A Compariso of Cirque Features i the Sierra Nevada ad Triity Moutais Iris Surame Bejami Holt Christopher Surame April, 26
INTRODUCTION Cirques are geerally semicircular, amphitheatre like bowls with a steep headwall ad ope side dowvalley. Cirques are formed by small glaciers (cirque glaciers) ear the sowlie. A cirque does ot have to be occupied by a cirque glacier to be a cirque. Rotatioal mass movemet of the glacier trasport ice ad sedimet dowvalley of the hollow. The itese freeze ad thaw cycles o the valley walls ad beeath the glacier make cirque erosio very effective. Small roud lakes (cirque lakes) are ofte preset with i the cirque boud dowvalley by a bedrock sill or a moraie. Cirque glaciers ad glacierets are very sesitive to small chages i climate due to their close proximity to the sowlie. This has very importat social implicatios as may commuities that surroud moutaious areas deped o the meltwater from cirque glaciers for drikig water i the dry summer moths, for healthy riparia habitats, ad recreatioal activities such as boatig ad fishig. We aalyzed various features of cirques amog the Sierra Nevada ad Triity Moutais to compare how these two geologic settigs iflueced cirque geomorphology. We aticipated that we would fid cirques at lower elevatios i the Triity s vs. the Sierra Nevada due to the proximity to a moisture source ad the higher latitude tha the Sierra Nevada. We expected more variability i cirque aspect at higher elevatio due to icreased amouts of sowfall, permittig cirque glaciers to withstad more su exposure to the south. Additioally, cirques with a southerly aspect are likely idicative of a regioal ice cap that covered parts of the respective rages. Sice the Sierra Nevada cotaied a more extesive ice cap compared to the Triity Moutais we predicted that we would fid a larger variatio i aspect there. We compared various features of cirques amog the Sierra Nevada ad Triity moutais. These features iclude cirque desity (umber of cirques per uit area), aspect, elevatio, ad latitude. We expected more variability i cirque aspect at higher elevatios. Also, we expected to see more ortherly facig cirques at lower elevatios. Betwee the two regios it was aticipated that there will be o differece i mea aspect. I the Sierra Nevada Moutais we aticipated that latitude will explai much of the variability i cirque desity ad will be icluded i our aalysis. We also believe that variability i desity ad aspect could differ betwee regios. DEFINITIONS Our workig defiitio of a cirque may foud i Evas (1977): a cirque may be defied as a hollow, ope dowstream but bouded upstream by the crest of a steep slope (headwall) which is arcuate i pla aroud a more getly-slopig floor... we might tetatively suggest that part of the head wall exceeds 3º ad the floor slopes less tha 2º.
We used the log axis defiitio to measure cirque aspect. Cirque log axis aspect is measured from the lie passig through the furthest poit o the head wall crest ad midpoit of the sill (Evas 1977): Figure 1: Log axis defiitio of cirque aspect. After drawig the log axis, we used a protractor to measure its agle from due North, or º, keepig track of the East ad West directios. A agle East of North was recorded as egative ad a agle West of North was recorded as positive. For reasos to be explaied i the comig sectios, we also measured what we call pod, or bowl, aspect whe cosiderig the Triity Moutais. Pod aspect is a feature of cirques that is measured i a maer similar to that of cirque aspect usig the log axis defiitio. The lie passig through the furthest poit o the lowest cotour withi the cirque ad midpoit of the sill determied our pod aspect measuremet. The lowest cotour is typically i cocert with the cirque lake topography. It is our belief that our defiitio of pod aspect is early syoymous to that of the media axis aspect from Evas (1977). SAMPLING DESIGN AND METHODOLOGY Our data was obtaied from Uited States Geological Survey (USGS) topographical maps. These maps iclude all of the Uited States i quadragles 7. by 7.. The logitudial dimesios do ot vary i legth however the latitudial dimesios do vary from orth to south, but is ot sigificatly differet amog the two regios we aalyzed. The average area represeted i a quadragle amog the two regios is 7 square miles. Every map coverig the state of Califoria is available i the library. For each map viewed, we couted the umber of cirques ad measured their aspect ad elevatio. The elevatio of each cirque was recorded as the elevatio of the lowest cotour lie withi the cirque. For each map we also made ote of the miimum latitude ad maximum elevatio. Sice each map has roughly the same area, cirque desity was measured simply as the umber of cirques for that map.
