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An air mass climatology of Canada during the early nineteenth century : an analysis of the weather records… Minns, Robert 1970

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AN AIR MASS CLIMATOLOGY OF CANADA DURING THE EARLY NINETEENTH CENTURY •An Analysis of. the Weather Records of Certain Hudson's Bay Company Forts by ROBERT MINNS B.A*, University of Lancaster, 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Geography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1970 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f G e o g r a p h y  T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e A p r i l 13, 1970 i i A b s t r a c t The post j o u r n a l s o f c e r t a i n Hudson's Bay Company f o r t s were examined f o r ev idence tha t a i r mass f requenc ies du r ing the f i r s t h a l f o f the n ine teenth century were markedly d i f f e r e n t from those of a modern (1955-1959) p e r i o d . The " p a r t i a l c o l l e c t i v e " techn ique of Bryson was used to determine the modern f requenc ies and to p rov ide the bas i s of the c o n d i t i o n a l p r o b a b i l i t y s t r u c t u r e employed to es t imate the h i s t o r i c a i r mass f r e q u e n c i e s . There i s ev idence from each s t a t i o n for which a n a l y s i s was performed of a g r e a t l y i nc reased presence of " A r c t i c " a i r , p robab ly as a consequence of a weakened zonal atmospher ic c i r c u l a t i o n and a decrease i n the eastward pene t ra t i on of " P a c i f i c " a i r . TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION . . . . • 1 II. AIR MASSES AND CLIMATIC CHANGE 3 III. THE MODEL 20 IV. THE ANALYSIS 29 BIBLIOGRAPHY 58 APPENDIX A. LIST.OF CONDITIONAL PROBABILITIES 62 B. PERCENTAGE DEVIATIONS FROM 1955-59 AIR MASS FREQUENCIES . 67 C. CONFIDENCE LIMITS FOR AIR MASS FREQUENCY 69 iv LIST OF TABLES TABLE PAGE I. Sources of Modern Data 31 II. Air Mass Temperatures and Sources for Edmonton 35 III. Air Mass Temperatures and Sources for Winnipeg 36 IV. Air Mass Temperatures and Sources for Fort William . . . . . 38 V. Air Mass Temperatures and. Sources for Fort Simpson . . . . . 41 VI. The Availability of Daily Records of Climate 44 V LIST OF FIGURES FIGURE PAGE 1. A i r Masses of North America . . 7 • 2. Trends of World Temperature 15 3. Schematic Drawing Showing the Eastward Penetration of P a c i f i c A i r During Times of High and Low Zonal Index . . 17 4. Temperature-Frequency Histogram for Baker Lake, February 1950-1959 22 5. Sketch of the Location of Fort Simpson and Possible A i r Mass Sources 40 6. Fort Simpson: Deviations from 1955-1959 A i r Mass Frequencies 48 7. Edmonton: Deviations from 1955-1959 A i r Mass Frequencies . 49. 8. Winnipeg: Deviations from 1955-1959 A i r Mass Frequencies . 50 9. Fort William: Deviations from 1955-1959 A i r Mass Frequencies . . 51 10. Wind Roses for Fort William 53 Acknowledgements Acknowledgement i s due to the Governors of the Hudson's Bay Company for permission to inspect certain of the records of the Company. Par t i c u l a r thanks are due to Dr. J. K. Stager for great assistance in formulating the project and bringing i t to a conclusion. The timely c r i t i c i s m s of Mr. J . E. Hay were also much appreciated. CHAPTER I INTRODUCTION A paper by Bryson (1966) introduced some interesting perspec-tives into the study of the Canadian climate, making clear for the f i r s t time the manner in which the climate of the central portion of the country may be interpreted in terms of the seasonal predominances of and interactions between " A r c t i c " , " P a c i f i c " and "United States" airflows. The relationship between Bryson's schema and the recognized structure of the mid-latitude general c i r c u l a t i o n of the atmosphere promotes speculation concerning the altered disposition of the three airflow types during a period with a rather d i f f e r e n t general c i r c u l a -tion regime. I t i s generally conceded (Lamb, 1963) that the strength of the mid-latitude westerly vortex was considerably less than that of the present during the f i r s t half of the nineteenth century--this thesis represents an attempt to deduce the consequences of such a weakened zonal c i r c u l a t i o n i n terms of Bryson's airflow model and to test the v a l i d i t y of these deductions using a previously untapped source of "meteorological", data for nineteenth century Canada. The data are contained i n the post journals of the forts of the Hudson's Bay Company and were abstracted for the period 1824-1851. These journals are ess e n t i a l l y a record of the d a i l y l i f e and transactions of the forts and, as a r u l e , contain some reference to the state of the weather. Thus, apart from t h e i r unique interest to the h i s t o r i a n , the journals contain material of use to the 2 climatologist concerned with recent secular climatic variation. The observations are mostly of a qu a l i t a t i v e nature, comments on wind d i r e c t i o n , precipitation type and the l i k e , although occasional more comprehensive meteorological journals are to be found. But to obtain useful results from data of this kind i t i s necessary to resort to rather unusual analytic procedures, developed within the framework of a careful experimental design in order to test a very s p e c i f i c research hypothesis. I t i s probably useful to provide a br i e f statement of this research hypothesis at this early point in the discussion. Recalling that the aim of the study i s to attempt to develop insights into the Canadian climate of the nineteenth century, consideration of the results of other researchers leads to the conclusion that the investigation can be performed to greatest effect by i n s t i t u t i n g tests of the following general research hypothesis That the strength of the zonal winter c i r c u l a t i o n of the atmosphere was weak r e l a t i v e to the present during the h i s t o r i c period. That this weakness was reflected in a decrease in the frequency of P a c i f i c a i r at stations to the east of the Co r d i l l e r a . A discussion which leads to the framing of this hypothesis i s contained in the subsequent, chapter. The remainder of the thesis consists of the development of suitable analytic procedures and the testing of the hypothesis on the data obtained from selected Hudson's Bay Company f o r t s . CHAPTER II AIR MASSES AND CLIMATIC CHANGE Ai r mass climatology i s no longer fashionable. Most cl i m a t o l -ogists would probably share Hare's (1960, p. 356) sentiments Instead of applying rules of thumb to the crude a i r mass and frontal concepts of yesterday, we now attempt to apply the baroclinic and barotropic theorems to the entire three-dimensional f i e l d of the atmospheric elements. The concept of a i r masses was a b r i l l i a n t approximation necessary in an age when upper a i r measurement was extremely d i f f i c u l t . Hare i s at pains to stress that modern climatology has not simply at-tached more euphonic or otherwise more impressive labels to older con-cepts, but rather i s attempting to apply the theory and associated laws of the relevant branches of physics towards an understanding of the atmosphere. A crudely stated a i r mass climatology, with a i r of homogeneous and d i s t i n c t i v e character located over a well-defined source region and clashing with a i r of a di f f e r e n t character at fron-tal "battle-grounds" cannot be part of this analysis (although i t i s an engaging and successful teaching device, as i s witnessed by i t s use in Strahler (1966) and Trewartha (1954), m i l i t a r y analogies and a l l ) . The suggestion that a i r mass climatology i s not an adequate model (or "approximation") in the context of contemporary knowledge of the atmospheric c i r c u l a t i o n and modern sophistication i n the measure-ment of atmospheric phenomena forms a convenient springboard for this section, since i t leads to the conclusion that some modification of a i r mass analysis might be an appropriate technique when measurement i s only of the crudest and most inconsistent kind. This leads to an examination of modern approaches to a i r mass analysis and the i r r e l e -vance to schemes of recent climatic change on the North American conti-nent. Thus j u s t i f i c a t i o n and background are provided for the mathema-t i c a l model developed and used i n subsequent sections. I t would appear to be a reasonable general pri n c i p l e that the coarser the data available, the cruder, although not necessarily the simpler, the analytic model that should be used. The data used in this thesis are p a r t i c u l a r l y coarse, consisting mostly of q u a l i t a t i v e assess ments of the state of the weather. Therefore the model used must be crude, although not necessarily without mathematical or conceptual elegance. We have learned above that a i r mass analysis i s "crude" and involves the use of "rules of thumb"; this in i t s e l f i s hardly s u f f i c -ient j u s t i f i c a t i o n for i t s inclusion in this study, these comments stress that the es s e n t i a l l y two-dimensional approach of conventional a i r mass analysis does not adequately r e f l e c t the state of the three-dimensional system that i s the atmosphere. But the data of this study were compiled by unskilled observers, r e s t r i c t e d to impressions of tem-perature, wind and p r e c i p i t a t i o n conditions in the lower layers of the atmosphere. The only observations d i r e c t l y representing processes oper ating in the t h i r d dimension are those of cloud formation, type and movement. Since the data used are mostly two-dimensional i t would not b e - r e a l i s t i c to reject the use of a i r mass analysis on the grounds that i t i s inherently two-dimensional. But the analysis may be made more r e a l i s t i c by the incorporation of contemporary knowledge of the general c i r c u l a t i o n and the d i s t r i b u t i o n of meteorological elements. 