJ +2" n..1.K’+§;n.sl.K <1 > - — '~ _ — _ — fl _ _ - — ‘ LINE Posmorv FOR FIRST OVERTONE or v. Q) E (“3413,43“)= [160+ +6X++ +2 X4s 35] +[4X1.1.+23:3f“2744;:{2'26351901 ”EMA , +[ Bel—é z «:35]([ZIAI +A3+(AI)"]-[2KAK new?) Ruin- C" 2 z 2 Hoe-goo 3,)+(2 3. 3+*Z’?++‘i;'245353(1KAK+(4K) ) —[Z°‘:](U*°UU*M+’J“ LK+AK)2) {24:} (E K+AKlz) ~[ DJ] (L {3+AT}{J+A3’+l}]2‘ [j{J+I311) —[ DIX] ([5 J +AIHI+AIHI { K+AK}"— {unokzfl —[ DK] ( t “M‘- K‘) ._ [ 2 '11:] (AKUI-l-AT} {HAIH} { K+AK}]) “[2W1K](AK[K+AK]3) (2) Figures 1 and 2 Using the fact that Aiz-ZAK (3) for this vibrational transition, we can write the general frequency expression as the polynomial in J,K, AJ’ andAK as expressed in Figure 2. This particular arrangement is not unique. It was chosen because it gave a simple form to the coefficients while maintaining linear independence between the various functions of J,K,AJ’ and AK . Once the spectrum is obtained and transitions are assigned to the measured frequencies, our problem is to: 1) find the best set of estimators for the molecular parameters as functions of the observed frequencies, 2) make a confidence statement about these estimators. ‘Hhen searching the literature, one finds disagreement as to the correct manner in which to proceed. Not only does one find disagreement as to the proper manner in which to obtain the estimators; but also, there are no, or at best misleading, statements concerning the reliability of these estimators. In order to find the correct answers and to understand their range of validity, the following statistical study was made of the general problem. In appendix I, there will be found a review of the introductory concepts dealing with probability and distribution functions presupposed in the following discussion. For those not familiar with past or present procedures used, reading the summary which is provided may give a quicker grasp of the problem and its solution. STATISTICAL STUDY (4,5) Let us suppose that we have n observations or measurements, y1, y2, ..., yn with their associated errors e1, e2, ..., cm In our case, the set 5y); will be the set of observed frequencies of 11 different spectral lines, which are assumed to be linear combinations of p molecular parameters, 6, , 62 , ..., 8P , that have well defined but unknown values. So that 715 2- 7-,3,+X¢2+ ' ‘ ‘ 1' X593? + 84' 4°=(!)2/'-;n) I (3')- where the EX“? are known constants. In our case, the EX“; are known functions of the quantum numbers 1 (J, K, 2.3, and AK) specifying the energy levels involved. The purposes of our study will be to: 1) find the best linear (i.e., linear function of the observations in? ) estimator of the unknown values of the parame te re {3;}? 2) find the best linear estimator of the value of any linear function of theEBJ'; 3) find an estimator for the variance (”‘2 of the set of errors 6e13,; these errors will be treated as random variables, and assuming the iefi to be normally distributed we shall: 4) find the simultaneous confidence intervals for ( J estimators of the g 5.? and any linear function of 6 the 56,3. 5) test the hypothesis that the values of (3‘; of a given subset of the flgj2are significantly different from a set of known (from microwave, other infrared spectra, or theoretical considera- tions) constants. Assumptions The minimal assumption which will always be made about the random variables gel? is that their expected or mean values are zero: E(€;)=O <‘=(I,a,~~,n) (4) e.g., on the average, a given measurement is Just as likely to be high as low. Here we shall also make the r \ assumption that the random variables ESQf are uncorrelated and have equal variance (:(G‘v 63: 3 2-" 7‘25; ' ' (Cb) — d J .4 J Later we shall show that there exists a transformation Which may be applied to any set of data so that the latter assumption always holds (i.e., the process of Weighting). The errors may arise from two physically different but statistically equivalent sources (measurement errors or incorrect model formula). First, even if we assume the value of the ith observation can be represented D exactly by g/‘g; 3; , experimentally it is impossible to measure the value exactly. However, any well-designed experiment will strive to satisfy the condition E (e1) = O. In the second case, the value is not represented exactly by ‘4ij G; . but by Q (XI'PH/ >:('P+2/ ' ' 2 X‘s) U P Ext). BI + If we assume that gnjej adequately represents the measurements and that the term 3(X,-D.LX;J,.2 “,X-q) acts as a random variable, then we could represent the ith measurement by ; X55 (3.; + 6?; . A large part of the testing we will be doing is to see if we need to include the explicit dependence on Xpu in the representation. Matrix Formulation If we identify the (Vi) and gel? as n dimensional column vectors 1 and g, and the 9‘13? as an n by p matrix X, and. the {ages a p dimensional column vector .8. ; we "‘ J are then able to cast equation 3) into the following matrix equation: Ll -: fig + $2, (a) (See I'Figure' 3) If we also define the operation of the linear operator E, the expected value operator, on a matrix Q as: 5(a) = H Efazgfllj QCX.~‘M£X‘s.NzX EX m.vx....sx...m.sx_ xx M%ses.—.¢NX\..NNX ~wA ix..12x.. .~.x ..x J .aofiadauos 53.! an 86E H 10 we are then able to write our assumptions 4) and 5) as: EKQBZQ ECQ m In the above relation, the ' denotes the transposed matrix, and ;_ an n by n identity matrix. From appendix 1, the covariance be tween two random variables is defined as: C C‘ V (V?) V!) :- E L (“‘2' - E<\J( ) ){\/f - E (‘45) 3 Let us define the covariance matrix as: 51v ZilfCDOV‘gQgfiH z, : EZCy—E*z‘;\.~(y-E’1 (>2) Note that ?V 3-” EV . .,_. d It can be seen that if we wish to transform from a set of variables V3 to a new set “i ,4:- Av (where AM are constants), " / \ O ‘l P "_ v. \- / *- "g,“- then 7“ = figvf: , (.J q ./-, —.-.- EL’Ae-EMW M-‘Ex‘vna -.- -4 ‘I., :: gfimv —§_E(;‘,\Ay--~m-‘4 E ‘3" ,. , 11 With these definitions we are able to write our fundamental assumptions 3), 4), and 5) as: C \ (I EC4)=Z_§ g3 =- 72; Estimator g; A Let b = ~. 3ubz) : 112p), denote the vector with which we estimate the fixed but unknown quantity g z (3:, 3,2. . u) (3p): Then the vector 1 - Q is the estimator of the error vector 2. Let us choose as the best estimator of A that value of b which minimizes the length of this estimator of the error vector; where (L) (I) - k (5;- ? o “q: - Z/I'x' 13), )5 h' 1:, " x. K; N 't} ’ U 1:” « I" , ‘ ~) 5 .. 6“” K .34 {:1 Ms SSE :: ,, is the squared length of the estimator. The conditions 9355—20 I=CI12/-"J'D) c) E 5 ‘ Will minimize this vector. However, let us use the following geometrical arguments in order to find the b to minimize y; - 29;. (See Figure 4.) The vector Xb is a p dimensional vector since it is a linear combination of the p linearly independent columns“ * -1 * See discussion on the existence of .11 . 12 3%, Khan-Hz min SSE (z .- mm; - :2) / lb. t 7/ / J7 ight angles general in the p§§ne‘-—Z--.. =32: --..Y .« and rewrite the above, we have flJb: Z_4 . (’3‘ In component form these are called the set of normal equations. If N" exists (to be more fully discussed later), then ____ I—J /’ .12- N 5.4 ‘" (If and using 9), 10), and 13) Zb : (fl-lfi’)zj (2:4qu 3:5 " (LnlfiqTaiw-wq/ 375: TaCE’Z’M; 1") but Since 35/5 :j—V W [112": We have Eb : 725-, (14> for the covariance of b. Lack _f_ Bias d Minimum Variance of Estimator To say that an estimator of a parameter is unbiased means that the expected value of the estimator is equal to the parameter. Now 15 demonstrating that our estimator is unbiased. To say that an estimatoris the best estimator means, that in the class of all estimators, this particular estimator has a minimum variance. We shall show that in the class of all linear estimators that b is the best unbiased estimator of Q . For our general linear estimator, let us take: 5*: fig and let fi=fi"g{"+g . For the expected value of pf we have ‘) +2. ‘. —‘ \ 1l #1) :1 E:- N';I:)/ _.rC ‘1‘]— . {\‘—I>K//E' L - l ‘ .I! ‘\ EC 1 3" ‘u 1 3 H E( KT 42:;- But to be unbiased, E(b_*) must equalfi , which implies OX :0 for all 9, which in turn implies 93g :: 9. For the covariance of 12*, using 9), we have / Z s“ = “IX“ 93 Zt: (E'EWKQ-C) ,- ~ , -17 2b: = rECP-J'Jv‘gC/ , 19% (C -<)’_+f\ .4. ...2 ’) a/fi “But above we had C__)_{_= 0; therefore, Zy— [£1 +L: ..3 The diagonal elements of Z); are the variances of the individual be. , Thus to minimize the variance of b*3, n ’2 We must minimize N” + K27! "K . 