Due to there beig a relatively low umber of cirques i the Triity rage, a exhaustive study was possible. O the other had, it was ecessary to take a radom sample whe cosiderig the Sierra Nevada Moutais due to the large umber of maps that cover the regio. A exhaustive study of the Sierras would have bee too time cosumig for a project of this scope. By radom sample we mea a collectio of USGS maps radomly chose from regios that iclude the Sierra Nevada Moutais. The map associated with each radomly chose grid square was obtaied i the library ad aalyzed i the fashio described previously. I the course of our data collectio it became ecessary to compare cirque aspect with pod aspect. The data collected for the Sierras due to error o the part of two group members was ot cirque aspect, but pod aspect. Cosequetly, whe cosiderig the Triity Moutais, pod ad cirque aspect were measured. We measured these two quatities as we suspected that they were closely related. From this relatioship we hoped that we could salvage the data we collected from the Sierras to say somethig about cirque aspect rather tha just pod aspect. I our aalysis we extrapolated the relatioship betwee cirque ad pod aspect for the Triities to the Sierras. We estimated the mea cirque aspect of the Sierras by feedig i the pod aspect data ito the model we costructed relatig pod ad cirque aspect for the Triity Moutais. Cosequetly, our estimate for mea cirque aspect i the Sierras is questioable. As regrettable as this may be, the relatioship betwee cirque ad pod aspect is itself quite iterestig, ad uexpectedly added aother very iterestig feature of cirques to our aalysis. Also, sice there seems to be a strog relatioship betwee these two quatities, we feel that our estimate for mea cirque aspect i the Sierras is a good oe i that it gives us a sese of the true mea cirque aspect. We also ote that this problem does ot affect our aalysis cocerig cirque desity. Fially, we suggest that pod aspect could be take as aother way to defie cirque aspect give the stregth of the relatioship observed. ANALYSIS We adopted two poits of view whe aalyzig our data. Oe was to cosider each cirque ad its various features as a data poit; aother was to cosider each map ad its features as a data poit. Whe cosiderig cirque aspect, the former poit of view was take. Whe cosiderig cirque desity, the latter poit of view was take. Aspect is a circular measuremet. Therefore, whe cosiderig either cirque or pod aspect, we used techiques that rely upo circular distributios. Calculatig the mea agle for a set of circular data is a bit more ivolved tha that for liear data. Also, there are several measures of spread for circular data, or agular dispersio, aalogous to that of stadard deviatio for liear data. Each has its advatages ad drawbacks. The measure of agular dispersio we chose to use here is oe that is aalogous to the stadard deviatio o a liear scale i that it rages from zero to ifiity. The measure we have chose is ofte referred to as the circular stadard deviatio ad we will also
adopt this termiology. The iterested reader may cosult the appedix i which the details of these calculatios are addressed. The techiques we use here may be foud i Zar (1996). To aalyze other features of cirques we used a geeral liear model to predict cirque desity i terms of elevatio ad latitude. Liear regressio aalysis was implemeted to fid the relatioship betwee cirque aspect ad pod aspect. RESULTS I a effort to salvage the results of our study of pod aspect rather tha cirque aspect i the Sierras, we discovered a iterestig relatioship, perhaps oe that should be explored more thoroughly. This relatioship surfaced from the regressio aalysis performed o our data set for the Triity Moutais. The regressio model produced from this aalysis ca predict cirque aspect from pod aspect. We may the extrapolate these results to say somethig about cirque features of the Sierra Nevada Moutais. We ote that it would be a iterestig study to compare the various measures of cirque aspect preseted i Evas (1977) ad the compare these results with pod aspect. If there is a strog relatioship, pod aspect could be take as yet aother measure of cirque aspect. Our regressio model of cirque aspect versus pod aspect is Cirque Aspect = - 4.72 +.768 Pod Aspect. The graph of our data for the Triity Moutais, alog with the regressio lie, is pictured i Figure 2. Usig this model we will estimate cirque aspect by extrapolatig this relatioship to the Sierra Nevada data kowig full well that this estimate is questioable. It is questioable for several reasos. Oe is that this relatioship may ot hold for data outside the rage of the Triity Moutai data. Aother is that we are usig a liear model of this relatioship as opposed to a circular model. Yet aother reaso, ad perhaps the most importat, is that we do t really kow how good this estimate is sice we do ot have a cofidece level associated with this estimate. Clearly, the coclusios of the aalysis are ot statistically reliable. However, for the sake of accomplishig what we set out to do, we used this model to obtai a estimate for cirque aspect of the Sierra Nevada Moutais. Also, give the stregth of the liear relatioship observed (p <. for the explaatory variable, R 2 =68.2 ), we feel that the estimate we obtai at the very least gives oe a sese of the true mea cirque aspect i the Sierra Nevada Moutais.