5 Realism may be introduced in two respects; f i r s t l y by considering the a i r mass concept in r e l a t i o n to present knowledge of the atmospheric c i r c u l a t i o n over the North American continent and secondly by r e a l i s i n g that the concept of the a i r mass i s essenti a l l y a s t a t i s t i c a l one. These improvements, largely due to Bryson (1966), lead to a conceptually a t t r a c t i v e mode of analysis. In i t s simplest form, a i r mass analysis assumes that a i r of a homogeneous character develops i n the "source regions", above the "polar" and " t r o p i c a l " oceans and continents. These a i r masses then meet in the zone of the mid-latitude westerlies at temperature and humidity discon-t i n u i t i e s known as "fronts", along which the depressions of mid-latitudes form. Bryson (1.966) develops this simple scheme in two ways, f i r s t l y by introducing the concept of the "climatic complex" and secondly by stress-ing the significance of the physiographic context of North America. The s i m p l i s t i c view of the a i r mass treats i t as a homogeneous body of a i r stretching from source region to f r o n t — t h e climatic complex involves the more limited conclusion that, at any particular s t a t i o n , a i r coming from a s p e c i f i c d i r e c t i o n w i l l tend to be associated with a r e l a t i v e l y narrow range of temperatures, with particular cloud condi-ti o n s , with certain pre c i p i t a t i o n types and amounts etc. This a i r may be tracked to i t s source region by any of a number of techniques (dis-cussed l a t e r i n this chapter) and an a i r mass climatology so constructed. Bryson offers a quite convincing discussion of the climatic complex as the determinant of the position of the northern boundary of the Canadian Boreal forest, stressing that i t i s the complex rather than i t s 6 individual elements which determine the location of this boundary. The a i r mass determinations for North America of Brunnschweiler (1952) are those included in the fa m i l i a r elementary climatology and physical geography texts. Firmly based on the European pattern, these determinations do not correspond well with the physiography and known airflow characteristics of North America. By means of a rather tedious trajectory analysis, Bryson was able to demonstrate that the contribution of P a c i f i c a i r to the Canadian P r a i r i e s and even to some parts of Ontario i s much greater than that shown by Brunnschweiler. Bryson (1966, pp. 230-234) chose to delimit four-source regions for Canada—the P a c i f i c , the A r c t i c , the United States and ( r e l a t i v e l y small) the A t l a n t i c . Figure 1 summarises Bryson's analysis; note i n particular the broad wedge of P a c i f i c a i r penetrating the prairies and central Canada, and the zone of contact between P a c i f i c and A r c t i c a i r , prominent in l a t e r discussion as the "Arctic Front". Conventional analysis tends to group together the A r c t i c and P a c i f i c a i r in the category "continental Polar", obscuring the r e a l i t y of the P a c i f i c intrusion. More e x p l i c i t acknowledgement of the s t a t i s t i c a l content of the a i r mass concept has been made. Discussion of the climatic complex above was phrased i n such terms as "associated", "tend" and "range", i l l u s t r a t i n g the nature of this change. Bryson (1966) has observed that frequency di s t r i b u t i o n s of da i l y temperatures are often multimodal or of a very irregular shape, leading to the hypothesis that the frequency di s t r i b u t i o n s represent the composite of several d i s t r i b u t i o n s , 7 FIGURE 1 AIR MASSES OF NORTH AMERICA. AREAS CONTAINED WITHIN THE HEAVY BROKEN LINES EXPERIENCE THE NAMED AIR MASS 50% OF THE TIME mT = maritime Tropical cT = continental Tropical 8 each representing the contribution of a part i c u l a r a i r mass. Frequency di s t r i b u t i o n s derived for a i r mass source regions were unimodal normal type curves, leading Bryson to the conclusion that the form of the frequency of temperature at a station outside of the source region represents a composite of several such curves. The consistency with which he was able to i d e n t i f y the separate components throughout the North American continent supports this conclusion. A detailed d i s -cussion of the techniques appropriate to the i d e n t i f i c a t i o n of the various components w i l l be given in a subsequent section; here the aim is to stress the consciously s t a t i s t i c a l approach employed. The use of this approach implies that the a i r mass climatology of a station should be discussed i n terms of the r e l a t i v e frequency of the various a i r masses, rather than in terms of the dominance of any one particular a i r mass. A further implication i s that the mean temperature of an a i r mass, as expressed by the mean temperature of the appropriate component of the composite d i s t r i b u t i o n , may exhibit a systematic spatial variation. I t i s clear that, in the Northern hemisphere, temperature should decline systematically from mid-latitudes northward to the pole in response to d i f f e r i n g i n s o l a t i o n , quite apart from any effects of the a i r mass * d i s t r i b u t i o n . One rather simple-minded attempt to incorporate this effect into the analysis was made by the author (Minns 1968) in a study to establish the relationship between the boundaries' of continuous and discontinuous permafrost and a i r mass d i s t r i b u t i o n in Canada. The data were divided into groups within and without the permafrost boundaries and an analysis of covariance was then performed, removing the l a t i t u d i n a l influence by li n e a r regression. Analysis of variance on the adjusted means revealed 9 The use of these techniques provides Bryson with a comprehensive series of maps i l l u s t r a t i n g the seasonal predominances of the various source regions and the locations of the major fronts. I t w i l l be re c a l l e d , however, that the aim of this study i s to unify in some quantifiable manner series of weather observations from the nineteenth century. Recall also the uneven quality of these records. Neither streamline nor trajectory analysis y i e l d an immediately obvious means of completing this task, at least in any way lending i t s e l f readily to adequate mathe-matical formulation. Bryson's th i r d technique i s that of "p a r t i a l c o l l e c t i v e s " , alluded to above in the discussion of the form of the histogram of frequency of temperature. The aim of this technique i s to break the histogram into i t s component distrib u t i o n s and to regard these as. c o n t r i -butions from separate a i r masses. This technique i s much better suited to the purposes of th i s study and w i l l be discussed below. Computational aspects and validation of the technique w i l l be discussed i n l a t e r sections; the present object i s to point out the implications of a successful application of the technique. The form of the histogram of temperature frequency may be regarded as a r e f l e c t i o n of the probability density function of temperature at the station under consideration. S i m i l a r l y , the individual normal distributions isolated by the technique of p a r t i a l c o l l e c t i v e s may be regarded as a r e f l e c t i o n s i g n i f i c a n t differences between the groups, which might with some j u s t i -f i c a t i o n be regarded as evidence of an a i r mass eff e c t . 10 of the probability density of temperature i n a pa r t i c u l a r a i r mass. It i s then possible to estimate probability of the occurrence of any a i r mass given a part i c u l a r temperature. Since the conditional probability of this temperature given wind, prec i p i t a t i o n and cloud conditions may also be estimated, i t i s possible to derive an e s t i -mate of the probability of occurrence of a particular a i r mass, given the prevailing meteorological conditions. Making certain assumptions of independence and constancy, i t i s possible to apply these results in order to obtain estimates of a i r mass frequency for certain c r i t i c a l locations during the nineteenth century. The especial virtue of the technique described i s that i t may be applied to records of uneven q u a l i t y , although, c l e a r l y , the less complete the record, the less precise the estimate of a i r mass frequency. I t has been shown that i f the data of this study are approached in terms of the a i r mass concept, they may be unified in some meaning-ful way by the application of the technique of p a r t i a l c o l l e c t i v e s . The intent of the early part of this chapter has been to show that modern objections to the use of a i r mass analysis are largely not meaningful, in the context of this study and i t s data sources. Bryson's analyses were discussed in order to demonstrate that the a i r mass i . e . , the conditional probability of temperature T, given the presence of a i r mass A (P(T|A)). ** a more detailed mathematical treatment i s given at the begin-ning of the next chapter. n can co-exist with contemporary emphasis on the three dimensional struc-ture of the atmosphere in general, and the westerly vortex i n part i c u l a r . To this moment a rather negative case has been made for not rejecting the use of a i r mass analysis. A more positive approach w i l l follow, since discussion of the nature of recent climatic change leads to the conclusion that analysis of a i r mass frequencies should provide a pa r t i c u l a r l y sensitive index of the state of the general c i r c u l a t i o n , so long as the analysis i s based on a careful experimental design. I t i s convenient to introduce this discussion with the following b r i e f quotation from Sawyer (1966, p. 226) The causes of such persistent anomalies in the c i r c u l a t i o n are not known, but the fact that i t i s quite common for the large-scale c i r c u l a t i o n to revert to a particular abnormal form several times i n one season (despite intervening more normal periods) does suggest that the more spectacular and persistent anomalies of the seasons are more than accidental vagaries of a rather unstable sys tern. An important implication of this statement i s that the atmospheric system contains within i t s u f f i c i e n t v a r i a b i l i t y to take on, at least for a short time, c i r c u l a t i o n patterns which seem to have predominated during periods of "climatic change". A further implication i s that the causes of such s h i f t s in the c i r c u l a t i o n patterns are unknown. A br i e f scanning of the nine papers contained i n "Theories of changes of climate" (Part i i i , UNESCO 1963, pp. 277-380) i s evidence enough of t h i s . For the purposes of this study, such questions are regarded as metaphysical; discussion i s r e s t r i c t e d to the apparent form of the recent cli m a t i c v a r i a t i o n . The f i r s t of these two implications i s the focus of the early discussion since i t leads to a c r