77 .» n '- C t‘ which necessarily positive, we must have 15'" K O , 9, which 0 for all 3 and k, which implies g = implies C k = J jg has the minimum implies A 2 34;, which shows that _b_* = 16 variance of all unbiased linear estimators of Li . One can use a similiar argument to show that the best unbiased linear estimator of any linear function, 3:; of 33 is given by‘tfb. However, this only holds for linear functions of £3 and is not true for functions in general. From the results of this section we see that we would have arrived at the same _b = y'g'y as our estimator of 5 if, instead of requiring that _b minimize the length of the vector y.- Eb, we had required that our estimator be that one with the smallest variance from the class of all unbiased linear estimators. If?» "0 Unbiased Estimator'gf the Variance In order to find an unbiased estimator of 0'23 , the variance of the random errors Eel? , let us find the expected value of SSE. Substituting b: N; 1,5 and 542,. simplifying, we have EQSE) = {2:34 I- fryvg’) g3 Remembering that E is a linear operator and that E (et 93.) :2 C2 ("(5 we have: l 'A to n;f H for the first part, 735$" n He's»): ECZ." are) = u. “‘N" 6‘». 1 7 7 and for the second part / -l '.’ _‘ -.., \ .{ N :7 .__ E (:._ 7 , “.4 .,, 7 . 7 ~~‘ * 1"‘< 'va' kz’~£<\g / E( M summing over 3 and putting ‘ng in front, we have: / -/ XCJ’ ”K3 K 'A' “i ) 4i 5‘ , h- \J — ~ A M 4 but if;_: §_(a p by p matrix) so, summing over i, we get 5(Q’XN4X’6) ~'—‘ 7‘2 Z /\ . K,” :2 3—3:? : 7- ‘ ~\. o'- k:<’ __ P’ (K 77x2! ‘& Combining these two, we have: ECSSEJ—t ;:9(r‘-;>) . And finally, since E" ,,- ”a is an unbiased estimator of \a . Unequal Variances (Weighting) Let us consider the case where the variances of the various members of the set of observations :yif are not equal to one another, but their ratios are assumed to be known. In fact, we shall include the more general case where the covariances between various members of the 3Y1? are not zero, provided their ratios are known. That is: 18 .- '— £?2;1 'where §_is the matrix of the known ratios Earn 532‘ is an unknown positive constant. As long as the §y1§ are linearly independent then E will be nonsingular and positive definite (6). / Since 27é==gig and is positive definite, there exists a nonsingular matrix §_such that / —— —L..c>’3 ‘F‘M~‘D -' :- and P1 '- '- L . Using this 2, let us make a transformation on the observa- tion 1 to a new variable ”if. For ”f, we have 7; rv __ ‘\ ‘. “‘__./.\—— (DY? '—/." EC1) —- Egg) - _S’E\1J — Jig; " My and I ’__._ O 2,1,;J:.P.HE :EEQPMP wt:‘"I -—-£- "-—9’ -.., ’— 'so we see that the transformed variable x satisfies our original assumptions 10). The quantity we wish to minimize is /‘~\.,/' — /:\J " £1 / 7/”.— />( F \ 39E —- \4f'_/.__I 5,, _,:.,.J/ .. w / \'w w/r /,->/r“ — LEuyiéL .—_. + '-=— N I r ”‘4‘ = g-XbDC--K%‘Y5J Aw,“ ' f ._ / -'l %‘ <5E= (m— 4:21 4 (J‘Xa) a c.._.. - If we consider the case where all the covariances are zero (i.e., §,is diagonal), then , To C: 2—: 62": '73.; LE P R) 1| \ 1 fl (A, . ..I If we call 7 2 72M),- n :> then SSE 2 2? v (2.- 2-7, .2 7; <':/ ‘ \\ JC '1‘ “‘( j J; x J‘.’ / which we call the weighted sum square error. The “i ’ 'which is inversely proportional to the variance of y1, is the weight, or relative number of times that yl is used. The transformation used in this case is - m» :0. rv --., ~ 5') " V‘ or ""1, '- )/ VVc' ti,’ and X',’ = V V—V—Z1 yd 1 U x. (J _ a Let us consider m /\ 0‘ \J {I H] /'\ #7} L ?‘><> EG) == ”Q‘fl g EQ) :- 3 Thus we see that the expected value offiE is $3 , independent of what fi,is used. Intuitively this infers that even if our choice of the weights is not correct, our determination of/Eiis an unbiased estimator of 43 . However, the inferences that we make about the variance of N ‘2 are no longer strictly true; and in fact, the variance of theiE determined will be larger when the E (the set of weights) is not the "correct" one. 20 From this point on, if assumption 5) does not hold for the original set of observations, we shall assume that the proper transformation has been performed on these observations, obtaining a new set for which the assumption does hold; hence, we shall drop the'“’ as a notation for the transformed quantities. Summary 937Results with Minimal Assumptions Up to this point we have, with the assumptions: ” a = P53 + § (:5) 2’ E <13) = 2&6. «44) found that our least squares estimator 3; =.}\_I_'1_}.{_'y is the best (smallest variance) unbiased linear estimator of £3 , and that ’- v \v 'v’ -—--‘..A. ., l o ‘ E G...— is an unbiased estimator of 37d. Confidence Intervals Now that we have found estimators for values of our :2' arianoe of unknown molecular parameters :2)? and the v ls the observations 3’2 , let us find confidence interva for these estimators. It would be desirable to be able to find a constant g_such that, given b, then the 21 probability is 1’C£ that 4: will be contained in the inter- val b izg, However, a statement such as this is incorrect because once 2 and 2 have been fixed, the probability that ‘fii is contained in the interval is either 1 or 0. That is to say, since J} is a constant and has some fixed but unknown value, it either is or is not in the specified interval. In order to ascertain the proper statement to make, let us consider the set of all possible sets of observations. We will designate the 2 determined from the kth set of observations as 2k or the set of all possible bfs asgékg. If the distribution function of 327% were known, then a g_could be found, such that the probability that a 9k taken at random from the set 2k be in the fixed interval _éfltg_would be 1'vi,0nce we have determined a b from a given set of observations, we then say that the fixed interval 2,: g_is a confidence interval for g: with a confidence coefficient of I-Cfi. The interpretation of this :13 that if a succession of different experiments are per- formed using a confidence coefficient of 1-5& then, on the average, (1 —o(,)-100% of the intervals will include _3. In order to derive confidence intervals for our estimators, we /‘ the set of observationsgyif 3-72,». This shall add the assumption that have a Joint normal distribution, i.e., 1_is N(;7 is combined with our previous assumptions _— * see appendix II 22 to give: Let us now use the statements and theorems of appendix II to find the distributions of b_and SSE. With 1 distributed NQgg , 02;), and since 13 = y'gg' x is a linear transformation of'y; we note that (PE) is =3 and rf’m'w) l<fl"2<.')' will": thus we have that p is mg @397") i.e., the {bag have a Joint normal distribution with a mean of ii and a covariance of {TQEf‘. However, this does not permit us to find a confidence interval for ii since the distribution of‘b is determined by the unknown parameter c%3. But we shall be able to find the distri- / u 7 / .. bution of the variable, W713..