Trend-Surface Analysis of Central Pennine Erosion Surfaces

Cuchlaine A. M. King

Transactions of the Institute of British Geographers, No. 47. (Sep., 1969), pp. 47-59.

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http://www.jstor.org Wed Sep 12 16:06:37 2007 Trend-Surface Analysis of Central Pennine Erosion Surfaces

(Reader in Geography, University of Nottinghani)

Revised MS. received 3 October 1968

ABSTRACT-Trend-surface analysis can be used to describe a three-dimensional surface mathematically. It can be used to reconstruct a partially dissected surface, to test a hypothesis, and as a search procedure. These applications of the method are illustrated by trend-surface analysis of the summit erosion surface of the Alston and Askrigg Blocks in the central . The blocks form well-defined structural units, separated by the syncline, and bounded on the north, west and south by a series ocmajor faults. The quadratic surface explains 89 per cent of the variability of the data in the Alston Block and the cubic surface explains 78.5 per cent of the variability in the Askrigg Block. The trend surface in the Alston Block slopes down from a high area adjacent to the faults to the north and south-east, while the major slope of the Askrigg Block cubic surface is eastward from an isolated area of high elevation in the west. The two surfaces link in a trough which approximates to the position ofthe Stainmore depression. The positive residuals represent monad- nocks rising above the warped summit surface. The negative residuals mostly lie along the major valleys and may represent minor downwarps that played a part in the establishment of the drainage pattern. The major streams flow nearly at right-anglcs to the trend-surface contours. They may have been influenced by the form of the uplifted and warped summit surface.

TREND-SURFACEanalysis is being increasingly used to study a wide variety of spatial problems in geography and geology (R. J. Chorley and P. Haggett, 1965; W. C. Krumbein and F. A. Graybill, 1965).The method can be used for a number of purposes, such as the reconstruction of a distribution that is no longer complete, the description of an areal pattern of any given variable, the testing of hypotheses, and as a search procedure. Trend-surface analysis is applied in this contribution to the upland erosion surface in the Alston and Askrigg Blocks of the central Pennines of northern . All the purposes mentioned can be illustrated in this

The Area and the Data

The area covered by the analysis is shown in Figure I. It is bounded on three sides by the great capital sigma-shaped fault system, which bounds this part of the Pennines. The northern boundary follows the Stublick faults along the Tyne valley. The western boundary follows the Pennine, and north Craven fault systems, while the southern boundary runs partly along the south Craven fault. The eastern boundary is shown by the dashed line on Figure I. The slightly warped Carboniferous strata dip gently eastward to disappear beneath the over- lying Magnesian Limestone near this margin. The area thus forms a clearly defined structural unit. It can be divided into two almost equal parts by the structural depression of the Stainmore syncline. The northern part forms the Alston Block and the southern part the Askrigg Block. These two distinct structural units were analysed separately in order to obtain a better fit of the surfaces and to reveal any significant differences between the two parts. The two blocks consist of relatively little disturbed Carboniferous strata, including limestone, shales and sandstone. Millstone Grit caps the , particularly in the northern part FIGUREI-MJP of the centr~lPennines, showing the Askrigg and Alston Blocks and thc major fault systems. Marginal figures refer to the National Grid in this and succeeding Figures. logical development of this region have been discussed elsewhere (C. A. M. King, 1963). One of the most conspicuous facets of the landscape is the summit surface, the Alston Block section of which was described by F. M. Trotter in 1929. He suggested that this surface had been PENNINE EROSION SURFACES 49 warped and uplifted, following early Tertiary peneplanation, during the late Tertiary period. This is the hypothesis that is tested by means of trend-surface analysis. The summit surface is well preserved in both blocks. The good state of preservation of the surface makes this area suitable for this type of analysis. The summit surface was defined for this analysis by the irregularly spaced heights of closed summit contours and summit spot heights as depicted on the I :63,360 maps. The heights were recorded to the nearest 10 feet (3 m) in the case of summit spot heights and to the nearest 50 feet (IS m) for summit closed contours. The major summits were all included in the analysis and as even a spread of summit and broad spur flats as possible was obtained. All the points used in the analysis were thought to lie on or above the summit surface. Later dissection of the raised land surface did not, therefore, affect the pattern of the trend surface. The east and north grid coordinates were used to define the positions of the points. These coordinates were measured from an origin to the north-west of the area analysed and they provide the x and y independent variables used in the analysis. These values are shown on the left and top of the Figures and the O.S. grid numbers are shown on the right and bottom. The total north-south extent of the area analysed is nearly IIOkm and its east-west extent is 70 km. The total number of recorded elevations was 333, of which 175 points were in the Alston Block and 158 in the Askrigg Block. The two units were also split into a northern and a southern part and separate analyses were carried out for these subdivisions.