i t i c a l and 12 open-minded consideration of the form and structure of climatic change. Thus prepared i t i s possible to embark upon a b r i e f survey of what i s known of climatic fluctuation during the nineteenth and twentieth cen-turies and to use the benefits of what has largely been a European experience to construct an adequate and appropriate experimental design for the present study. One of the most refreshing developments in the study of climatic change is due to Curry (1962). He i l l u s t r a t e s that large fluctuations from the mean are not at a l l unlikely in series of random variables having a number of simple probability distributions.. He therefore con-cludes that climatic series might encompass si m i l a r large fluctuations about the mean, without any profound change in the controlling para-meters. His models serve only as very crude approximations to climatological r e a l i t i e s , but the approach i s an interesting one, insofar as i t i s a contrast to the conventional deterministic approach of the meteorologist. However, Sawyer's words, quoted above, warn against ascribing climatic fluctuation to "accidental vagaries", stressing that the atmosphere appears to alternate between a number of d i s t i n c t states, rather than the random fluctuation envisaged by Curry. Nevertheless, both authors seek to comprehend climatic fluctuation in terms of the action of contemporary processes, Curry favouring the suggestion that observed secular variation may l i e within the expected v a r i a b i l i t y of the process producing the climatic s e r i e s , Sawyer preferring to analyse the probable response of the atmosphere to changes in external energy supply. 13 As a res u l t of this analysis, Sawyer stresses that a climatic variation should not be expected to take the form of north-south s h i f t s of the f a m i l i a r c l i m a t i c b e l t s , but rather alterations in the intensity of east-west pre c i p i t a t i o n and temperature gradients, in response to changes i n the strength of the atmospheric c i r c u l a t i o n . In support of this contention he shows that the poleward transport of heat by the general c i r c u l a t i o n varies i n the r a t i o 7:1 between winter and summer, yet the poleward s h i f t of the subtropical high i s only of the order of six degrees of l a t i t u d e . Dzeerdzeevskii (1963, p. 285) states that d a i l y synoptic analyses may be grouped into four basic c i r c u l a t i o n patterns for the period 1899-1954, tra n s i t i o n a l types being present only 2 per cent of the time. The four patterns are as follows 1. Well formed polar anticyclone; zonal c i r c u l a t i o n in high l a t i t u d e s ; two to three intrusions of southern cyclones into high l a t i t u d e s . Over the greater part of the hemis-phere the zonal transfers are preserved. 2. "Violation of zonality"--a single intrusion of a r c t i c a i r masses over a hemisphere; zonal flows are preserved in a l l other sectors. 3. Two to four simultaneous intrusions over a hemisphere; this i s the group of meridional c i r c u l a t i o n types. 4. Development of cyclonic a c t i v i t y in high latitudes and over the A r c t i c Ocean. Inflows of "southern cyclones" reach far to the north and often cross the north pole region. (Dzeerdzeevskii, 1963, p. 225) Other workers, notably W i l l e t t (1950) and Butzer (1957), have prefer-red to talk in terms of the so-called "index-cycle"; e s s e n t i a l l y considering fluctuation between Dzeerdzeevskii's state 1, strong zonal flow, and Dzeerdzeevskii's state 3, disturbed zonal flow. A b r i e f 14 review of the use of zonal index models i s contained in Barry (1967, pp. 110-112) and w i l l not be repeated here, except to echo Barry's conclusion (p. 112) that"this concept, o r i g i n a l l y developed for synop-t i c purposes, appears better suited to a discussion of palaeoclimatol-ogy. I t i s generally held that strong zonal flow (high index) i s associated with milder weather i n mid-latitudes, while disturbed zonal flow (low index) i s associated with more extreme conditions, especially in continental i n t e r i o r s (Butzer, 1959; W i l l e t t , 1950). Figure 2 i l l u s t r a t e s Mitchell's (1963) determinations of world temperature trends since 1840. Note the general warming trend in the annual values and the pronounced warming trend in the winter values, although there i s some evidence of a decline since 1940. Mitchell (p. 163) states that, for the period of more r e l i a b l e data (1890-1960), the warming trend to 1940 and the subsequent cooling are both s t a t i s t i c a l l y s i g n i f i c a n t . I t i s necessary to r e c a l l Sawyer's warning that cl i m a t i c change is unlikely to occur as a uniform increase or decrease of temperature over the entire earth as might be implied by the previous discussion of Mitchell's r e s u l t s . Mitchell's maps of the twenty year change 1900-1919 to 1920-1939 indicate, for North America, some cooling within the zone shown by Bryson (see figure 1) to be dominated by A r c t i c a i r , and warming over the remainder of the continent. This would be consistent with a hypothesis of increasing zonal flow and hence an increasing incursion of P a c i f i c a i r into the continental i n t e r i o r . This hypothesis i s supported by the work of Lamb (Lamb and Johnson, 1959; Lamb, 1963; Lamb, Lewis and 15 annual winter - -0-6 1840 1960 FIGURE 2 TRENDS OF WORLD TEMPERATURE SHOWN FOR SUCCESSIVE PENTADS RELATIVE TO THE 1880-1884 PENTAD. SOLID CURVES • REPRESENT RELIABLE WEIGHTED AVERAGES, BROKEN CURVES ARE WILLETT'S 1950 DETERMINATIONS, BASED ON LESS EVIDENCE (Figure and text after M i t c h e l l , 1963) 16 Woodruffe, 1966). Lamb (1963, p. 128) reviews the evidence for what i s commonly called the L i t t l e Ice Age, which he dates at approximately A.D. 1430-1850. He ci t e s expansion of the A r c t i c ice pack, colder sea temper-atures in the North A t l a n t i c , g l a cier advance in Europe, Asia Minor and North America and many other features as evidence for the existence of this phase. A subsequent meteorological analysis of the available evidence (pp. 129-140) leads to the following conclusion: Increases of strength of the zonal c i r c u l a t i o n have been found in January and July over widely separated parts of the globe, apparently being quite general from around the middle of the l a s t century, and culminating around 1930 in the Northern Hemisphere and 1900-1910 i n the Southern Hemisphere . . . The increases from c i r c a 1800 to 1930 in the strength of the zonal c i r c u l a t i o n over the North A t l a n t i c in January amounts to between 5 and 10 per cent, but in general i s probably less than t h i s . Lamb suggests that this increased westerly flow was associated with a northward s h i f t of the wind belts in the northern hemisphere. The consequences of a westerly flow weak re l a t i v e to the present should include a decrease in the amount of a i r reaching l o c a l i t i e s to the east of the C o r d i l l e r a . Figure 3 shows Bryson's (1967) conjecture of the a i r mass pattern during times of low arid high zonal index. The low index pattern should bear some relationship to the patterns prevalent during the nineteenth century, although perhaps in a less extreme form. Subsequent analysis should therefore be designed so as to test for the existence of this kind of low-index pattern during the nineteenth century. The stations for which the analysis was performed were chosen for t h e i r location on or close to the winter position of the "Arctic FIGURE 3 SCHEMATIC DRAWING SHOWING THE EASTWARD PENETRATION OF PACIFIC AIR DURING TIMES OF HIGH AND LOW ZONAL INDEX (Figure after Bryson and Wendland, 1967) / 18 Front" between A r c t i c and P a c i f i c a i r (see figures 1 and 3). The stations chosen were Fort Simpson, Edmonton, Winnipeg and Fort William, each marked on figure 1.' The i n i t i a l intention was that the analysis should be performed for three stations only, Fort Simpson, Winnipeg and Fort William. The stations were chosen c a r e f u l l y so as to test the general hypothesis of a zonal c i r c u l a t i o n weak r e l a t i v e to some modern control period; each station having i t s own p a r t i c u l a r relevance to this hypothesis. Fort Simpson was included in order to investigate Bryson's (1966, p. 267) assertion that the A r c t i c Front i s . . . topographically anchored year round at the northern end of the C o r d i l l e r a near Aklavik, but swings north and south with the seasons in the continental i n t e r i o r where the terrain i s r e l a t i v e l y f l a t . Presumably this r e s u l t may be extended by analogy to periods of weakened or strengthened zonal c i r c u l a t i o n . A i r mass frequencies for Fort Simpson might thus be expected to be r e l a t i v e l y more stable than those at stations i n the continental i n t e r i o r . Winnipeg, lying (figure 1) on the northern edge of s i g n i f i c a n t flow from the P a c i f i c would appear to be a p a r t i c u l a r l y suitable station to use in the construction of some measure of the degree of penetration of the westerlies during the h i s t o r i c period. Unfortunate-l y , data for Winnipeg were r e l a t i v e l y scarce and of generally poor q u a l i t y , so data for Edmonton were added to the analysis as a further measure of the strength of the westerlies. Fort William was included as a further check on the depth of penetration of the P a c i f i c a i r and, assuming a decrease i n the 19 frequency of a i r from that o r i g i n , to determine the extent to which i t was replaced by a i r of A r c t i c or of United States, o r i g i n . The next chapter outlines the framework of the analysis devised for testing the general hypothesis of a weakened zonal c i r c u l a t i o n during the h i s t o r i c period. CHAPTER III THE MODEL The use of the term "model" implies only that the framework of the analysis will be exposed and that basic assumptions will be stated and discussed. Questions which may be more justly considered as computational are delayed until the final chapter. It is f i r s t of a l l necessary to present more precisely some of the material discussed in the previous chapter. The suggestion was made that the form of the histogram of temperature frequency for any one month at some location would represent contributions from two or more air mass source regions. Suppose there to be two significant sources, A and B, then freq(T) = A(T) + B(T) where: freq(T) = frequency of observations of temperature T A(T) = frequency of air mass A at temperature T B(T) = frequency of air mass B. Bryson (1966, p. 237) points out that, in an a i r mass source region, the histogram of temperature frequency closely approximates the form of the density of a normally distributed random variable, i.e. f(T) = (2nb3rh exp (-(T-b2)2/2b3) where: f(T) = probability density of temperature in the source region by = the mean of this density, i.e., the mean temperature in the source region b 0 = the variance of the distribution. 