-” 7 whose distribu- tion does not depend on any unknown parameters. ‘In order to find this distribution, we shall make transformations to new sets of variables which possess special properties; however, these variables need not be used for calculations, but only for their theoretically useful properties. Recalling the arguments used to minimize the vector x,- Xb; let us first construct an orthonormal basis (3.7327. . 3549) to span the p dimensional space spanned . A) 23 by the p columns of 5. Now let us extend this orthonormal basis to span the whole :1 dimensional space of all possible 1's. In terms of the new basis, we can write :31: 2 225g 7 or y : 53 where the n columns of A are Recalling that _xp was the projection of 3; upon the plane spanned by the columns of x; we have 2f}; 2- g =7 . Haw <=/ since (Tack/“5?) spans this plane. Thus *2 P / f) 3 SSE: (22,27 —:ze Hz- 7‘m-Z‘Ze4) (2/ — (7/ —( i:( (_( I (~ , ‘)3L: (2 *)("\!’) (Z E(HO\L> c P-r “' ¢=F+t "’ giving f? SSE: Z; 2,“ . (I H That is, SSE is a function of the E". with i>p. p We had 55 z E 2'05; , By multiplying on the left - c':, .- C by '11"? and remembering that Edy}; = I, we have \ a: N“’2_<’ b l- 2‘. 31‘ . That is, _b is a function of the 2’7. with is p. Since g = A'x is a linear transformation of a normally distributed vector, 3; itself is normally distributed with a covariance 2?: CFZA’IA : «TS-I- Hence as stated in appendix II, the £22? are statistically independent. Since SSE is a .4 function of 2'. with 1 >13. whereas 2 is a function of 3+”; .i m. n... 24 i€ P; 2 and SSE being functions of mutually exclusive sets of statistically independent variables, are statistically independent themselves. We can see that -5-;-3S~5ji is distributed Kin-p) by noting that the E; with i> p are distributed N(O,I ). That E625) = 0 for 17p can be seen from the following But E(1) 2 3g _3_ and E(.1_(_b) = 95.3 ; therefore, E<’i’.')=0 for i > p. q Since £2 = 02;) we see that ECg‘é‘) :/ for i>p. From appendix II, we know that the sum of the squares of the set of k statistically independent normally distributed variables with expected values of zero and variances of one has a chi-square distribution with k degrees of freedom. Thence 53,525- is distributed Kin - p). Finally, let us find the distribution of ( .‘_ t.) I x J " . 7, -— .1... ~ e-N ' end We recall, as stated earlier, that the covariance matrix of any set of linearly independent variables is nonsingular, symmetric, and positive definite. Since this is true, -.. _ _ there exists a nonsingular matrix gt; such that 32’s __ and BB : :75”, Since 2 is IJ(@_)U’2N") then (_b_ -B.) must be N(9_, waif"). 1:13 (_b_ .3.) where Let us transform to a new variable 25 then 1 is N(_q,;[_) giving I P W5? ‘m watts-5) =x’x = .' ,vc'a 2 is X (P) . Let us also note that, since 33 is statistical- ly independent of 503.5 , then (Q'§)6"2fl<é_‘§) will be statistically independent of ‘33—, which is distributed II 2 as X (n - 13). Referring to appendix II; if we are given two statistically independent functions. u distributed X20!) and v distributed X20) , then the variable 14 k _\_/_ X " I ‘ SSE “1&4.“ 18 NP. n - P); where x12: '37:". If we mean by 13(p, n «- p), the constant such that the is distributed F(k.1). Thus we see that probability is 1~0k that F(p, n - p) (x Fadp, n - p), then we can say that the probability is l-d that W ‘< P SKEW?) 74?. for any _b_ chosen at random-from the set of all possible 2's. The above relation may be thought as an ellipse centered at fl in a p dimensional space. It can be shown (4.7) that the above requires the probability is 1-0K that simultaneously for all ha in {bi}; ”DJ. 67) S 84b; 26 where S : ([me (p, n-pf and A57: W . Thus 12 2t Sgb is our confidence interval forfl with a confidence coefficient of I-d . The confidence interval for any linear function, 1.225 of .3 is given by @242 ‘z-f’iS‘é S»\’£’fl"$" If we wish to test whether an esz‘zmded 83 is significantly different from a particular value 807 I we use lb; - Bojlssflm If the inequality holds, we say that the 53 is not significantly different from £03 with a (l -o\) 100% confidence. In particular, if the ratio ”XV/457 is less than S, then Bj is not significantly different from zero. Tables of F“ (k,1 ) for various 0( , k, and ,4 may be found in many standard statistics texts such as (5) or (6). we find that s is ~ 3.5 to 4 for (X: .05 and the range of k and fl. used in this study. Linear Independence 93 3g Let us examine the relation P( [b5 - Bilésflm): ;- q for all 3 simultaneously. This can be viewed as the component-wise relations requiring that the probability be 1 - at that the point _b_ is contained in the p dimensional ellipse (Q-EYSE’Q-E 5 82 , -1 where $242!)" is the least squares estimator of éb 472!- 27 It can be seen that $13 a measure of the eccentricity or ratios between the lengths of the various axes; whereas given :82, , $2 is a measure of the length of the semi-major axis or the volume of the ellipse. We note that 27;, is determined by 2-1 up to the scale factor ,42- Let us assume we had approximately 1000 observations at our disposal, but only wished to use about 100 of them ' (because of time, effort, money, etc.). We see that by carefully choosing our 3g we can shorten our confidence intervals. This does not mean that one should eliminate any observation points already measured from 3;, but rather that one should be certain that enough correctly chosen points are included to adequately determine the {53} under investigation. Intuitively, when dealing with spectra, this implies one needs transitions identified as uniformly as possible from all subbands. One can show that a necessary and sufficient condition that 3-1 exist (i.e., E be nonsingular), is that the columns of 3; be linearly independent. It can also be shown that the ratios of the diagonal elements of _bl" depend on the degree of near linear dependence between the columns.of 3;. This near linear dependence can also cause round-off errors in the inversion process. As an example of this, let us choose J1(J1 - l) for our X52, J1(J1 + l) for X”) and l for X10. We see that N“: 23:3 + 23’; and s Noa ’5 ZS}? - 23;. . These terms are of the order of J3”, while their difference is of the order of Jlgnax, 28 When the individual terms of a matrix are large, while their differences are small, the matrix is said to be poorly conditioned and there may be large round-off errors in its inversion. If we choose 1, J1, and J? as our X10, X11, and X12, respectively, we will find the round-off problems minimized. If, during numerical com- putations, round-off still seems to be a problem, there are two numerical aids for the solution of these round-off problems. First, there exists the simple expedient of carrying more figures during the computations (i.e., double precision arithmetic, which was used in our case). Also there exist various iterative schemes for increasing the number of significant figures retained in an inversion calculation. EXPERIMENTAL The spectrum was obtained employing a self-recording Pfund-Littrow type vacuum spectrometer described in references (8), (9), and (10). A Bausch and Lomb Certified Precision 600 lines/mm.echelette grating, blazed at 28°41', was used in the first order. The infrared detector was an Eastman Kodak type P lead sulfide cell cooled to -50°C by circulating an acetone-dry ice mixture. Slit widths of 18'microns were used. A ten gram sample of CD31 was purchased from the Volk.Radio Chemical Company, Chicago‘m3,Illinois. The sample was placed in a White-type multiple traverse cell (11) adjusted for an' 03'qu meter absorbing path length. A modification of the fore-optics permitted a calibra- tion beam to be interchanged with the infrared beam whenever desired. Before reaching the DOlnt of interchange the infrared beam had passed through a cell containing the CD31 under investigation, whereas the