The Resirlts of the Analysis Trend-surface analysis is a technique by which the square of the vertical distance between the data points and the fitted surface is reduced to a minimum. The simplest surface is the linear one, but the higher order surfaces usually provide a better fit in an area of complex relief. The goodness of fit of the surfaces will be considered first. Then the form of the linear, quad- ratic and cubic trend surfaces will be described and, finally, the pattern of the residuals will be . .

The Fit of the Surfaces The goodness of fit of the surfaces can be defined as the percentage variation of the data accounted for by each surface. The results of the analysis for the two units are given in Table I, which also gives the F value for the surfaces. The F value indicates the degree of significance of the surface. If the F value is higher than the tabulated value for the number of data points used, then it is unlikely that the surface could have been generated by chance. In all the surfaces shown in Table I, the F values are very high. The surfaces may, therefore, be taken as real and not chance ones. TABLEI

Askrigg Block F value Alstoir Block F valrre

Linear surface 60 per cent 116 62 per cent 141 Quadratic surface 64 per cent 55 8g per cent 278 Cubic surface 78.5 per cent 60 go per cent 164

The values given for the percentage variation show considerable differences between the two structural units. The linear surface fits the two blocks almost equally well, explaining only 60 to 62 per cent of the variability. With the quadratic surface, there is a very great increase in the percentage explained in the Alston Block, but a much smaller increase in the Askrigg Block. The cubic surface, which explains nearly 90 per cent of the variation in the data of the Alston Block area, can be considered to describe best the pattern of the sumnlit erosion surface of this area. The high F value also shows that the surface is highly significant. When the Alston Block was analysed in two separate parts, the linear surface of the northern section accounted for 83 per cent of the variability, and the linear surface of the southern section accounted for 84 per cent. The two individual linear surfaces, however, had very different directions of slope. For this reason, when the linear surface was fitted to the whole of the Alston Block, no one linear surface could fit the data closely. The quadratic surface, which has a double curvature, could, however, fit the data more closely. The conclu- sion is reached that the Alston Block summit surface has a fairly simple form that can be described by the quadratic trend equation with a high degree of precision. Of the 89 per cent variability accounted for by the quadratic surface in the Alston Block, the variable x2 accounts for 53.4 per cent and y2 accounts for a further 15.4 per cent. The importance of the squared values of x and y can be explained by the form of the surface, which has a rapidly increasing gradient from the south-west corner of the area downward both to the north and east. The division of the percentage explanation among the independent variables enables the most important of these to be recognized. If the for111 of the surface is the result of deformation of an originally fairly flat surface by earth movements, then the importance of the variables probably reflects the nature of the earth movements that have caused the deformation. The most important directions of tilting can be picked out by this means. The results agree with other evidence which has been discussed by Trotter (1929) and more recently by M. H. P. Bott (1967). It is not necessary to consider the cubic surface in the Alston Block, because this surface only adds I per cent to the explanation of the variability of the observed data and is very similar to the quadratic surface in form. The Askrigg Block differs from the Alston Block in that the quadratic surface in the Askrigg Block only adds 4 per cent to the amount of explanation given by the linear surface. The pattern of the summit surface is evidently more complex in the Askrigg Block and the cubic surface must be considered. The greater complexity could result from several factors. The originally fairly flat surface could have been warped in a more complex manner. It could have had greater relief before it was warped, or it could have been more deeply dissected and its summits lowered since it was uplifted and warped. The pattern of residuals provides the best information for studying these possibilities, which will be considered in a later section. The cubic surface, however, provides a reasonably good description of the summit surface. The F value for this surface is highly significant, so that it is a genuine one. In all three surfaces fitted to the Askrigg Block data, the x variable accounts for by far the greater part of the explanation. I11 all three surfaces, the x value accounts for 58 per cent of the variability. In the cubic surface, the x3 variable also adds a significant amount to the total. The easterly tilt of the surface in the Askrigg Block is as conspicuous on the ground as it is in the trend surface. The subdivision of the Askrigg Block into a northcrn and a southern part for trend-surface ailalysis shows that the northern part is simpler than the southern. The quadratic surface of the northern part accounts for 83 per cent of the variability, while this surface in the southern part only accounts for $5 per cent of the variability. PENNINE EROSION SURFACES The Trend-Surface Maps The contours (100-foot (30 m) intervals) on the trend-surface maps are drawn so as to reduce the squared distances between the data values and the surface to a minimum. The equations defining the three surfaces for the two blocks are given in Table II. The equations