21 It therefore seems not unreasonable to postulate that the contribution from any source in another location should retain a density function of this kind, although with some change in parameter values. Confirm-ation of this postulate will be provided in the following chapter—an example of a curve from a source region is figure 4. This curve is from the sub-Arctic station Baker Lake for the month of February during the years 1950-1959. Bryson's streamline map for February (Bryson, 1966, p. 255) shows this station to be influenced primarily by a north westerly outflow from an anti-cyclone centered over the western Arctic, but also by a more southerly variety of Arctic a i r . The bimodal form of the curve is indicative of a situation of this kind. Given that such a histogram does represent a composite of several, say two, normal distributions, then i t is possible to write freq(T) = bftl exp [-[T-b A 2] 2/2b A 3]+b B 1 exp [-[T-bB2]2/2b where: freq(T) = as before b»i = the frequency of the mean of the distribution of observations from air mass A b«2 = the mean temperature of air mass A, at this  location b» 3 = the variance of temperature for air mass A at this location. Generalisations to situations with three or more source regions are obvious. Given equation 1, adopting a "relative frequency" definition of * probability and making one rather gross assumption, i t is possible to The terminology and elementary manipulations of probability 22 f r e q u e n c y " 4 2 -16 t e m p e r a t u r e ° F FIGURE 4 TEMPERATURE-FREQUENCY HISTOGRAM FOR BAKER LAKE, FEBRUARY 1950-1959. SOLID LINE IS THE FITTED CURVE 23 write the probability of air from source A on any particular day, given that the temperature is T P(A|T) VA(T)/freq(T) where: A(T) and freq(T) are defined as before P(A|T) = the conditional probability of air from source A given that the temperature is T. The "gross assumption" noted is that the day to day probabilities are independent of one another, that the probability does not vary from day to day according to the temperature and air mass source recorded for the previous day, or for the previous few days. In view of the very marked persistence effects evident in climatic series i t is necessary to undertake some discussion of this assumption. Firs t recall the definition of conditional probability P'(A|T) = P(AnT)/P(T) where: n = the symbol for set intersection P(AnT) = the probability that the temperature will be T and that the a i r mass will be A P(T) = the probability that the temperature will be T. i.e., the conditional probability is the quotient of the probability that the a i r will both have temperature T and come from source A, and the probability that the a i r will have temperature T. Now i t is clear that both of these probabilities will vary from day to day, that the probability of temperature T will increase i f the temperature of the used here may be found in any introductory text; see for example Feller, W., An Introduction to Probability Theory and its Applications, Wiley, New York, 1957. 24 previous day was close to T and that the probability that the air mass source will be A will increase i f A was the source on the previous day; indeed this is one of the principal justifications of the air mass concept. But i t is also clear that these probabilities will tend-to vary together, thus minimising the effect of their variability on their quotient. A numerical example may help to c l a r i f y this statement. Suppose that the a i r can take on one of two temperatures, T and T , and that the following holds P(T|T) = the probability of temperature T i f the previous day's temperature was T = 0.8 P(T|TQ) =0.2. Now hypothesise that P(A|T) is a constant (that is the assumption stated above), and that this constant is 0.5. Then, from the definition of conditional probability, P(AnT) must equal 0.4 i f the previous day's temperature was T, and 0.2 i f the previous day's temperature was T Q, i.e., as the probability of T increases so does that of A"T. It is intuitively obvious that a relationship of this kind exists outside the scope of this simplified example, and i t seems not unreasonable to propose that the conditional probability of an a i r mass, given the temperature, is sufficiently stable to permit meaningful analysis. A second plea for this assumption is a pragmatic one, stated here by Arbib (1964, Preface, no page number) We apply (original i t a l i c s ) mathematics to derive far-reaching (sic; conclusions from clearly stated premises. We can test the adequacy of a model of the brain (or climate) by expressing i t in mathematical form and using mathematical tools to prove general theorems. In the light of any discrepancies we find between 25 these theorems and experience we may return to our premises and reformulate them, thus gaining a deeper understanding of the workings of the brain. To this point, an" estimate has been made of the conditional probability P(A|T). Our present purpose is to relate this estimate to other meteorological phenomena, in order to f a c i l i t a t e estimates of air mass frequencies without consistent records of temperature. The procedure is most easily demonstrated by the following much simplified example. Suppose we have the following distribution of temperature (in 5° classes) T P(T) T P(A|T) P(B|T) 30 0.10 30 1.0 0.0 35 0.40 35 1.0 0.0 40 0.20 and 40 0.7 0.3 45 0.25 45 0.2 0.8 50 0.05 50 0.0 1.0 Clearly, air mass A, the colder of the two, is the predominant contrib-utor to this distribution. Let us further suppose that records of the frequency of winds from the various quarters have been kept. By associating these with records of temperature, the following values for the probability of temperature T on a day with a north wind have been derived T P(T|N) 30 0.15 35 0.60 40 0.15 45 0.05 50 0.05 26 It is clear that the north wind is strongly associated with air mass A--to make this notion more precise note that P(A|T|N) " = P(A]T).P(T|N) where: P(A|T|N) = the probability of air from A given temper-ature T, with the probability of T conditioned by the occurrence of the north wind. For the example T P(A|T|N). P(B|T|N) 30 0.150 . 0.000 35 0.600 0.000 40 0.105 0.045 45 0.010 . 0.040 50 0.000 0.050 The final step is to calculate the probability of a ir from A, given a north wind, P(A|N), which is given by P(A|N) = I P(A|T|N) =0.865 and similarly P(B|N) =0.135 Once again, generalisations to cases with more than two air masses and where meteorological elements other than wind direction are measured are obvious. The records of the Hudson's Bay Company include a number of post journals which have f a i r l y continuous series of climatic observations for the f i r s t half of the nineteenth century. Sometimes these are measures of temperature, but more often they are of wind direction, amount of cloud cover, the occurrence and. type of precipitation and other similar quantities. It is suggested that the application of 27 similar procedures to those described above should lead to a useful assessment of the relative frequencies of the various air masses at these posts. But this application requires one further assumption; that the conditional probability of any air mass source", given any particular wind, precipitation or cloud conditions remains relatively constant in the long term. Under this assumption, and those discussed above, and assuming that the frequency distribution can be successfully decomposed into the contributions of the various source regions, an estimate of air mass frequency for the historic period can be made. The problem of f i t t i n g the appropriate curve and of the form the e s t i -mate should take are discussed in the following chapter; this chapter concludes with consideration of the implications of the assumptions of constancy. Consider the example of Arctic air and the north wind. An increase in the frequency of Arctic air would presumably represent an increase in the frequency of the north wind—there appears to be no reason to suppose that the degree of association between the north wind and Arctic a i r should alter radically without some sort of catastrophic alteration in the major determinants of the atmospheric circulation, the disposition of incoming solar radiation, the orbit of the earth or the distribution of the continents. The argument previously applied for the short term may be applied again; referring once more to the definition of conditional probability, P(A|N) = P(AnN)/P(N) where a l l quantities are defined as above. P(AnN) and P(N) should 28 vary together in time, hence their quotient should remain relatively stable. A pragmatic argument was offered as a justification for the assumption in the short term case; this assumption may be regarded as a reflection of the wish of meteorologists to comprehend recent climatic fluctuation in terms of the action of contemporary processes. Recall the discussion of the work of Sawyer and Curry in the previous chapter, both of whom stress that we should seek the mechanisms of recent clim-atic change within observed contemporary pattern and process. In addition, i t is clear that i f the assumption is not valid, then the error will be such as to bias the values of air mass frequency calculated for the historic period towards those of the present. Therefore any change detected may be regarded as doubly significant in view of the probable "inertia" or conservatism in the estimation process. If the records were of an even or homogeneous character, the analysis might be performed by the use of more conventional techniques —calculation of mean temperatures, construction of wind roses, determination of precipitation amounts, etc., but the technique out-lined here may be applied to records of quite uneven qua!ity--although the less adequate the record, obviously the less precise the estimate of a i r mass frequency. With limited data i t is only to be expected that the analysis will be subject to limiting assumptions or conditions such as those stated above. CHAPTER IV THE ANALYSIS The analysis follows the order in which the model was developed in the previous chapter. 