calibration beam had passed through a cell containing a mixture of CO and I20. In the region of interest, Rank et. al. (12,13) have made highly precise measurements of the frequencies of the absorption lines of CO and N20. Twenty-seven sharp absorp- tion lines from the CD31 band were calibrated employing the CO and N20 absorption lines as standards. These twenty-seven lines in turn were employed as internal standards to calibrate the whole CD31 spectrum. The 29 3O frequency of an individual absorption line was determined by interpolation between these standards using Edser- 'Butler fringes (14). This results in a constant frequency separation between the fringes. he observed a shift of 0.03 of a fringe separation in the frequencies of the fringes recorded before and after the recording of the CD31 spectrum. Measurements made under similar conditions demonstrated such a shift was uniform (linear) with respect to time and could be considered to be the same over the extent of the whole band at any given time. Thus, by assuming the shift was uniform with time, and recording the spectrum continuously, no additional experimental uncertainty should be introduced by this shift. The record employed for the final measurements consisted of two individual spectral recordings. The pressures of the absorbing CD31 gas were 8.5 and 2.0 cm of Hg respectively. The spectrum recorded at a pressure of 8.5 cm of Hg permitted one to identify and measure the less intense transitions occurring in the wings of the band, whereas that recorded at a pressure of 2.0 cm of Hg was more useful in the regions of strong absorption. Comparison of the observed spectrum with the position and intensities of one calculated after initial analysis showed that some of the lines had to be assigned to two transitions. A breakdown of the number of transitions assigned on the two records is: 31 number of transitions on chart I 187 number of transitions on chart II 284 number of lines on chart I 145 number of lines on chart II ' 203 number of transitions only on chart I 31 number of transitions only on chart II 128 number lines only on chart I 29 number lines only on chart II 87 number of transitions on both charts I and II 156 number of lines on both charts I and II 116 total number of transitions 315 total number of lines 232 The variance for each line was assigned by estimating the uncertainty in measuring each line according to its signal to noise ratio , amount of blend with neighboring lines, and involved personal judgement. This uncertainty, called 4)) , was then squared to give the variance. Appendix III gives the assignment, calculated frequency, observed frequency, All , and the observed minus the calculated frequency for each transition observed on the two records. As can be seen, the lines calculated from the constants obtained in the analysis agree very well with the observed ones. (Viz. .057 and .C 07 cm" are the largest and mean differences respectively.) Here the standard notation of P, Q, and R for A? andAK being.) -1, 0, & +1 respectively has been retained. Figure 5 shows a highly condensed representation of the spectrum. The strong absorption on the left belongs to the parallel (liKier) band and the strong peaks throughout the band 32 are the centers of each of the perpendicular (AK zii) band subbands. The lines used in the analysis are the many smaller ones between the strong peaks and they were mea- sured on a record which was approximately forty feet long. 33 752$ asapomam um 2de TE 0 QQQ$ ...EU 00%.? ANALYSIS OF DATA The selection rules for electric dipole transitions are 3 AK-zo 03:11”: K10 A‘Jzi') «=0 Ak=il 1Q3=i 0 L In the past, standard practice has been to consider the transitions AK :21 (perpendicular bands) completely separate from those with AiK.= 0 (parallel bands). In addition, within these divisions, all transitions from a given K were written separately as the Kth subband. The bands were usually analyzed on about twenty envelopes of many transitions which have neither well defined theore- tical nor experimental positions. In the few cases where the spectrum was well enough resolved so that the substruc- ture could be measured, the data was analyzed subband by subband. The parameters determined for each subband were fit to expressions giving parameters for the whole band(15). Some of the major objections to this method are: 1) Such an analysis scheme does not take into account the physical fact that some of the constants determined for each subband are, in fact, the same for all subbands, (6.8., D7); 2) any transition which belongs to an unanalyzed subband (because too few transitions were identified) 3# 35 are lost to the analysis. These problems are avoided when the expression for the frequency is written as in equation 2) where the actual changes in the quantum numbers are not explicitly evaluated but are left as unspecified variables. Even though their ranges are only - 1, 0, and +1; AJ and AK are treated as full variables on the same footing as J and K. By substituting the explicit values of J, K, AJ and AK into equation 2), we can evaluate the frequency of any transition. That is, we have one expression for all transitions. This is to be contrasted with the past practice of writing several apparently irreconcilably different expressions for the different types of transitions occuring in various portions of a spectrum (16). However, if we wish to estimate any coefficient, we must insure that the variable belonging to this coefficient is linearly independent of the other variables (see discussion of the existence of fif‘). For example, consider the coefficient of (4102. In order to estimate this coefficient, (AK)2 must take on the values of both 0 and 1. That is, we must include lines from both parallel and perpendicular bands. However, the stepwise procedure is correct for a preliminary analysis. The first disadvantage now becomes an advantage by serving as a check on the correctness of the assignment of the various transitions. The observed intensities of the various transitions provide another aid 36 for us to make the proper assignments. For an unperturbed spectrum (which this one seemed to be), finding the proper assignment is the key to analyzing the complete spectrum. The perpendicular band (OK-31') was at first attacked in this stepwise procedure until the parameters determined gave a satisfactory representation of the whole band. These parameters were used to calculate a theoretical parallel band(Al<-=o). This theoretical spectrum enabled us to identify the series of lines belonging jointly to K = 3 and K = 5. An analysis of the complete spectrum was made by combining the transitions with those already identified from the perpendicular band. Quite often spectroscopists have used combination differences (the combining of two transitions originating (or terminating) on the same level) to analyze a spectrum. However, this method could not be applied to this perpendicular band because the intensities were such that the second member for a pair was at best weak and/or blended. In the parallel band, combination differences were used to confirm the assignment given to the various transitions. MISTIC (Michigan State Integral Computer) was program- med to do the numerical calculations involved in analysis. Specified at the input of the program was the form of the Xij's (their functional dependence on J, K,AJ, andA K) and the values of any parameters known from other sources (in our case, Bx, DJK' and DJ from microwave data (17)). The set of data was punched on IBM cards, one datum point 37 per card, in the form: transition assignment, frequency,/_\.