TABLEI1 Trrrid-Sur/ace Equations

Alsfott Block Linear: z = 2007.9337-25.43 x+ 17.19 y Quadratic: t = - 59.4907+ 18.473 x+ 123.3359 y-0.2624 x2-0.866 xy- 1.1954 y2 Cubic: t = - 166.8847f37.762 x+ 118.119y- 1.145 x2-560.6~xy- 1.469 Y~+O.OIOI X~-O.OI~~ X~Y+O.OIIIxy2 -0.0033 y3 Askr&g Block Linear: z = 2426.25 -26.754 r + 5.0023 y Quadratic: z = - 1598.1jo- 52.23 x-44.47 y-0.1776 x2 -0.514 xy-0.400 y2 Cubic: z = -20189.62+477.07 x+ 525.49 y- 5.609 x2 - 5.733 xy-4.192 y2 +0.029 x3 +o.0096 xZy +o.0309 xy2 +o.m85 y3 enable the elevations of each point on the three surfaces to be calculated and contours can be drawn for the surfaces. Figures 2 and 3 show the linear and quadratic trend-surface maps for the Alston Block. The -linear map shows a plane surface sloping down to the north-east. The quadratic sur- face has a double curvature. The increasing gradient to the south-east 20 in the southern part of the area and to the north in the northern part reflects the importance of the x2 and y2 variables. The high area from which the surface slopes down is centred on Cross . , which is nearly 915 m high, and

Mickle Fell to its south-east, lie near 360 370 380 390 4W 410 420 the in the west margin FIGURE2-L~near trend-surface map of the Alston Block. The of the area. Uplift took place along major faults are s~~own. this fault line in the late Tertiary or early Quaternary. The warping of the summit surface during uplift IS apparently closely reflected in the form of the summit surface. The uplift appears to be related to the structure of the underlying basement in which the granite has a mass deficiency, which has allowed elevation to take place (Bott, 1967). The summit erosion surface, as represented by the trend surface, is closely related to the structure. A few contours on the base of the Great Limestone are included in Figure 3. These contours run almost parallel to the trend-surface contours. There is, however, a steeper gradient on the structure surface than on the trend surface, the difference in level being 120 m in the west, increasing to 400 m in the east. The ridge whose axis slopes down to the north-east represents the anticline, which Trotter showed to be an important element of the structure of the uplifted peneplain. He indicated that this structural element of the area has affected the drainage pattern of the western part of the Alston Block in particular. It can be seen by comparing the main drainage lines, shown on Figure I, with the contour pattern on Figure 3 that the drainage is closely related to the trend-surface contours. The illajor streams flow very nearly at right-angles to the trend- surface contours. The South-Tyne, Derwent and Upper Tees exemplify the relationship. A possible reason for the relationship is the modification of the drainage pattern by the earth movement that led to the uplift and 10 20 30 40 50 60 70 --am warping of the surface. Trotter sug- lo- 560 this in his analysis of the -200 drainage development. If the relation- ship between the drainage pattern and 20- the trend surface implies that the streams flowed down the slopes of 30- ' ~540 the warped surface, then it follows that the surface prior to warping