1) Contemporary data a) f i t t i n g the composite curves b) establishment of the relevant condi-tional probabilities 2) Historic data - determination of probable air-mass frequencies. The organisation of both sections will be the same—firstly a discus-sion of the data used, their relevance to the problem and their source, and secondly an outline of the techniques and f a c i l i t i e s used in computation, and f i n a l l y presentation and discussion of the results obtained. The chapter will conclude with a summary of the results and a discussion of their significance to the hypothesised model of climatic change. It will be recalled from the second chapter that four stations were used in this analysis, Fort Simpson, Edmonton, Winnipeg and Fort William and that winter months were to be the focus of the analysis. The months of January, February, March and April were considered, since the Arctic Front is located along the line of the four stations during this period, and in order to investigate any delay in the development of the spring circulation pattern. It was intended i n i t i a l l y that analysis of contemporary conditions should be 30 performed on data for the period 1950-1959 but d i f f i c u l t i e s in obtaining some of the relevant information caused this period to be reduced to . the pentad 1955-1959, inclusive. A problem which arose, probably as a result of this reduction, will be discussed later in this chapter. The analysis requires series of daily temperature for the fi t t i n g of the composite curves and series of cloud, precipitation and wind observations for the establishment of the relevant conditional probabilities. Table 1 (page 31) l i s t s the sources used. The cloud, precipitation and wind series, Table I, were modified so as to make them as compatible as possible with the historic records. Maximum temperature was chosen following Bryson (1966, p. 235) since he obtained significant results using this quantity and because he suggests that maximum temperature is an especially characteristic quality of any particular type of a i r . Bryson (1966, p. 236) outlines a technique for the f i t t i n g of a . composite curve consisting of the sum of several normal distributions to a histogram of temperature frequency. This graphical approach seemed rather unsatisfactory, leading Bryson to investigate the develop-ment of a least squares f i t t i n g routine. Since, given the equation of the distribution, estimates of the parameters may be made by conventional curve f i t t i n g procedures this approach has been followed here. Computation was performed in the Computing Centre of the University of British Columbia, using that centre's IBM 7044 f a c i l i t y and i t s successor, an IBM 360/67. The program used in the f i t t i n g of 31 TABLE I SOURCES OF MODERN DATA Element Characteristic Units Source Temperature Precipitation Wind Cloud Daily maximum Occurrence & type Direction Amount °F 1/2 Bearing 1/2/3 Monthly Record Monthly Record Table 14 Table 14 Sources: Monthly Record, published monthly by the Meteorological Branch, Department of Transport, Canada. Table 14, Weather Summary, Synoptic Hours also available from the Department of Transport. Units: Precipitation - trace or none = 1; measurable amount =2. Cloud - 0,1,2 tenths = 1; 3,4,5,6,7 tenths = 2; 8,9,10 tenths =3. 32 the curves was obtained through the SHARE program 1ibrary—routine SPA 3094; Least squares estimation of non-1inear parameters, fami1iarly known as NLIN2. Input for NLIN2 consisted of temperature frequency histograms, calculated separately for the five year period for each month at each station. Bryson (1966, p. 235) used a 2°F class interval in the calculation of similar histograms; experiments by the writer confirmed that, over the ten year period, this class interval gave relatively smooth curves and clearly differentiated peaks. However, for the five year period used in this study, a 3°F class interval yielded more regular curves. The equation for distributions comprising the composite of several normal distributions was given in the previous chapter. This equation includes the sums of several exponentials; such functions * are notoriously d i f f i c u l t to f i t , therefore, subsequent to the calcu-lation of the histogram, the frequencies were subjected to a "center-weighted" f i l t e r of the form y\ = (yj.-, + 2 Y i +y 1 + 1)/4 where: ' y J = the adjusted ith frequency ^ i - l = ^ n e observed i - l t h frequency y. = the observed ith frequency y.+1 = the observed i+lth frequency The purpose of this f i l t e r is to minimize the effect of random error The system library program for least-squares curve f i t t i n g at the University of B.C. was unable to f i t this function. 33 on the form of the histogram and thus to f a c i l i t a t e the f i t t i n g of the curves. In effect, the f i l t e r "smooths" the curves and removes a l l but the major peaks. Justification for the use of the f i l t e r is twofold; f i r s t l y , and primarily, Bryson obtained significant and consistent results through its use and secondly i t would be almost impossible to f i t the curves without prior smoothing, in view of the nature of the function involved. It should also be remembered that one application of the least-squares procedure is the smoothing of curves (McCalla, 1967, chapter 7), therefore i t can be argued with some conviction that a least-squares solution of the best f i t t i n g curve for the observed histogram should not dif f e r considerably from the least-squares solution for the filt e r e d histogram. The algorithm for NLIN2 is presented by Marquardt, 1963; a readable account of the differential correlation technique on which i t is based is given in McCalla (1967, pp. 255-261). No discussion of this technique will be made here. Suffice i t to point out that input for the routine consists of the temperature frequency histogram and i n i t i a l , f a i r l y accurate, estimates of the values of the parameters of the individual normal distributions. The program contains a plotting option which makes i t particularly suitable for terminal operation in those cases where i t was d i f f i c u l t to establish a suitable f i t . The program converged to a solution on only about 50 per cent of the occasions on which i t was used, since poor estimates of the parameters will lead to a rapid propagation of error through the program. 34 But before NLIN2 is used i t is necessary to decide on the * number of normal distributions represented in the histogram. To this end, the frequency of winds from each direction for each temperature class was calculated. Comparison of these results with the histogram of temperature frequencies usually reveals that a certain range of temperatures is associated with a certain range of wind directions. Two or three groups of predominant wind directions are usually apparent. With justification these groups may be regarded as representing separate a i r masses and the curves fi t t e d accordingly. Inter-month consistency is probably the most important criterion of the validity of this analysis. From the analysis of chapter II, i t would appear that the curves for each station should reflect the presence of Pacific, Arctic and United States a i r . Consider f i r s t the curves for Edmonton. For each month, three components have been f i t t e d ; from their temperature and wind associations these have been identified as Arctic, United States and Pacific, as indicated in table II (page 35). The interconsistencies in the monthly results are an encouraging index of the appropriateness of the analysis. Table III (page 36) presents the temperatures and source regions for Winnipeg. Once again westerly, southerly and northerly components indicate the presence of air from each of the three source regions. Two differences are to be seen between these figures and those for Edmonton; f i r s t l y , whereas the Pacific a i r was the warmest at Edmonton, for Winnipeg the US air becomes the warmest. Two factors probably 35 TABLE II AIR MASS TEMPERATURES AND SOURCES FOR EDMONTON JANUARY FEBRUARY #• Mean T Winds Source # Mean T Winds Source 1 -1 N Arcti c 1 10 N Arcti c 2 13 S US 2 28 S US 3 34 w Pac 3 41 w Pac MARCH APRIL # Mean T Winds Source # Mean T Winds Source 1 16 N Arcti c 1 30 N Arcti c 2 29 S US 2 43 S US 3 42 w Pac 3 57 w Pac (Mean T is mean maximum daily temperature in Fahrenheit degrees) 36 TABLE III AIR MASS TEMPERATURES AND SOURCES FOR WINNIPEG JANUARY FEBRUARY # Mean T Winds Source # Mean T Winds Source 1 2 N&W Arc/Pa 1 1 N Arcti c 2 22 S US 2 17 W Pac 3 32 S US MARCH -- APRIL # Mean T Winds Source # Mean T Winds Source 1 18 N Arctic 1 45 N Arctic 2 32 W Pac 2 64 W Pac 3 45 S US 3 78 S US 37 account for this change, the prolonged travel and modification of the Pacific a i r on its track across the Prairies and the inhomegeneity of the United States a i r ; Winnipeg experiencing a mixture of Gulf and interior U.S. a i r , whereas Edmonton would probably be restricted to the interior variety. The second change is that for January, i t was impossible to distinguish between Arctic and Pacific a i r . This is an indication of the cause of the tendency to attribute large frequencies to the classification "continental Polar". Data for a longer period and the use of a smaller class interval in the calculation of the histogram would probably make i t possible to separate the two kinds of air . As table IV (page 38) shows, the histograms for Fort William appear to comprise two components, one associated with westerly, north-westerly and northerly winds, the other with south-westerly, southerly and south-easterly winds. The wide range of wind-directions associated with the former implies that i t represents the sum of Pacific and Arctic flows, whereas the latter appears to represent primarily United States (Gulf) a i r . However, a marked decrease in the frequency of south-west winds in A p r i l , a month in which Bryson (1966, p. 258) notes a very considerable weakening in the eastward penetration of the westerlies, leads to the suspicion that the Pacific a i r is also represented in the warmer of the two distributions. Indeed, Bryson's maps of the circulation in the f i r s t four months of the year indicate that both Arctic and Pacific a i r reach this location in a predominantly westerly flow, Arctic mostly from the north-west and Pacific mostly from 38 TABLE IV AIR MASS TEMPERATURES AND SOURCES FOR FORT WILLIAM JANUARY FEBRUARY # Mean T Winds Source # Mean T Wi nds Source 1 8 NW Arc/Pa - 1 15 NW Arc/Pa 2 26 S US 2 