\). The output consisted of a calculated spectrum, deviations from the observed spectrum, the calculated estimators, b3, along with their standard deviations, 22b: (square root of their variances), and the estimator of mm standard deviation of the errors in the measurements, During the preliminary analysis, only the terms representing the main effects were used in the theoretical expression; but once the proper assignments were found smaller terms were tested and added to the expression one by one, in the decreasing order of their significance, until no other significant terms could be found, as shown 6' a by the ratio' "l/“b; . Here Vs) XMQ/GMB (nag) were used as main effects; and dfqzlqwoq: and. ad’swere tested n43) and added in that order. The parameters found to be significant, with their corresponding confidence intervals (13,45, {or 06.01;; S: 3.78) are given in Table I. J Note that a b’j‘is included in the results that does not appear in equation 1) for the energy through the third- order. This is a term from the fourth-order twice trans- formed Hamiltonian of Goldsmith et al. (18) which is the coefficient of (Vs+é3,)J(J + 1)K2 and gives the change in DJK with vibration. While we do not know all the terms 7 that come in the fourth-order, we did test for those that ) we do know (v12. 8:, fifand X11?) and kawas the only one found to be significantly different from zero. ' ( Some difficulty was experienced with the interpreta- l ...__J. 38 the m3 0.... m.wm 0. 0b 0. £0 042.0." 0.303 0.40:..3N .sa \ QB. omooo. Nwoooooo. «400000. «00008. .3880. £608. ‘H +1 30 080.0 «8088. u .888. .. «H0008. #8308. .820. cease.n cmm~.am 93H. 33 .24 madame»— emaooooo. condom. o 2 ARM/Naivfimm : 3:. m oh. eaaoooooo. as aaaaooooo. om a masses Hues Nassau. mm 2.3 spec 05.5.83 38M use all e e e e e o e e 95 iv . . C . . C O . Han-h . has . 1.3 . . 0 . . I . é a. so 0 e e e KJKulmur u sweecwwwecmfismm Emmy ...,. m-mm to.‘ I. . limit .. mmeech -32 -MwammiT is: a .3: xx 0 m . . :05“ mmaex Ni. ..xfssm d 39 tion of the results concerning DK and mm. Referring to equation 2), we see that DK is the coefficient of (K + A K)4 - K4, whereas mm is the coefficient of AK(K+AK)3. A careful examination will show that if AK: i 1 only, then AK(K + A K)3 is linearly dependent on the other variables; and only when AK :2 0 is included does AK(K +AK)3 become linearly independent. However, in the parallel band (AK = 0), only two different K values (3 and 5) were identified. Thus we should not be too surprised that we experienced difficulty with r2“, With n“, assumed to be 0, DK had the value recorded. When the dependence on ”411. was included in the expression, the value of DK became negative (believed to be unlikely for physical reasons) and its variance increased to the point where the ratio le'/’4Q5" 85' S flbNPFok‘Pfi'P) for all j simultaneously. In the range that we work, we find \Ifidm‘z-p)‘ N 2 and VPFg(p,n-P) ~ 4' . 41 Thus we see that our confidence intervals will be about twice as wide as when the F test is used; but we have the assurance that they apply to all the parameters simultaneous- ly. SUMMARY We have investigated the problem of estimating many parameters from a large number of observations and have found that 1) If we can write an expression for the value of each observation as a linear function of the parameter: plus a measurement error (i.e., y; 35 + g). 2) if we can assume that the expected (mean) value of the error is zero (i.e., 3(3) = _O_ or E(y_) zgfi ), and 3) if we can assume their errors are all uncorrelated and have equal variance, or if unequal, their ratios are known, (i.e., E(g_e_') = we; )5 then the least squares estimator b_= 371gfy_is the best unbiased linear estimator of Q , and that A2 : -,§-§—§— is an unbiased estimator of ch . If we add the assumption that the errors (and the observations) have a normal distribution, then we are able to give simultaneous confidence intervals for the estimators as 2'; 8495 where 3 ‘”¢bj:: /d,le;; £3 :: \ffiTFaxpyrv-pi‘ 43 with p as the number of parameters and n as the number of observations. (i.e., If we use these confidence intervals from experiment to experiment, then only $10076 of the time should the true value of fl fail to fall within the specified inter- val and conversely, it will be in the interval (1 ~05) 100% of the time.) These confidence intervals also may be used to test if the inclusion of the dependence on another term makes a significant improvement in the representation (fit) of the observations. If “DH/db; s S J then we say no sigificant improvement is made to the representa- tion by including the dependence of 83 to the representa- tion. ‘Whereas, if one only compares the magnitudes of the txvo SSE's, the apparent fit will always be better. We also may use the confidence intervals to test if any of the 8; are significantly different from some known Constant (3,) If lb; “@Wb; S S) then we say the value of B" is not significantly different from $0; . The same frequency interpretation made earlier can be made about these tests (i.e., we will be wrong only (“00% of the time). We have used the infrared absorption spectrum of of CD31 to demonstrate the recommended procedure to follow when using the above results. We shall assume that all the assignments have been made and verified. Any incorrect assignments will usually become quite obvious in the course of the analysis. One should correct these and start the analysis procedure over, skipping” any obviously 42+ unnecessary steps. Step I Assign uncertainties to each observation. If no other information is available, one may assume that all the uncertainties are equal, and assign equal weights to each observation. In our case, we used the signal to noise ratio and amount of blend with neighbors to form a personal Judgement as to the correct set of uncertainties. Once these uncertainties are assigned (let us call them A») then a weight which is proportional to 0/4»)? is assigned to each of the observations. As was shown earlier, the fbj§ determined will be unbiased even when the set of weights are not the correct ones; but only when we have the correct set, will theibag have a minimum variance. It can be inferred that the nearer the set of weights are to the correct ones, the smaller the variance of the {bag should be. Therefore, we should use all available informa- tion in order to select the best set of weights. Step II Form the set of all possible terms which might appear in the analytical expression for the values of the observations. One uses both theoretical and intuitive considerations to identify all of these terms. The analysis scheme assumes that we know the forms of all the terms (i.e., the form of g) on which the value of the observations might depend and the scheme will reject any extraneous ones included; however, it is not clear in the 45 general case what the effect of neglecting one of these dependences is. One can demonstrate the effect of neglect- ing a particular dependence (say quadratic) from a given expression (say linear), but the effect is too dependent upon the particular term and parameters involved to make any general statement to cover all cases. Often, especially in simple cases, one is able to examine the residuals and have an indication as to the form of any dependence neglected. Here (for the CD31 spectrum), in order to form the set to be considered, we used the third order energy expression plus three terms from the fourth order whose form we know. As the analysis progressed, the residuals did not suggest any apparent further dependence which may have been omitted. Step III Build up the best representation, one term at a time, by adding the terms in order of decreasing significance as indicated by the ratio 'myzbj. First, we assume that (1":Xt"'B;+e" for each 3 individually, and consider the ratios lbqflab; for all J's. We retain that ha for which the ratio is the largest and call its 3,0. he then assume that «1": xzogo-v-XL'J- 8; +8; . Again we consider the ratios ”4Z4!”