Cirque Aspect vs Pod Aspect i the Triity Moutais 1 Cirque Aspect - -1 - - -1 - Pod Aspect 1 Figure 2: A liear regressio model for Cirque Aspect versus Pod Aspect. Sice a exhaustive study of cirques i the Triity Moutais was carried out, we report the parameters of mea cirque ad pod aspects ad a measure of agular dispersio aalogous to that of stadard deviatio for liear distributios. The true mea cirque aspect for the Triity Moutais is α C,T = 13. 712, or 13.712 degrees East of North. The circular stadard deviatio of cirque aspect is 42.91º. SIN(Cirque Aspect) vs COS(Cirque Aspect) for the Triity Moutais 1. 3 3 Cirque Aspect i the Triity Moutais SIN(Cirque Aspect).. -. Percet 2 2 1-1. -1. -... COS(Cirque Aspect) 1. -18-12 -6 Cirque Aspect 6 12 18 Figure 3: Agular dot plot with the associated liear histogram of cirque aspect for the Triity Moutais. The true mea pod aspect for cirques i the Triity Moutais is α P,T = 12.297, or 12.297 degrees East of North. The circular stadard deviatio is 4.8º.
SIN(Pod Aspect) vs COS(Pod Aspect) for the Triity Moutais 1. 3 3 Pod Aspect for the Triity Moutais SIN(Pod Aspect).. -. Percet 2 2 1-1. -1. -... COS(Pod Aspect) 1. - -1 - Pod Aspect 1 Figure 4: Agular dot plot with the associated liear histogram of pod aspect for the Triity Moutais. Our estimate for the mea pod aspect of cirques i the Sierras is α P,S = 4.8, or 4.8 degrees East of North. A circular 9% cofidece iterval for the mea pod aspect of cirques, α P,S, i the Sierra Nevada moutais is -3º< α P,S <21º. The circular stadard deviatio for the pod aspect of our Sierra sample is 78.7º. Sice the mea pod aspect i the Triity Moutais falls withi the above cofidece iterval, there is o statistically sigificat evidece that mea pod aspect differs betwee the two regios. Usig our model for cirque aspect versus pod aspect, we estimate the mea cirque aspect α C,S from our estimate for the mea pod aspect α P,S. So, α C,S - 4.72 +.768 α P,S = - 8.23º, that is, our estimate for the mea cirque aspect is 8.23º East of North. Agai, we emphasize that we do ot have a cofidece level associated with this estimate. SIN(Pod Aspect) vs COS(Pod Aspect) for the Sierra Nevada Moutais 1. 3 3 Pod Aspect for the Sierras o a Liear Scale SIN(Pod Aspect).. -. Percet 2 2 1-1. -1. -... COS(Pod Aspect) 1. - -1 - Pod Aspect 1 Figure : Agular dot plot with the associated liear histogram of pod aspect for the Sierra Nevada Moutai Rage.