29 S US MARCH APRIL # Mean T Winds Source # Mean T Wi nds Source 1 24 NW Arc/Pa 1 46 NW Arc/Pa 2 34 S US 2 67 S US 39 the south-west. Therefore, the results from Fort William must be treated with caution. The predominant winds at Fort Simpson are from the north-west and south-east; the fact that there is no difference in temperature associations between them implies some sort of topographic control of the flow of the wind. Confirmation is provided by the location (see figure 5) on the Mackenzie River between the Horn Mountains to the north-east and the Mackenzie Mountains and the Cameron Hi l l s to the south-west. At this point the Mackenzie flows from south-east to north-east, in the direction of the predominant winds. It appears reasonable to suppose that the predominant air mass, associated with the south-east and north-west winds is an Arctic type entering the valley either to the south or to the north of the Horn Mountains (see figure 5). This air is designated "Mid Arctic" in table V (page 41), to distinguish i t from a colder variety which Bryson (1966, p. 254) shows as developing over the north of the Dist r i c t of Mackenzie. This a i r is here designated "Full Arctic" and is apparent in the tabulations for January and March, primarily in association with calms and northerly winds. Fort Simpson is situated at the confluence of the Liard and Mackenzie rivers, the Liard flowing into the Mackenzie from the south-west. Bryson (1966, p. 238) distinguishes a type of air which he designates "Yukon Pacific", crossing the Cordillera through the gap made by the valley of the Liard River. It is this kind of air which must be represented by the warmest component present in the histograms of temperature frequency, associated with westerly, south-westerly FIGURE 5 SKETCH OF THE LOCATION OF FORT SIMPSON AND POSSIBLE AIR MASS SOURCES FA = f u l l A r c t i c MA = mid A r c t i c P = P a c i f i c Stippled areas are land above 1000 feet 41 TABLE V AIR MASS TEMPERATURES AND SOURCES FOR FORT SIMPSON JANUARY FEBRUARY # Mean T Winds Source # Mean T Winds Source 1 -24 N&C F. Ar 1 -3 NW&SE Arcti c 2 -5 NW&SE M. Ar 2 19 W Pac 3 10 W Pac MARCH APRIL # Mean T Winds Source # Mean T Winds Source 1 0 N&C F. Ar 1 35 NW&SE Arctic 2 22 NW&SE M. Ar . 2 44 W Pac 3 40 W Pac (F. Ar = f u l l Arctic, M. Ar = mid Arctic) 42 and north-westerly winds. In summary, the curves fitted for Edmonton, Winnipeg and Fort William accord moderately well with the Arctic/Pacific/United States model proposed in the second chapter, and are therefore appropriate to this analysis of changes in the relative frequencies of these components. The curve for Fort Simpson can be given a r e a l i s t i c physical interpretation, but topographic control of the airflow . hinders its use in the subsequent analysis. Nevertheless, a l l stations were included in the tests for climatic change to be described in the following paragraphs. The next stage of the analysis is the establishment of the conditional probabilities required to implement the model developed in previous chapter. The following table l i s t s these probabilities and their estimators. PROBABILITY ESTIMATOR P(A|T) fA(T)/f(T) P(T|W) fW(T)/f(W) P(A|W) E P(A|T).P(T|W) where: P(A|T) = the probability of air mass A given temperature T P(TjW)•= the probability of temperature T given wind W P(A|W) = the probability of air mass A given wind W 43 fA(T) the frequency of air mass A at temperature T f(T) the frequency of occurrence of temper-ature T. fW(T) the frequency of wind W at temperature T f(W) = the frequency of wind W. These probabilities are listed in Appendix A for each of the four stations and for each of the four months at each of the stations. Under the assumptions discussed in the previous chapter, these estimated probabilities may be used to produce an estimate of air mass frequencies during the historic period. The writer spent the major portion of the summer of 1968 examining the records of the Hudson's Bay Company in the Public Archives of Canada, Ottawa. In particular, the post journals of Fort Simpson, Edmonton, Winnipeg and Fort William were inspected to deter-mine the availability of regular series of climatic observations. The records are generally f a i r l y continuous after the union of the Hudson's Bay Company and the North-West Company in 1821, so attention was concentrated on the period 1820-1851. Table VI (page 44) l i s t s the ava i l a b i l i t y of daily records of climate for each five year period between 1820 and 1850. The table shows clearly that Fort Simpson has the most complete record, and that the records of Edmonton and Winnipeg span the earlier part of the period and those of Fort William the latter part. The quality of the records is most uneven; at times there being a complete meteorological journal (e.g., Fort Simpson 1839), but more 44 TABLE VI THE AVAILABILITY OF DAILY RECORDS OF CLIMATE PERIOD Fort Simpson Edmonton Winnipeg Fort William 1820-1824 No Yes Yes No 1825-1829 Yes Yes Yes No 1830-1834 Yes Yes No Yes 1835-1839 Yes No No Yes 1840-1845 Yes No No No 1849-1851 No No No Yes 45 often a simple comment on the state of the wind and weather. There may be rich material in these records for those whose interests l i e in the standardization and homogenization of historic temperature series, but for this study the quantitative records of temperature are too isolated, the units of measurement too uncertain and the exposure of the instrument too obviously faulty to warrant their inclusion. Despite the uneven quality of the records i t was found that the daily observations could be categorised into one or more of the following classes. TEMPERATURE PRECIPITATION CLOUD WIND Cold Fine Clear Direction Mild Rain Half Hot Snow Overcast These classes correspond to those qualities for which the conditional probability of the occurrence of the various air masses was calculated. Therefore, bearing in mind the discussion of chapter III, for each ~day of the historic period with a meterorological observation i t is possible to estimate the probability of occurrence of the various air masses. Individually these estimates can provide l i t t l e information as to the climate of the nineteenth century, and consequently i t is necessary to aggregate them in some meaningful way. By assumption, and of necessity, these estimates are considered to be independent of one / 46 another, i.e., their joint probability is given by the product of the individual probabilities. It is therefore theoretically possible to determine precisely the probability distribution of the proportion of days dominated by each air mass. But such a determination would require an astronomical amount of calculation, so a more convenient approximate solution must be sought. A method of determining confidence limits for estimates of the relative proportions of the various air masses in the historic period was found by application of the Central Limit Theorem. Since the mathematical details of this discussion are at a considerably more advanced level than those in other parts of this thesis, the discussion has been relegated to appendix C. The result of this discussion is that the following expression gives an approximate confidence limit for the relative frequency of an air mass during a period of length n days P(p-z , 2 (p(l-p)) % < Sn/n < p + z , 2 (p(l-p))^) = a ' n ' n where: p = (p-| + p2 + ... + P n)/ n p. = the probability of air mass A on the ith day z / 0 = the standard normal variate at a/2 a/2 Sn = the number of days on which A occurs. Setting a = 0.05, i.e., constructing 95 per cent confidence limits, P(p-1.96 (pd-p))^ < Sn/n < p+1.96 (p(l-p))^) = 0.05 n n Significance of a change from the 1955-59 control period was measured as the probability of obtaining a value for the relative frequency of 47 air mass A as extreme as the present, using the confidence limits described above. Note that the width of the confidence band depends primarily on the size of n, the length of the record. Before 1955-1959 can be used as a control period i t is necessary to establish the nature of the climate during that time. In general,, averages of temperature are slightly below the long-term average temperatures published by the Department of Transport for January, February, and March and slightly above for A p r i l . Large positive deviations from the normals are evident in January of 1958, apparently associated with a strong eastward penetration of Pacific a i r . In contrast, large negative deviations from the normals are recorded in January of 1959, especially at Fort Simpson and Edmonton, probably representing an increased presence of Arctic a i r , cutting off the westerly flow to Edmonton. But the temperatures are sufficiently close to the normals to permit valid comparison with the nineteenth century data, and since such deviations as exist are generally towards a colder regime, such as that hypothesised for the early nineteenth century, indications of a climate significantly colder than that of the control period would be doubly significant. The results of the analysis are presented in figures 6 to 9 and are tabulated in appendix B. Five per cent confidence bands are included on each diagram. The results of the analysis for Fort Simpson are shown diagram-matically in figure 6. The most prominent feature of this diagram is the increased frequency of Arctic a i r in comparison with the 1955-1959 control period. With the exception of February 1824-1829 and April F A 48 M A J—-L " I a " i b " " I c I S T " ! J A N FA = f u l l A r c t i c MA - mid A r c t i c A = ( f u l l & mid) A r c t i c P._=_Pjicific F A i o t c i < n F E B a b c 1824-1829 1830-1834 1835-1839 T 1 ' 1 M A I b I c I d I M A R FIGURE 6 FORT SIMPSON: DEVIATIONS FROM 1955-59 AIR MASS FREQUENCIES (35) J A N A = Ar c t i c US = United States P = P a c i f i c F E B 1820-1824 1825-1829 1830-1834 -5-U S M A R A P R FIGURE 7 EDMONTON: DEVIATIONS FROM 1955-59 AIR MASS FREQUENCIES (35) J A N A = A r c t i c F E B a = 1820-1824 US = United States b = 1825-1829 P = P a c i f i c M A R A P R FIGURE 8 WINNIPEG: DEVIATIONS FROM 1955-59 AIR MASS FREQUENCIES (%) 51 J -1=3 I 1 A P U S f^TTT J A N f " a i — s i r AP = Ar c t i c & P a c i f i c US = United States F E B a = 1825-1829 b = 1830-1834 c = 1849-1851 A P U S A P U S A P R FIGURE 9 FORT WILLIAM: DEVIATIONS FROM 1955-59 AIR MASS FREQUENCIES (%) 52 1830-1834, the frequency of this air (the sum of "Mid" and " F u l l " Arctic) appears to have increased markedly, at the expense of the warmer Pacific a i r . Similar results were obtained for Edmonton (see figure 7). Arctic a i r appears