- individually and we pick the next largest as b1. This procedure is repeated until all likely terms are exhausted or until the ratio [my/dab; becomes less than some predetermined constant (viz. S). Quite often, there exists a set of terms which we know from experience to be significant. he will call these our 46 main effects. The procedure outlined may be shortened if, instead of testing each of the three terms individually, we assume at first that L15: 2 X58)- +8,' where the sum extends over the main effects. We now start adding the dependence on the other (is (our smaller effects) and testing one at a time in the above manner. In the case of CD31, we used vinxhh )dnd 83(H234) as our main effects and added af—gg‘j 0,9ch J ’24:, and 5’3," in that order as our smaller effects. The test used (i.e., “Dive,“ 3 S) indicated that values of X:- and X: determined were not significantly different from zero. The results concerning '(4k and DK were more complicated (see earlier discussion of results). It should be emphasized that as each new term is added, the value of p increases by 1 in S=T§Edswdu+a'“d‘n is the marginal distribution function of the subset of p random variables yi i=(1,p). Conditional distribution function: 401:,‘12,’ '3 ‘jp'bjpw, ‘jpfzf' 'z‘jk) : fiflLfl/‘tjfl'd— ('16 ‘3 #0 g((JP’D‘JPIZJ ”143“) is the conditional distribution function of the subset of the random variables , e3; 3 ’='( I, P) o 51 Statistical independence: A set of random variables y1,y2,...,yk are jointly statistically independent if and only if the joint distribution function equals the product of the marginal distribution functions;i.e., if 40311112,”; 11):) : fic‘j.)+2(¢jz)"' 4,..(‘3F3 where f1(y1) is the marginal distribution of Vi: Note 1) if Y1 and y2 are statistically independent then COV (ngyz) : 00 Note 2) Statistical independence implies “wt/um 23;-.,t..,o~,w = my) . APPENDIX II COMMONLY USED DISTRIBUTION FUNCTIONS AND THEIR INTERRELATIONS Normal distribution: y is N(/u , qr? ) means that y has _1 _(--2 ‘f0j)"\nifiq;fip G? ‘iifigé; for a distribution function with an expected value of>o¢ and variance of g~2. Chi-square distribution: u is Kin) means that u has «gm-2) a "7" '25? 1(2) 6 as a distribution function with n degrees of freedom F distribution: u is F(m,n) means that u has r1 T” "‘ 9u5(’”‘32 F(i°)P(-E)(z+ £29 @4507”) 4%}fl(u):: as a distribution function with m degrees of freedom in the numerator and n degrees of freedom in the denominator. There exist extensive tables of these distributions and their oft used functions for various values of the 52 53 parameters involved. In particular, we make use of the tables of oo 0‘. : “Fm”. (Z1)db( - C((mm) which gives PC FCm/Mé Fawn/M) = l- O( . Joint normal distributign: 'n’ith the set {5'1} represented by the vector y, y is INA) Z: ) means that the set {371; has {(3 32"":1n)‘: __ 1 e -Z’(g-,a)’§,(a ‘4'.) 1 2 , 2V)’)’a’ga‘I/2 as its joint distribution function with a set of expected values (9"? represented by/zg and a set of covariances represented by g? Note if 2:, is diagonal, implying g?" is diagonal, then f(y) can be written as: 2 I _ " Liza) I , {(%) = T e “2’ 20:; Whe'c 3;? = (23/1? h. u_ . a) = 4-2.41 giving that the y1 are statistically independent. Relations Between Various Distribptions 2 If y is distributed MW) (1'2 ). than V = $5.4 is distributed X20), 54 If the set of 11 random variables {yig are independently - A distributed N(0,1), than V = . I 3‘2 is distributed X201). X2011) If u and v are independently distributed u 15 and fin) respectively, then ‘7— is distributed F(m,n), 71' ) then any linear transformation Ifyis “(14, Fig 0f L E=L113 NUS/y, £23510 RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR TABULATION 0F OBSERVED AND CALCULATED SPECTRUM K9 39 39 39 39 39 39 3910 3911 3912 3916 3917 3918 3919 3924 3925 3926 3927 3928 3932 3933 3934 3935 49 4 49 8 49 9 4910 4911 4912 4916 4917 4918 4919 4932 4933 4934 4935 5916 5917 5918 5919 5922 \OmflO‘WIPL. CALCULATED 460598847 460594837 460696856 460790855 460794850 460798841 460892828 460896811 460990790 461096666 461190626 461194581 461198532 461398229 461492157 461496081 461590001 461593917 461699543 461793440 461797333 461891223 461291891 461397892 461491882 461495868 461499850 461593828 461699700 461793658 461797612 461891562 462392549 462396444 462490335 462494223 462392182 462396139 462490091 462494039 462595860 APPENDIX 111 CHART I OBSERVED 460598813 460594811 460696818 460790844 460794810 460798706 460892847 460896856 460990943 461096797 461190743 461194595 461198363 461398054 461491852 461496029 461590002 461593880 461699610 461793610 461797552 461891558 461292111 461398054 461491852 461496029 461590002 461593880 461699610 461793610 461797552 461891558 462392340 462396288 462490298 462494250 462392340 462396288 462490298 462494250 462595808 55 DELNU 08$-CALC 90120 90120 90120 90140 90120 90140 90120 90120 90140 90200 90200 90140 90140 90170 90170 90200 90200 90200 90170 90170 90170 90200 90200 90170 90170 90200 90200 90200 90170 90170 90170 90200 90170 90170 90170 90170 90170 90170 90170 90170 90200 “90034 '90026 '90038 “90011 -90039 -90134 90019 90045 90153 90131 90117 90014 -90169 -90175 -90305 -90052 90002 -90037 90067 90170 90218 90335 90220 90163 -90030 90161 90152 90052 -90090 -90048 '90060 -90004 ~90210 -90156 -90037 90027 90157 90149 90207 90211 -90052 K9 J CALCULATED RR 5! 6 462594328 RR 69 7 462598324 RR 69 8 462692317 RR 6915 462990145 RR 6922 463197770 RR 6923 463291699 RR 6924 463295625 RR 6925 463299546 RR 6926 463393464 RR 6927 463397377 RR 6931 463592990 RR 6932 463596883 RR 6933 463690772 RR 7. 7 463199691 RR 79 6 463293682 RR 7. 9 463297668 RR 7910 463391651 RR 7.11 463395629 RR 7912 463399603 RR 7915 4635.1499 RR 89 6 463894470 RR 69 9 463898455 RR 8910 463992436 RR 8912 464090384 RR 8915 464192275 RR 8916 464196229 RR 8917 464290180 RR 8919 464298067 RR 99 9 464498655 RR 9.10 464592634 RR 9911 464596608 RR 9.12 464690579 RR 9915 464792462 RR 9916 464796415 RR 9’17 4648.0363 RR 9918 464894306 RR 9919 464898245 25 9921 464996109 R 9.22 465090035 22‘9923 465093956 RR10910 465192236 RR10911 465196209 9910’12 465290177 RR10.15 465392053 RRI0916 465396003 RR10.17 465399948 Raxg’13 465493889 RRl .19 265997824 RR1§911 465795204 991 ‘12 “65799170 1‘14 465897087 ifiii‘is “559.1038 ‘ 6 465994985 56 OBSERVED 462594339 462598312 462692266 462990323 463197955 463291739 463295817 463299690 463393519 463397438 463592945 463596730 463690527 463199564 463293607 463297704 463391720 463395693 463399667 463591595 463894223 463898280 463992292 464090409 464192254 464196274 464290271 464298015 464498711 464592648 464596597 464690598 464792362 464796316 464890209 464894158 464898164 464996142 465090042 465094067 465192138 465196010 465290116 465391806 465395923 465490082 465493811 465497833 465795260 9657.9295 465897039 565990967 465994952 DELRU OBS-CALC 90120 90120 90120 90280 90280 90120 90120 90100 90120 90100 90140 90200 90200 90140 90140 90120 90120 90120 90140 90140 90140 90200 90200 90200 90140 90140 90140 90120 90140 90140 90120 90140 90140 90120 90140 90140 90140 90200 90200 90200 90200 90200 90200 90280 90280 90280 90280 90280 90140 90200 90200 90120 90120 90011 -90013 *90051 90178 90186 90039 90192 90143 90055 90061 ‘90045 ‘90153 490246 ’90127 ‘90075 90036 90069 90065 90064 90096 “90246 ‘90174 -90143 90024 ‘90020 90045 90092 ‘90053 90056 90014 ‘90012 90019 “90101 “90099 -90154 -90148 '90081 90033 90008 90111 ‘90098 ‘90199 *90061 ‘90248 -90080 90134 -90078 90009 90056 90126 ‘90048 -90072 '90033 ‘ {'6’ - _. ”...,. '1 v K0 J CALCULATED RR11017 465908927 8811018 «660.286« RR11.21 «661.