The mea cirque elevatio i the Triity Moutais is 6421.38 feet while the circular stadard deviatio of pod aspect for the Triity Moutais is 4.8º. Our estimate for the mea elevatio of cirques i the Sierras is 1464.1 feet. A 9% cofidece iterval for the true mea cirque elevatio i the Sierras is (124.8 feet, 1687. feet). Our estimate for the circular stadard deviatio of pod aspect for the Sierras is 78.7º. These data for pod aspect versus elevatio for both the Triity ad Sierra Nevada Moutais are summarized by Figure 6. As suspected, this figure, Figures 3, 4 ad, alog with the above estimates, strogly suggests that variability i pod aspect icreases as elevatio icreases. Hece, based upo our model for cirque aspect versus pod aspect, we see this as evidece (ot statistically sigificat) that variability i cirque aspect will also icrease as elevatio icreases. 2 Pod Aspect vs Elevatio Rage Sierra Nevada Triity 1 Pod Aspect -1-2 4 6 7 8 9 Elevatio 1 11 12 Figure 6: Variability i pod aspect icreases with elevatio. Aother feature of iterest is how cirque desity varies with elevatio ad latitude. We suspected that observed variability i cirque desity could be explaied by these two variables. Table 1 below cosists of p-values associated with ELEVATION ad LATITUDE whe both of these explaatory variables are cosidered i the geeral liear model. These aalyses were performed o the data collectios of iterest. Notice that a separate aalysis was performed o the Sierra Nevada data whose elevatio raged from 9 ft. ad above. This was icluded because all of the maps whose maximum elevatio was below 9 ft. were void of cirques. Havig so may zero cirque desity poits sigificatly skewed the Sierra Nevada data set. Elevatio became less sigificat as a result of istitutig the lower boud o the Sierra Nevada data. For every data set, ELEVATION was a statistically sigificat explaatory variable, while LATITUDE was ot.
ELEVATION LATITUDE Sierra Nevada Moutais p-value <. p-value =. Sierra Nevada Mts. 9 ft. p-value =.4 p-value =.281 Triity Moutais p-value =.4 p-value =.386 Both Moutai Rages p-value <. p-value =.496 Table 1: P-values of iterestig data sets for each explaatory variable i the geeral liear model DENSITY ~ ELEVATION + LATITUDE. Table 2 cosists of p-values for each explaatory variable whe they are cosidered idividually i the appropriate geeral liear models. Notice that over each data set ELEVATION became more sigificat whe it was cosidered aloe. This was also the case for LATITUDE over the Triity Moutai ad Sierra Nevada Moutai data sets. Note the strikig shift i sigificace of the p-value for LATITUDE over the Triity Moutai data i Table 2 compared to that i Table 1. This pheomeo ca be attributed to a uique feature of the Triity Moutai Rage itself. The highest elevatios are foud o the orth side of the rage, ad it has bee see that cirques are foud at high elevatios. So, eve though it seemed that LATITUDE could be a sufficiet predictor for cirque desity, really, it was iflueced by the overwhelmig predictig power of ELEVATION. Note figure 7, a scatterplot represetig the correlatio betwee elevatio ad latitude. From this figure, we ca see that said tred is i place. A p-value <. from the geeral liear model ELEVATION ~ LATITUDE supports this claim that latitude is a fair predictor of elevatio. ELEVATION LATITUDE Sierra Nevada Moutais p-value <. p-value =.383 Sierra Nevada Mts. 9 ft. p-value =.3 p-value =.312 Triity Moutais p-value <. p-value =.9 Both Moutai Rages p-value <. p-value =.8 Table 2: P-values of iterestig data sets for each explaatory variable whe cosidered idividually i the geeral liear models DENSITY ~ ELEVATION ad DENSITY ~ LATITUDE.