to have been of greater frequency during 1820-1834 than during the control period, largely at the expense of air of Pacific origin, at least during the months of January, February and March. In contrast, the frequency of southern, United States, air appears to have been greater during A p r i l . At Winnipeg too (see figure 8), Arctic a i r appears to have been more prominent than during the control period, especially during the f i r s t three months of the year. However, unlike Edmonton, this increase seems to have been largely at the expense of United States a i r . There is no evidence of any marked decline in the frequency of air from the west. The results for Fort William were disappointing. There is no obvious consistency or pattern in the fluctuations indicated, a surprising result in view of the relative uniformity of the results for the other stations. Since the record for Fort William is a very f u l l one, at least during the decade 1830-1839, i t was possible to investigate the reason for these surprising results. Figure 10 shows wind roses for the control period and the two pentads 1830-1834 and 1835-1839. These wind roses indicate a quite remarkable change in wind regime; during the control period south-westerly and westerly winds FIGURE 10 WIND ROSES FOR FORT WILLIAM 54 predominated in each of the four months, but during both the historic pentads winds are primarily from the north-east, north and north-west. Presumably this reflects'an increase in air of Arctic origin. But why was this change not detected by the regular analysis? The answer to this question must l i e in the comparative rarity of north and north-east winds during the control period, leading to very poor estimates of the relationship between Arctic air and northerly winds. Furthermore, i t will be recalled that i t was not possible to distinguish between the Pacific and Arctic airstreams in the decomposition of the temperature frequency histogram. Similar problems did not arise at other stations since no equivalent shifts in the dominant wind direction took place. There is considerable evidence of an increase in the relative frequency of Arctic a i r at a l l stations in the period 1820-1850. This is consistent with the discussion of chapter II where i t was suggested that the nineteenth century was a period during which the zonal atmospheric circulation was markedly weaker than that of the present time. A consequence would be a southward extension of Arctic a i r and a decreased penetration of Pacific air into the Prairies. The analyses for Fort Simpson, Edmonton and Fort William are entirely in accord with this hypothesis, but there is no evidence of any decline in the frequency of westerly wind at Winnipeg. First of a l l i t was necessary to ascertain whether this result was not due to any inappropriateness in the analysis. Inspection of that part of the record with an adequate number of wind observations 55 showed no apparent decline in the frequency of winds from the west and south-west and, unlike Fort William, the relationships between the dominant wind directions and air mass type appear to have been well established. Therefore i t is not possible to conclude that there was any marked decrease in the frequency of Pacific a i r at Winnipeg during the historic period. The model of climatic change must take account of this. To this end, note Lamb's (1963, p. 134) suggestion that there was a southward displacement of the major wind systems in the northern hemisphere during the early nineteenth century, in association with a decline in the strength of the westerly vortex. Bryson's (1966, p. 249) map of air mass dominances shows that the Pacific a i r present at Edmonton arrives by means of a relatively northern track through the Cordillera. A southward shift of the westerlies of only a few degrees would lead to a very marked decline in the frequency of Pacific air at this location. Similar reasoning may be applied to the decline in the frequency of Pacific a i r at Fort Simpson. Winnipeg receives Pacific a i r from both the northerly track and from the main core of the westerlies, and would therefore not respond to the same degree to a southward shift of the wind belts. An example of a situation of this kind arose in January 1959, with average temperatures at Fort Simpson and Edmonton more than 10°F below the long term normals, whereas the Winnipeg mean was only some 5°F below the normal. It was pointed out in chapter III that the process for estimating the historic a i r mass frequencies probably possesses a strong 56 bias towards the control period—what might be called an "inertia". In view of this factor we must conclude that the decline in the strength of the westerlies at this station has not been proven, rather than that i t did not take place. The data for the other stations supports the hypothesis of a weakened westerly vortex; i t is particularly unfortunate that the data for Winnipeg were the least f u l l of any of the stations, since that station had been intended to provide the prime index of the state of the westerlies. The results were calculated separately for each pentad in order to demonstrate that the changed climatic regime is characteristic of the entire period, rather than any isolated group of years within that period. Similarly, the regularity of the results.obtained for each of the four months are further evidence of a uniformly different climatic regime during the historic period. It will be recalled that a general hypothesis of a weakened westerly vortex was presented in chapter II; we have seen above that the evidence for Winnipeg precludes an unqualified acceptance of this hypothesis, although the results for the other stations are as expected. It was also suggested that the frequencies recorded for Fort Simpson might be relatively more stable than those for the other stations in view of the topographic anchoring of the Arctic Front in this area. This has been shown not to be the case, although possibly as a result of specific-local conditions, notably the presence of the Liard gap. The above are such as conclusions as may be drawn with respect to the existence of a secular climatic fluctuation in Canada during the 57 early part of the last century; what conclusions may be drawn as to the practability of the technique employed? F i r s t l y , i t should be noted that the five year period used for the calculation of the modern results was probably too short for the accurate establishment of some of the relevant conditional probabilities. The problems encountered in the analysis for Fort William is a case in point. A longer period might permit a more precise separation of air masses which share relatively similar air mass characteristics. A considerable amount of time is necessary to establish the computational procedures for the analysis, although once these have been developed the analysis proceeds relatively smoothly. It would probably not be worthwhile to use the analytic tools of this study unless the data was of a similar kind and quality. Wherever regular observations are available, simple procedures such as averaging and graphing would probably yield at least as much information to the skilled interpreter. However, such records were not available for this study; the data could only be analysed by some specialist technique such as that developed here. In view of the lack of other records for the same period and the lack of previous Canadian studies of this period, i t is suggested that the results obtained here are interesting and are not without climatological significance. BIBLIOGRAPHY BIBLIOGRAPHY Arbib, M.A. (1964) Brains, Machines and Mathematics. New York: . McGraw H i l l . ; Barry, R.G. (1967) Models i n Meteorology and Climatology i n Chorley, R.J. and Haggett, P. [ed.]. Physical and Information Models in Geography. London: Methuen, pp. 97-135. Bryson, R.A. (1966) Airmasses, Streamlines and the Boreal Forest Geographical B u l l e t i n , 8 (3), pp. 228-269. and Wendland, W.M. (1967) Tentative Climatic Patterns for some Late Glacial and Post Glacial Episodes in Central North America, Technical Report #34. Madison: Department of Meteorology. Brunnschweiler, K. (1952) The Geographic Distribution p.f A i r Masses i n North America. Vierteljahrschr. Naturforsch. Ges. Zurich, 97, pp. 42-49. Butzer, K.W. (1957) The Recent Climatic Fluctuation i n Lower Latitudes and the General Circulation of the Pleistocene. Geografiska  Annaler, 39, pp. 105-113. Curry, L. (1962) Climatic Change as a Random Series. Annals of the  Association of American Geographers, 52, pp. 21-31. Dzeerdzeevskii, B.L. Fluctuations of the General Circulation of the Atmosphere and Climate in the Twentieth Century in Unesco (1963) pp. 285-296. Hare, F.K. (1960) The Westerlies. Geographical Review, 50 (3), i pp. 345-367. Lamb, H.H. (1963) On the Nature of Certain Climatic Epochs which Differed from the Modern (1900-39) Normal, i n Unesco (1963), pp. 125-50. and Johnson, A.J. (1959) Climatic Variation and Observed Changes i n the General C i r c u l a t i o n . Geografiska Annaler, 41, pp. 94-133. .Lewis, R.P.W. and Woodruffe, A. (1966) Atmospheric Circ u l a -tion and the Main Climatic Variables Between 8000 and 0 B.C.; The Meteorological Evidence in Proceedings of the International  Symposium on World Climate 8000to 0 B.C. London: Royal Meteorolog-i c a l Society. M i t c h e l l , J.M. (1963) On the World Wide Pattern of Temperature Change i n Unesco (1963), pp.' 161-182. 60 Marquardt, D.W. (1963) An Algorithm for Least Squares Estimation of Non-Li near Parameters. Journal of the Society for Industrial and  Applied Mathematics 11 (2), pp. 431-441. McCalla, G. (1967) Introduction to Numerical Methods and FORTRAN  Programming. New York: Wiley. Minns, R. (1968) A i r Masses and Permafrost Boundaries. Unpublished Seminar Paper, Department of Geography, University of B r i t i s h Columbia, Vancouver. Peterssen, S. (1956) Weather Analysis and Forecasting (Two Vols.). New York: McGraw HiTT Sawyer, J.S. (1966) Possible Variations of the General Circulation of the Atmosphere in Proceedings of the International Symposium on World CIimate 8000 to 0_ B.C. London: Royal Meteorological Society. Strahler, A.N. (1966) Introduction to Physical Geography. New York: Wiley. Trewartha, G.T. (1954) An Introduction to Climate. New York: McGraw H i l l . Unesco (1963) Proceedings of the WMO/Unesco Rome 1961 Symposium^on Changes of CIimate. Paris. W i l l e t t , H.C. (1950) The General Circulation at the Last (Wurm) Glacial Maximum. Geografiska Annaler, 32, pp. 179-187. APPENDICES APPENDIX A LIST OF CONDITIONAL' PROBABILITIES 63 FORT SIMPSON • WIND N ' NE E Jan. F. Arctic M. Arctic Pacific 0.15 0.64 0.21 — 0.17 — 0.38 ~ 0.45 Feb. Arctic Pacific 1.00 0.00 1.00 0.92 0.00 0.08 Mar. F. Arctic M. Arctic Pacific 0.67 0.33 0.00 0.13 0.43 0.87 0.55 0.00 . 