40 must have been smooth and that the warping was fairly rapid. Only an exceptionally smooth surface would allow the streams to follow the slope

360 37~ 31~) 340 4ba 4io 420 produced by warping. Rapid warp-

FIGURE3-Quadratic trend surface of the Alston IJiocl. The lng prevent by the contours (~nfeet) on the Great Limestone are shown by dotted Streams in a former course and lines thus help to establish streams related to the slope of the surface. All three trend-surface maps for the Askrigg Block show the importance of the x variable in the strong easterly component of slope. Figure 4 shows the linear surface sloping down to

20 30 40 5,O 60 70 the east-north-east. The quadratic 1 zi2f?%r3% -' 5, --1 4'??-- i--- surface, shown in Figure maintains \ D - 60. -510 +.-:- " -o O a '1% the easterly slope, but there is also a ' I southerly element coming in at the I +R south-western part of the map. This -50C southerly element reflects the presence \ \ of the Craven faults in this area \\ \ Uplift took place along these at about the same time as the uplift along the Pennine fault to the west of the M+ -480 Alston Block. The cubic surface, shown in Figure 6, differs consider- / ably from the quadratic trend-surface -470 ,/ map. The easterly tilt is still dominant I in the north-eastern section of the ,/' / I, ,,, area, but a closed area of high eleva- 410 4 20 tion is now found in the central FIGURE4-Linear trend surface of the Askrlgg Block The major faults and the boundary of the reglon are shown Heights In feet western part of the Block, centred around the hills of Shunner Fell, East and Dodd Fell (Fig. I). The contours on the base of the Main Limestone, some of which are shown in Figure 6, do not fit so closely as those of the Alston Block The easterly PENNINE EROSION SURFACES s3 tilt is, however, readily apparent in the structure. The presence of the Wensleydale granite accounts for the uplift of the Askrigg Block in 3 similar way to the Alston Block.

In the Askrigg Block, there is 20 30 4p 50 60 70 also a strong relationship between the pattern of the contours on the cubic trend-surface map and the major streams, which flow almost 70 at right-angles to the contours. The streams that flow normal to the

contours include the Ure in 80~ Wensleydale, the Wharfe, the Ribble, the Clough and the Eden, all of which drain from the enclosed area of high elevation. This high area was also the centre from which the local Dales ice-cap emanated. The Dales ice mass was ? k,m

sufficiently powerful to prevent any llo. far-travelled ice from encroaching 370 380 3 90 400 410 4 20 FIGURE5-Quadratic trend surface of the Askrigg Block. Heights in into the Askrigg Block area. feet. It has been shown that the summit surface of the Alston Block and Askrigg Block is warped in varying degrees of complexity. It is now necessary to indicate how the surface can be linked from one block to

the other. The map, Figure 7, shows 2,0 30 40 50 60 70 the link of the cubic surface of the

Askrigg Block with the quadratic 60 surface of the Alston Block. The adjustment necessary to link the two surfaces is shown by dotted lines. The position of the trough formed by the linkage is shown on the Figure, as ,,- well as the line of the Stainmore trough, the Cotherstone syncline and the Middleton Tyas anticline, follow- 90- ing Bott (1967) and Trotter (1929). The position of the Stainmore trough closely approximates to the position lW- of the trough formed by the linking of the two trend surfaces. The axis of this V-shaped trough slopes down to the east-north-east through ~~~~~~d FIGURE6-Cubic trend surface of the Askrigg Block. The contours (in feet) on the Main Limestone are shown by dotted lines. Castle. The contours suggest- - a col between the two blocks at a height between 490 111 and 550 m at G.R. 385512 approximately. This point lies close to the position of the Stainmore Pass. The trend-surface trough is followed in part by the River Greta and the Tees below Barnard Castle. The trend-surface analysis confirms the structural control of the landscape. The up-arching of the Teesdale anticline, shown on the Alston Block quadratic surface, is matched by the down- warping of the Stainmore syncline at the junction of the Blocks, shown by the form of the link between the two trend surfaces.