«6«8 R811.22 «661.8566 RR11023 «662.2«80. RR11.2« «662.6389 RR11.25 «663.0293 RR12.12 «663.75«8 RR12014 466405460 RR12015 4664.9008 RR12016 «665.3352 RR12017 «665.7290 RR12018 «666.122« RR12021 «667.2997 RR12.22 «667.6911 RR12023 «668.0821 RR12.2« 466804725 RR12.25 «668.8625 RR12.26 «669.2521 RR1«.1« «676.0326 RRl«.15 «676.6266 8R15015 «682.0735 RR15016 «682.«668 R815.17 «682.8595 R816.16 «688.0682 8816017 468804405 RR17.17 4693095534 RR17.18 «694.3465 RR17019 «69«.7373 R“7020 «695.127« 2212018 «699.7938 019 «700 .8« RR18.20 «70015732 RR18021 «700.9626 QR 3’ 3 “547.5209 QR 3' 5 «5«8.3227 OR 3. 6 454807230 °R 3’ 7 «5«9.1229 QR 3’ 8 454905224 Q“ 31 9 «5«9.9215 g: 3’19 «550.3203 0R 3’11 «550.7186 QR 8.15 «552.3082 Q 3.16 «552.70«6 3‘17 «553.1006 R 3.18 «553.«962 3.19 «553.891« R 3.20 .55#.2862 3.21 «556.6807 Q 3'22 «555.07«7 3023 055500680 1’26 «556.6«70 ’27 «557.0391 57 OBSERVED 465908856 466002852 466104641 466108560 466202366 466206488 066300213 466307676 066405526 466409531 566503446 466507389 066601313 466702975 466706877 466800802 466804665 066808565 466902469 467600386 467604381 468200760 468204651 468208594 468800452 468804218 469309557 469403500 069407346 469501128 469907924 470001862 570005815 470009651 454705238 454803233 454807252 454901202 454905235 454908990 455003029 455007049 455202936 '455206927 455300930 455304871 455308828 055402773 455406728 055500544 455504545 455606340 455700180 DELNU 00120 00120 00280 00280 00280 00280 00200 00140 00120 00120 00120 00120 00120 00140 00120 00140 00120 00100 00200 00140 00200 00140 00140 00200 00200 00200 00200 00200 00200 00200 00280 00280 00280 00280 00250 00170 00170 00170 00170 00150 00150 00170 00150 00150 00170 00170 00170 00170 00170 00170 00170 00200 00200 OBS-CALC -00071 -00012 -0000? ‘00007 -00114 00099 -00080 00128 00066 00123 00095 00099 00089 -00021 -00034 -00018 ’00061 “00060 -00052 00062 00116 00025 “00018 -00001 -00030 “00187 00004 00034 -00026 ‘00146 -00014 00022 00079 00025 00028 00006 00022 -0002? 00011 ‘00226 -00174 “00138 “00145 ‘00119 “00076 -00091 ‘00085 ‘00090 -00078 ‘00203 ‘00130 '00131 ‘00211 K0 J OR 3031 OR 3.32 08 3033 OR 3034 OR 3035 08 3036 OR 3037 0R 3038 OR 3039 OR 3040 OR 3041 OR 50 6 OR 50 7 GR 50 8 OR 50 9 GR 5010 OR 5011 OR 5012 OR 5016 OR 5017 OR 5018 OR 5019 OR 5020 OR 5021 QR 5022 OR 5023 OR 5024 GR 5027 OR 5028 08 5032 GR 5033 08 5034 OR 5035 QR 5036 QR 5037 QR 5038 QR 5039 QR 5040 08 5041 08 5042 CALCULATED 455806038 455309940 455903839 455907734 456001625 456005513 456009397 456103278 456107155 456201029 456204899 454803163 454807161 454901155 454905145 459909131 455003113 455007091 455202964 455206922 455300876 455304826 455308772 455402714 455406652 455500586 455504516 455606282 455700197 455805816 455809711 455903603 455907490 456001374 456005255 456009132 456103005 456106875 456200741 456204603 58 OBSERVED 455805929 455809745 455903620 455907531 456001493 456005380 456009174 456103043 456106882 456200857 456204780 454803233 454807252 454901202 454905235 454908990 455003029 455007049 455202936 455206927 455300930 455304871 455308828 455402773 455406728 455500544 455504545 455606340 455700180 455805929 455809745 455903620 455907531 456001493 456005380 456009174 456103043 456106882 456200857 456204780 DELNU OBS-CALC 00150 00150 00120 00120 00150 00150 00150 00150 00150 00150 00170 00240 00240 00240 00240 00210 00210 00240 00210 00210 00240 00240 00240 00240 00240 00240 00240 00280 00280 00210 00210 00170 00170 00210 00210 00210 00210 00210 00210 00240 ‘00109 -00196 -00220 “00203 -00132 ‘00133 “00223 “00235 “00273 ~00171 -00119 00070 00090 00047 00090 ‘00142 -00084 -00043 '00027 00005 00054 00045 00057 00059 00077 '00042 00029 00057 '00017 00113 00034 00017 00041 00119 00125 00042 00038 00007 00117 00176 .___._._______...._, x; ' r . .7 - 0 CALCULATED 449706726 449702647 449608564 949604478 999506296 449502199 4“19908100 449403997 999309891 949305782 449301670 499207555 449203438 449109317 452002293 451908229 451904162 451900091 451806017 451801940 451707859 451609688 451605598 460504837 460508847 460602854 460606856 460700855 460704850 460708841 460802828 460806811 460900790 961104581 221108532 A 90608 461500001 221503917 1507 461609525 461703440 461707333 461801223 461805109 461405868 461409850 461503828 461507802 461609700 461703658 461707612 59 CHART II OBSERVED 449706833 449702651 449608603 449604402 449506446 449502194 449408071 449404028 449309914 449305794 449301692 449207464 449203463 449109352 452002573 451908270 451904131 451900109 451805893 451801671 451707469 451609517 451605025 460504891 460508872 460602859 460606754 460700833 460704819 460708751 460802828 460806874 460900837 461104568 461108388 461406060 461500055 461503888 461507921 461609615 461703552 461707525 461801514 461805499 461406060 461500055 961503888 461507921 461609615 461703552 461707525 DELNU OBS-CALC 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00280 00100 00100 00200 00100 00120 00100 00120 00100 00120 00200 00120 00120 00170 00170 00140 00170 00140 00140 00170 00170 00170 00140 00170 00140 00170 00140 00140 00170 00107 00004 00038 “00076 00151 ‘00006 '00028 00031 00023 00012 00022 ”00091 00026 00034 00280 00040 -00031 00018 -00124 -00268 -00390 -00171 -00573 00054 00025 00006 ‘00102 '00022 -00031 -00090 00000 00063 00048 -00013 “00144 -00021 00054 ‘00029 00092 00073 00113 00192 00291 00390 00192 00205 00060 00119 -00085 “00106 “00087 1 04“ 3—03.28- 'V warden-i" _ .__ ”_" ‘1‘ I .‘9 0’ ~.. 0 ‘ ‘ , B . «1' ‘ "0 RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR K9 J CALCULAT 4919 ED #920 4931 4932 4933 4934 4935 4936 5915 5916 5917 5918 5919 5920 5922 5923 5924 5925 5926 5927 5928 69 6 69 7 69 8 69 9 6910 6911 6912 6915 6922 6923 6924 6925 6926 6927 6928 6931 6932 6933 79 7 79 8 79 9 7910 7911 7912 7915 7916 7917 89 8 89 9 8910 8911 8912 461891562 461895508 462298651 462392549 462396444 462490335 462494223 462498106 462298222 462392182 462396139 462490091 462494039 462497984 462595860 462599792 462693720 462697643 462791563 462795479 462799391 462594328 462598324 462692317 462696305 462790288 462794268 462798244 462990145 463197770 463291699 463295625 463299546 463393464 463397377 463491286 463592990 463596883 463690772 463199691 463293682 463297668 463391651 463395629 463399603 5635.1499 4635. Q 5455 463599409 463094470 463898455 453992436 463996412 964090384 60 OBSERVED 461891514 461895499 462298445 462392319 462396288 462490253 462494246 462498084 462298445 462392319 462396288 462490253 462494246 462498084 462595878 462599701 462693592 462697461 462791431 462795172 462799304 462594300 462598297 462692293 462696253 462790214 462794312 462798352 462990249 463197959 463291834 463295729 463299693 463393552 463397421 463491388 463592932 463596818 463690685 463199609 463293654 463297648 463391683 463395633 463399613 463591451 463595438 463599416 463894275 463898263 463992315 463996238 464090346 DELNU OBS-CALC 90170 90170 90170 90140 90140 90140 90170 90170 90170 90140 90140 90140 90170 90170 90120 90120 90120 90140 90200 90200 90200 90120 90120 90120 90120 90200 90200 90200 90140 90120 90100 90100 90100 90100 90100 90120 90120 90120 90120 90120 90120 90140 90120 90120 90120 90140 90120 90120 90120 90120 90140 90200 90140 ~90048 ~90009 -90205 -90231 ‘90156 '90082 90023 ‘90022 90223 90136 90149 90162 90207 90101 90018 '90091 ‘90127 -90183 “90132 ~90308 -90087 -90028 -90020 ‘90023 -90052 '90075 90044 90108 90104 90190 90134 90104 90147 90008 90044 90101 -90058 “90065 -90087 -90082 '90028 ‘90020 90032 90004 90010 ”90048 ~90018 90008 “90195 '90192 '90120 ~90174 ‘90039 ifl‘t - inf—1,232". '.; 2 I . K9 J CALCULATED RR 8915 RR 8916 RR 8917 RR 8918 RR 8919 RR 8920 