0.02 Apr. Arctic Pacific 0.94 0.06 0.99 0.78 0.01 0.22 CLOUD 1 2 Jan. F. Arctic M. Arctic Pacific 0.27 0.64 0.09 0.25 0.63 0.12 Feb. Arctic Pacific 0.90 0.10 0.94 0.06 Mar. F. Arctic M. Arctic Pacific 0.31 0.55 0.14 0.38 0.47 0.15 Apr. Arctic Pacific 0.76 0.24 0.90 0.10 SE S sw w NW C 0.09 0.65 0.26 0.10 0.75 0.15 — 0.10 0.79 0.11 0.09 0.68 0.23 0.38 0.53 0.09 0.91 0.09 0.98 0.02 — 0.68 0.32 0.93 0.07 0.87 0.13 0.30 0.52 0.18 0.16 0.74 0.10 0.00 0.01 0.99 0.48 0.42 0.10 0.37 0.56 0.07 0.31 0.54 0.15 0.74 0.26 0.39 0,61 — 0.86 0.14 0.81 0.19 0.80 0.20 PRECIPITATION 3 FINE RAIN SNOW 0.05 0.18 0.00 0.12 0.66 0.65 0.00 0.64 0.29 0.17 1.00 0.24 0.93 0.90 0.00 0.95 0.07 0.10 1.00 0.05 0.38 . 0.35 0.00 0.41 0.50 0.50 0.05 0.54 0.12 0.15 0.95 0.05 0.89 0.80 0.81 0.99 0.11 0.20 0.19 0.01 64 EDMONTON WIND Jan. A r c t i c U.S. P a c i f i c Feb. A r c t i c U.S. P a c i f i c Mar. A r c t i c U.S. P a c i f i c Apr. A r c t i c U.S. P a c i f i c CLOUD Jan. A r c t i c U.S. P a c i f i c Feb. A r c t i c U.S. P a c i f i c Mar.. A r c t i c U.S. P a c i f i c Apr. A r c t i c U.S. P a c i f i c N , NE E 0.44 0.42 0.14 0.31 0.68 0.01 0.21 0.54 0.25 0.77 0.03 0.20 0.92 0.07 0.01 0.84 0.04 0.12 0.41 0.27 0.32 0.54 0.23 0.23 0.50 0.35 0.15 0.52 0.35 0.13 0.55 0.41 0.04 0.35 0.40 0.25 SE S SW 0.52 0.44 0.04 0.18 0.47 0.35 0.03 0.44 0.53 1.00 0.00 0.00 0.56 0.16 0.28 0.18 0.07 0.75 0.27 0.41 0.32 0.16 0.32 0.52 0.13 0.01 0.86 0.23 0.39 0.38 0.04 0.53 0.43 0.06 0.28 0.66 W NW C ' 0.16 0.24 0.60 0.33 0.34 0.33 0.36 0.63 0.01 0.43 0.01 0.56 0.56 0.02 0.42 1.00 0.00 0.00 0.15 0.17 0.68 0.34 0.23 0.43 0.27 0.51 0.22 0.29 0.41 0.30 0.13 0.57 0.30 0.01 0.28 0.71 PRECIPITATION 1 2 3 FINE RAIN SNOW 0.29 0.34 0.22 0.25 0.00 0.27 0.37 0.31 0.50 0.41 0.02 0.52 0.34 0.35 0.28 0.34 0.98 0.21 0.74 0.55 0.55 0.56 0.01 0.86 0.07 0.07 0.09 0.10 0.01 0.10 0.19 0.38 0.36 0.34 0.98 0.04 0.35 0.25 0.29 0.22 0.03 0.52 0.21 • 0.21 . 0.34 0.25 0.08 0.41 .0.44 0.54 0.37 0.53 0.89 0.07 0.16 0.19 0.37 0.20 0.38 0.96 0.46 0.48 0.40 0.47 0.47 0.04 0.38 0.23 0.23 0.33 0.14 0.00 ( 65 WINNIPEG WIND Jan. Arc/Pac U.S. Feb. A r c t i c P a c i f i c U.S. Mar. A r c t i c P a c i f i c U.S. Apr. A r c t i c P a c i f i c U.S. N , NE E 0.67 0.33 0.86 0.14 0.38 0.62 0.51 0.45 0.04 0.20 0.79 0.01 0.13 0.80 0.07 0.61 0.32 0.07 0.82 0.18 0.00 0.33 0.26 0.41 0.82 0.17 0.01 0.65 0.15 0.20 0.55 0.32 0.13 SE S SW 0.20 0.80 0.33 0.67 0.99 0.01 0.05 0.81 0.14 0.21 0.61 0.18 0.32 0.51 0.17 0.19 0.41 0.40 . 0.33 0.31 0.36 0.44 0.51 • 0.05 0.74 0.15 0.11 0.45 0.31 0.24 0.39 0.61 0.00 W NW C 0.86 0.14 0.86 0.14 — 0.38 0.43 0.19 0.56 0.43 0.01 — 0.70 0.19 0.11 0.80 0.17 0.03 0.02 0.39 0.59 0.67 0.32 0.01 0.76 0.23 0.01 0.73 0.27 0.00 CLOUD 1 2 Jan. Arc/Pac 0.75 0.53 U.S. 0.25 0.47 Feb. A r c t i c 0.45 0.51 P a c i f i c 0.48 0.45 U.S. 0.07 • 0.04 Mar. A r c t i c 0.61 0.31 P a c i f i c 0.20 0.30 U.S. 0.19 0.39 Apr. A r c t i c 0.68 0.78 P a c i f i c 0.26 0.15 U.S. 0.06 0.07 PRECIPITATION 3 FINE RAIN SNOW 0.48 0.66 0.00 0.49 0.52 0.34 1.00 0.51 0.16 0.40 0.00 0.27 0.62 0.48 0.00 • 0.61 0.22 0.12 1.00 0.12 0.45 0.42 0.03 . . 0.65 0.35 0.27 0.44 0.31 0.20 0.31 0.53 0.04 0.85 0.80 0.81 1.00 0.11 0.16 0.16 0.00 0.04 0.04 0.03 0.00 66 FORT WILLIAM WIND N ' NE E Jan. Arc/Pac 0.57 0.22 0.26 U.S. 0.43 0.78 0.74 Feb. Arc/Pac 0.29 0.33 0.08 U.S. 0.71 0.67 0.92 Mar. Arc/Pac 0.33 0.26 0.26 U.S. 0.67 0.74 0.74 Apr. Arc/Pac 1.00 0.84 0.89 U.S. 0.00 0.16 0.11 CLOUD 1 2 Jan. Arc/Pac 0.66 0.55 U.S. 0.34 0.45 Feb. Arc/Pac 0.60 0.42 U.S. 0.40 0.58 Mar. Arc/Pac 0.25 0.20 U.S. 0.75 0.80 Apr. Arc/Pac 0.82 0.85 U.S. 0.18 0.15 SE S sw W NW C 0.05 0.95 0.24 0.76 0.50 0.50 0.60 0.40 0.50 0.50 0.22 0.78 0.06 0,94 0.21 0.79 0.46 0.54 0.51 0.49 0.81 0,19 0.42 0.58 — 0.00 1.00 0.16 0.84 0.25 0.75 0.44 0.56 0.19 0.81 0.29 0.71 0.66 0.34 0.85 0.15 0.99 0.01 0.62 0.38 PRECIPITATION 3 FINE RAIN SNOW 0.33 0.58 0.00 0.35 0.67 .0.42 1.00 0.65 0.33 0.53 0.05 0.33 0.67 0.47 0.95 0.67 0.26 0.22 0.10 0.33 0.74 0.78 0.90 0.67 0.84 0.79 0.93 1.00 0.16 0.21 0.07 0.00 APPENDIX B PERCENTAGE DEVIATIONS FROM 1955-59 .AIR MASS FREQUENCIES FORT SIMPSON JANUARY FEBRUARY MARCH APRIL F.A. M.A. P A P F.A. M.A. P A P 1824-29 1830-34 1835-39 1840-44 +7.12 +3.28 +3.50 +10.07 -2.13 +1.03 +1.24 -3.50 -5.00 -4.31 -4.75 -7.56 -1.17 +2.80 +2.98 +3.20 +1.16 -2.80 -2.98 -3.20 -0.66 +4.39 +0.72 +1.94 +4.12 +3.27 +2.95 +3.14 -3.48 -7.66 -3.67 -6.10 +7.27 -5.86 +7.38 +5.19 -7.27 +5.86 -7.38 -5.19 EDMONTON A U.S. ' P A U.S. P A U.S. P A U.S. P 1820-24 1825-29 1830-34 +5.12 +4.27 +6.13 +1.20 +1.64 +2.54 -6.32 -5.83 -8.67 +6.47 +6.58 +6.46 -3.45 -3.02 +0.67 -7.25 -0.61 -5.85 +3.85 +7.22 +6.51 -1.35 +1.38 +2.14 -2.50 -8.50 -8.67 -14.81 -8.18 -3.74 +9.24 +11.18 +9.60 +5.65 -3.22 -5.87 WINNIPEG - • A/P U.S. A P U.S. A P U.S. A P U.S. 1820-24 1825-29 +10.35 +11.81 -10.35 -11.81 +6.78 +3.56 -3.90 -3.88. +2.24 -5.80 +6.07 +13.33 -0.70 +0.38 -5.37 -13.72 +9.65 -2.08 -4.64 +5.23 -5.01 -3.15 FORT WILLIAM A/P U.S. A/P U.S. A/P U.S. A/P U.S. 1830-34 1835-39 1849-51 +0.90 +2.07 -4.42 -0.90 -2.07 +4.42 -7.88 -8.20 +3.41 +7.88 +8.20 -3.41 +3.74 +3.40 +2.15 -3.74 -3.40 -2.15 +1.92 +4.92 +6.10 -1.92 -4.92 -6.10 CO APPENDIX C CONFIDENCE LIMITS FOR AIR MASS FREQUENCY 70 Confidence 1 inn'ts for a i r mass frequencies Consider the following f a i r l y general form of the Central Limit Theorem. THEOREM I f X-j >. - - »X are a sequence of mutually independent random variables with f i n i t e expectations and variances, and i f Sn = X, + X 9 + ... + X I I n Mn = y 1 + m2 + . . . + \in R 2 _• 2 . 2 . • 2 Bn = a-j + a 2 + ••• + a where: \i. = the expected value of X.; 2 a. = the variance of X. then P ( S/ n" Bn n < a ) * N(a') as n + ». so long as the following conditions hold 1) The random variables X. are uniformly bounded 2) Bn -»• « as n -> 0 0 . where: a = some fixed real constant N(a) = the standard normal d i s t r i b u t i o n function c l t d ) 1 a 6 • $ N(a) = [I (2n)" % exp (- t 2/2) dt This rather imprecise statement of the theorem follows the discussion of F e l l e r (1957, chapter X). For a very precise statement of the theorem and a proof reference might be made to Lamperti (1966, pp. 69 f f ) . The proof i s not a simple one and w i l l not be included here. 70 Confidence 1imits for a i r mass frequencies Consider the following f a i r l y general form of the Central Limit Theorem. THEOREM I f X-j ,X2 X n are a sequence of mutually independent random variables with f i n i t e expectations and variances, and i f Sn = X1 + X 2 + ... + X n Mn = y.j + y 2 + ... + y n R n2 _ 2 . 2 . . 2 Bn = + a 2 + ... + o n where: y. = the expected value of X^ 2 = the variance of X. then P ( S" n" B^ n < a) + N(a) as so long as the following conditions hold 1) The random variables X. are .uniformly bounded 2) Bn •* °° as n ->• » where: a = some fixed real constant N(a) = the standard normal d i s t r i b u t i o n function N(a) = [I (2n)"% exp (- t 2/2) dt This rather imprecise statement of the theorem follows the discussion of F e l l e r (1957, chapter X). For a very precise statement of the theorem and a proof reference might be made to Lamperti (1966, pp. 69 f f ) . The proof i s not a simple one and w i l l not be included here. 71 The frequency of air mass A during some period of length n days may be treated in the following manner. Define the random variables X. X- =1 i f the air mass present was A = 0 i f the air mass present was not A. The X.. may be considered as a sequence of Bernoulli t r i a l s and we may define P(Xi = 1) = Pi hence the expectation of X^  E(X,> and the variance var (X.) = a\ =p. (1-p.) These random variables satisfy the conditions of the theorem since 1) E(X^) and var (X^) are f i n i t e 2) The t r i a l s are independent (by assumption) 3) For a l l X^ the uniform bound 0 <_ X. <. 1 exists 4) z. var (X^) •+ », as n + » To make use of the Central Limit Theorem we require estimates of Mn and Bn. The most convenient estimate for Mn follows directly from its definition: Mn = u 1 + u 2 + . . . + y n Mn = p1 + p2 + ... + Pn The pooled variance could be estimated in a similar manner, but i t was found more convenient to make use of the following lemma: 72 LEMMA If X.j are a sequence of mutually independent random variables, defined as above, and Sn is the partial sum of the f i r s t n members of the sequence then var(Sn) '.= max i f p1 = p2 = ... = pn = p = z p./n for any constant value of p. PROOF The variance of is given by var(X.) = p i(l-p i) Then the variance of Sn var(Sn) = z p i(l-p i) in view of the assumption of independence. The problem is clearly to maximise z.p^(1-p^) subject to the constraint zp^/n = p. To this end, form the function G (p1,p2,...,pn) = z Pi - z p ^ + A z (Pi - np) where: A - some appropriately chosen real constant or Lagrange multiplier. Then var(Sn') will have an extremum where | | = 0 for a l l 1 3 p i Now | i = 1 - 2 P i + A = 0 implies Pi = "* 2 X = const. Therefore the extremum occurs where p1 = p2 = ... = pn = Zi Pi/n It is easy to check that the extremum is a maximum, and then the lemma • is proved. 73 The s ign i f i cance of th is lemma l i e s in the fac t that i f p-j. = p 2 = . . . = p n then the d i s t r i bu t i on of Sn i s the fam i l i a r binomial d i s t r i bu t i on with variance var(Sn) = np(l-p) Therefore we have an upper bound for the variance of Sn in the case where the p.. are not equal . The preceding d iscussion may be summarised in the fo l lowing expression: Lim P(z < (Sn - np) / (np( l -p) ) J s < z ) <_ a n-*» -a/2 a/2 using the Central L imi t Theorem and the expression for the variance de-r ived above. where: z l 0 = ' a / 2 W~h e _ t / 2 dt a/2 -°° Manipulation of th is expression y i e l ds ' P ( p . z (fiUiEl)* < | H < p + z ' ( f i i l z E l ) ^ x r a/c n n r a/c n fo r s u f f i c i e n t l y large n. We may use th is as a confidence l i m i t fo r Sn /n , the re la t i ve frequency of a i r mass A. S ign i f icance of deviat ions from the modern values was measured as the p robab i l i t y of obtaining a value as extreme as that of the control per iod , given the estimates of the p.. obtained by previous ana lys i s . REFERENCES CITED: F e l l e r , W. An Introduction to Probab i l i t y Theory and I ts App l i ca t ions . New York: Wi ley, 1957. Lampert i , J . P robab i l i t y : a Survey of the Mathematical Theory. New York: Benjamin, 1966. 

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