Elevatio vs Latitude i the Triity Moutais 9 8 7 Elevatio 6 4 3 2 4.4 4. 4.6 4.7 4.8 4.9 Latitude 41. 41.1 41.2 41.3 Figure 7: Elevatio versus latitude i the Triity Moutai Rage with a regressio lie. Also, the adjusted sum of squares for LATITUDE was always cosiderably lower tha that of ELEVATION whe lookig at each data set over every type of geeral liear model metioed. Number of Cirques vs Elevatio For the Sierra Nevada Moutais Number of Cirques vs Elevatio for the Triity Moutais 2 2 Number of Cirques 1 Number of Cirques 1 2 4 6 8 Elevatio 1 12 14 2 4 6 8 Elevatio 1 12 14 2 Number of Cirques vs Latitude for the Sierra Nevada Moutais 2 Number of Cirques vs Latitude for the Triity Moutais Number of Cirques 1 Number of Cirques 1 3 32 34 36 Latitude 38 4 42 3 32 34 36 Latitude 38 4 42 Figure 8: Cirque desity versus latitude ad elevatio for both regios.
SUMMARY AND CONCLUSIONS The data cocerig aspect suggest that pod, ad hece cirque aspect, is more variable at higher elevatios as was suspected. Field aalysis of potetial cirques could yield more data poits ot detectable by a mappig survey aloe. Particularly, a field aalysis would likely yield more data poits at lower elevatios, where cirque features o the map are difficult to iterpret due to their less developed ature. This additioal data would likely show that at lower elevatios aspect is less variable. There is o statistically sigificat evidece to rule out the possibility that the mea pod aspects for the two regios are differet. A course for further study would be to survey agular-agular regressio techiques ad use them o this data, comparig the various measures of cirque aspect to as they relate to pod aspect. We ote that the agular-agular correlatio for our cirque vs. pod aspect data is r aa =.7629 which further bolsters our belief that pod aspect gives us a good sese of cirque aspect. Cocerig cirque aspect i the Sierra Nevada Moutais, our suggestio would be to simply measure cirque aspect correctly. Elevatio seems to be a decet predictor of cirque desity. I every case latitude was a poor predictor of cirque desity. APPENDIX Cosider a collectio of agular observatios α 1, α 2,, α. The mea agle is calculated i the followig way: Compute X= i= 1 cos(α i ), Y= i=1 si(α i ), ad r= X 2 +Y 2. The mea agle, a, is the the uique solutio to the pair of trigoometric equatios cos( a)= X r si( a )= Y r. This procedure may be rephrased as a vector average. If v 1, v 2,, v are vectors o the uit circle such that v i =(cos(α i ),si(α i )), the 1 v i = 1 (cos(α i=1 i ),si(α i )) i= 1 1 ( cos(α i ), si(α i )) i= 1 i=1 ( 1 cos(α i ), 1 i= 1 si(α i )) i=1 ( X,Y )
The above vector poits i the mea directio of v 1, v 2,, v. We make it ito a uit vector by multiplyig by the reciprocal of the magitude, that is 1 X 2 +Y 2 ( X,Y )= 1 r ( X,Y )=( X r, Y r ). Sice this vector is o the uit circle, we may the say our mea agle, a, satisfies ( X r, Y r )=(cos( a),si ( a)) which is the above result stated i vector form. The more cocetrated our data is, the closer r is to 1. Therefore 1 r may be iterpreted as a measure of spread for our data. S 2 =1 r is called the circular variace. A variatio of this idea is the agular variace, s 2 =2(1 r ). The circular stadard deviatio (the measure of agular dispersio we chose to use i our aalysis) i radias is give by s = 2l(r ). I degrees the circular stadard deviatio is s = 18 π 2l (r). This measure of spread rages from zero to ifiity which makes it more ituitively appealig tha the other two i that it is more aalogous to the liear stadard deviatio we are already familiar with. REFERENCES Ia S. Evas, 1977, World-Wide Variatios i the Directio ad Cocetratio of Cirque ad Glacier Aspects, Geografiska Aaler, 9A, pp. 1-17. Jerrold H. Zar, 1996, Biostatistical Aalysis, 3 rd Editio, Pretice Hall, Upper Saddle River, New Jersey.