RR 8921 RR 8922 RR 8923 RR 99 9 RR 9910 RR 9911 RR 9912 RR 9915 RR 9916 RR 9917 RR 9918 RR 9919 RR 9921 RR 9922 RR 9923 RR11911 RR11912 RR11914 R311915 RRllols RR11911 RR11918 RR11919 RR11922 RR11923 RR11924 RR11925 RR12912 RR12919 RRI2915 RR12916 RR12.17 RRIZ918 RR12919 RRI2920 RR12921 RR12922 RR12923 RRI29209 RR12925 RR12926 GP 3947 Q” 3945 GP 3’“ 464192275 464196229 464290180 464294126 464298067 464392005 464395930 464399866 464493791 464498655 464592634 464596608 464690579 9947.2492 064796415 464890363 464894306 404898245 464996109 465090035 465093956 465795204 465799170 465897087 055991038 465994985 465998927 466092864 466096797 066193566 056292480 465296309 466390293 466397548 466495460 “66499408 466593352 466597290 466691224 466695153 155699077 66792997 265796911 #66aooaz1 466094725 466898625 066992521 4526.9579 4527.0119 452794859 452798999 l’51’093139 452097276 61 OBSERVED 464192248 464196306 464290136 464294121 464298001 464391945 464395891 464399750 464493796 464498665 464592632 464596595 464690586 464792403 464796341 464890167 064894169 464898158 464996179 465090060 465094069 465795197 465799285 465896991 465990970 465994096 465998903 466092732 466096771 466198668 466292506 466296444 060390259 466397758 “66495548 466499484 466593499 466597408 466691313 466695302 466699052 466793036 966796857 466890736 066894731 466898535 466992366 052096632 452790719 052795017 452799067 052893173 452897227 DELNU OBS-CALC 90140 90140 90120 90140 90120 90200 90120 90140 90120 90140 90140 90120 90100 90100 90120 90140 90100 90100 90120 90120 90120 90100 90120 90120 90100 90100 90120 90100 90120 90140 90140 90140 90140 90140 90120 90100 90100 90100 90100 90120 90120 90120 90120 90120 90120 90120 90120 90150 90150 90150 90150 90110 90110 '90027 90075 -90044 ‘90005 '90066 .90060 ‘90047 ‘90116 90006 90010 -90001 -90014 90000 -90060 -90074 '90196 -90137 -90086 90069 90025 90113 -90007 90115 '90096 -90069 -90089 “90024 '90133 '90026 90102 90026 90055 -90035 90210 90088 90076 90147 90117 90088 90149 -90026 90039 -90054 ‘90084 90006 -90090 ‘90154 90055 90000 90150 90060 90036 ‘90049 K9 J CALCULATED GP 3941 GP 3940 GP 3939 GP 3938 GP 3937 GP 3935 GP 3934 GP 3933 GP 3932 GP 3931 GP 3930 GP 3929 GP 3928 GP 3927 GP 3926 GP 3925 GP 3924 GP 3923 GP 3922 GP 3921 GP 3919 GP 3918 GP 3917 09 3916 GP 3915 GP 3914 GP 3913 Q? 3911 09 3910 GP 39 8 OR 39 3 OR 39 5 OR 39 6 QR 39 7 QR 39 8 9 452991412 452995548 452999683 453093816 453097948 453196206 453290333 453294459 453298582 453392703 453396823 453490940 453495055 453499168 453593278 453597386 453691492 453695595 453699695 453793793 453891980 453896070 453990156 453994239 453998320 454092397 454096471 454194610 454198675 454296793 454795209 454893227 454897230 454991229 454995224 454999215 455093203 455097186 455293082 455297046 455391006 455394962 455398914 455492862 455496807 455590747 455594684 455696470 455790391 455896038 455899940 455993839 455997734 62 OBSERVED 452991445 452995491 452999685 453093774 453097986 453196169 453290292 453294554 453298616 453392746 453396882 453491098 453495345 453499173 453593312 453597353 453691617 453695632 453699826 453793954 453891964 453895998 453990137 453994285 453998391 454092333 454096467 454194671 454198555 454296716 454795400 454893304 454897388 454991294 454995305 454999227 455093201 455097157 455293085 455297049 455391022 455394965 455398926 455492865 455496778 455590709 455594596 455696570 455790416 455896027 455899941 455993795 455997635 OELNU 08$~CALC 90110 90110 90110 90110 90110 90110 90110 90110 90110 90110 90110 90170 90170 90170 90120 90120 90120 90120 90120 90120 90120 90150 90120 90170 90150 90170 90120 90120 90120 90170 90250 90170 90170 90170 90170 90150 90150 90170 90150 90150 90170 90170 90170 90170 90170 90170 90170 90200 90200 90170 90150 90120 90150 90033 '90057 90003 “90042 90039 -90037 “90041 90096 90034 90043 90059 90158 90291 90005 90034 '90033 90125 90037 90130 90161 ‘90016 ~90072 -90019 90046 90071 ‘90064 ‘90004 90061 -90119 -90077 90190 90078 90158 90065 90081 90011 ‘90002 '90029 90004 90003 90016 90003 90012 90002 ’90029 ‘90038 ‘90087 90100 90025 '90012 90001 ‘90044 '90099 9“.““14 i- . r ‘_;-!;7 ._———_ 2.. I“ K9 J. CALCULATED OR 3935 OR 3936 OR 3937 OR 3938 OR 3939 QR 3940 OR 3941 OR 3942 GP 5946 GP 5945 GP 5944 GP 5943 GP 5942 5941 P 5940 GP 5939 GP 5938 P 5.37 0: 5939 593 GP 593; GP 5932 P 5931 GP 5930 Q: 5929 5928 GP 5927 Qp 5926 GP 5925 GP 5924 GP 5923 GP 5922 09 5921 GP 5920 GP 5918 GP 5917 GP 5916 GP 5915 GP 5914 GP 5913 GP 5912 0? 5910 GP 59 9 GP 59 7 OR 59 6 OR 59 7 OR 59 8 OR 59 9 OR 5910 OR 5911 OR 5912 QR 5916 QR 5917 456091625 456095513 456099397 456193278 456197155 456291029 456294899 456298766 452696595 452790739 452794881 452799023 452893164 452897304 452991443 452995580 452999716 453093850 453097983 453196243 453290371 453294496 453298620 453392741 453396860 453490977 453495092 453499204 453593313 453597421 453691525 453695627 453699726 453793822 453892006 453896093 453990177 453994259 453998336 454092411 454096483" 454194616 454198677 454296789 454893163 454897161 454991155 454995145 454999131 455093113 455097091 455292964 455296922 63 OBSERVED 456091599 456095515 456099360 456193117 456197034 456290992 456294913 456298660 452696632 452790719 452795017 452799067 452893173 452897227 452991445 452995491 452999685 453093774 453097986 453196169 453290292 453294554 453298616 453392746 453396882 453491098 453495345 453499173 953593312 453597353 953691617 453695632 453699826 953793954 453891964 453895998 453990137 453994285 453998391 454092333 454096467 454194671 454198555 454296716 454893304 454897388 454991294 454995305 454999227 455093201 455097157 455293085 455297049 DELNU OBS-CALC 90150 90150 90150 90150 90150 90150 90170 90170 90210 90210 90210 90210 90160 90160 90160 90160 90160 90160 90160 90160 90160 90160 90160 90160 90160 90240 90240 90240 90170 90170 90170 90170 90170 90170 90170 90210 90170 90240 90210 90240 90170 90170 90170 90240 90240 90240 90240 90240 90210 90210 90240 90210 90210 -90026 90002 -90037 “90161 -90121 -90037 90014 -90106 90038 ‘90020 90135 90043 90009 “90078 90002 -90089 '90030 “90076 90003 '90074 -90079 90058 -90004 90005 90021 90121 90254 -90031 ‘90002 '90067 90092 90005 90100 90132 ‘90041 -90095 -90040 90027 90055 ‘90078 -90015 90055 ‘90122 ~90073 90141 90226 90139 90160 90096 90088 90066 90122 90127 “ -..L ‘. ~/. I , 9 Vat ~...— “7 ...— x. J CALCULATED HEIGHTED STANDARD DEV QR 5,15 6553.0876 on 5.20 0553.0772 OR 5.21 «550.2715 OR 5.22 6550.6652 OR 5.23 4555.0586 OR 5.20 6555.0516 0R 5.21 6556.6202 OR 5.20 0557.019? on 5.32 4558.5816 3R 5.33 6556.9711 R 5.30 655903603 8: 5.35 «559.7090 OR 2.36 456001376 on .37 656005255 OR 5.30 0560.9132 OR 3.39 456103005 QR 5040 656106375 OR .01 0562.0701 OR 5.02 656206603 5.03 0562.0063 g 3 #53301413066371 8 2 302556531727 8 3 .6716073559 8 4 .0126000226 B 5 00000380877 8 6 .0000912458 B 7 -.0000027110 -.0000001217 STD DEVIATION OF THE INVERSE 50 RT OF AVE 64 OBSERVED 455301022 «553.4965 655308926 «554.2865 455606778 455500709 655504596 455606570 455700615 455806027 455809961 455903795 655907635 456001599 456005515 656009360 656103117 656107034 456200992 656206913 456208660 DELNU OBS-CALC 00240 00260 00240 00240 00240 00200 00240 00280 00280 00240 00210 00170 00210 00210 00210 00210 00210 00210 00210 00240 00240 50 50 50 SD 50 50 50 SD DEV DEV DEV DEV DEV DEV DEV DEV POINTS IATION HT 00018802841 00021978805 00001756780 00000186256 00000005398 00000023840 00000005925 00000000308 00118606693 .0106617826 100006191862 00146 00139 00154 00151 00126 00123 00080 00288 00219 00211 00230 00192 00165 00225 00260 00229 00112 00160 00251 00310 00197 CONSTANTS 1N SAME ORDER AS TABLE I "an. ‘ ‘1 '0 I. r" I“ H 1 '- :6 din-‘0' ~ In" WW .~ 0. an: "‘001001“ —