The Pattern of Residrrals The residuals are tlie differences between the calculated heights, using the trend-surface equations, and the observed heights used in the analysis. The quadratic surface residuals for the data points in the Alston Block have been plotted and contoured. The result is shown in Figure 8. The residuals are small over much of the north-eastern part of the area, but to the south-west there are deviations ,,, exceeding 120 tn in a few places. The positive deviations are situated near the western border of the area. They represent the higher hills, , Cross Fell and . These hills were interpreted by -560 Trotter as monadnocks rising above the summit surface and trend-surface 370 380 390 400 410 420 analysis confirms this interpretation. FIGURE?-The junction between the quadratic trend surface of The negative deviations, exceed- the Alston Block and the cubic trend surface of the Askrigg . ing 120 m, are largest in the Tees Block. The linking of the two surfaces is shown by dotted lines and the structural features by dashed lines. Heights in feet. valley above Barnard Castle. A belt of negative anomaly also runs along the line of the Greta valley, following close to the Stainnlore syncline. There is a more extensive area of negative anomalies in the north-eastern part, running from south-east to north-west, but these residuals are mostly fairly small. Rather larger negative residuals also occur in the upper part of the South Tyne valley and the Wear valley above Bishop Auckland. The close association of the strongest negative anomalies with the main river valleys is also found in the Askrigg Block and will be considered later. The residuals for the Askrigg Block from the cubic surface are shown in Figure 9. The slightly poorer fit of this surface gives rather larger residuals, which exceed 180 m in places. The major belt of positive residuals extends east to west across the area south of Wensleydale. The maxi~numvalue of over 180 111 applies to the summit of Great . This and those farther west that also have positive anomalies, including and Whernside, represent monadnocks rising above the summit surface of the Askrigg Block. There is also a stretch of positive anomalies on the high ground between Swaledale and Wensleydale, extend- ing across the headwater area of the Eden. This belt in part follows the Middleton Tyas anti- cline. The slope of the residual surface is very steep in the Eden headwater area, changing in elevation from -60 m to f60 m in a very short distance. The steep gradient at this point shows that the actual summits vary in height more than the trend surface, which also slopes north at this point. It is significant that it is in this area that the River Eden, flowing north, has captured the south-flowing headwaters of the River Ure near Aisgill at G.R. 783966. The main area of negative residuals is linear and follows the valley of the River Ure. PENNINE EROSION SURFACES 5 5

Negative residuals exceed 120 m between Aysgarth and Hawes, and 60 m throughout the length of Wensleydale and along the upper part of Garsdale, an area which also formerly drained to the Ure. Another area of negative residuals extends north-eastward into Dentdale from Newby Head. A series of cols in this area links the drainage of the Ribble, Greta and Ure valleys. The negative residuals cover the area of the Wensleydale Granite, as shown in Figure 9, following Bott (1967), while the belt of magnetic basement rocks is associated with positive anomalies. The relationship between the deep structure and the trend-surface residuals is probably made through the influence of the basement on the structure of the Carboniferous rocks, which is reflected, to a certain extent, in the form of the trend surface.

FIGURE8-The residuals from the quadratic trend surface in the Alston Block. Contours are at zoo feet (60 m) intervals.

The belts of negative residuals that lie along the major valleys in the Askrigg Block and the Alston Block can be variously explained. They could result from minor warping across the general trend of the uplifted surfaces. Subsidiary warping of this type would then account for the positions of the major valleys. Alternatively, the hills on either side of the major valleys could have been lowered relative to those nearer the major interfluves by processes of slope recession or glacial erosion. There is some evidence in favour of the first hypothesis, in that the level of the Underset Linlestone outcrop is lower on either side of Wensleydale than it is on the interfluve between the Swale and the Ure and between the Ure and the Wharfe. The difference in level is of the order of 30-60 m, but even warping of this extent could explain the exact location of the major valleys if they were initiated on the general slope of the erosion surface as suggested by the trend-surface analysis. Conclusion In the introduction, four purposes of trend-surface analysis were mentioned. The conclu- sions to be drawn from this example of trend-surface analysis will be reviewed under these four headings. Description. Trend-surface analysis supplies an objective mathematical description of the summit erosion surface in the Central Pennines. In the Alston Block this surface can be best described by the quadratic trend-surface equation, which defines a surface having a double curvature, sloping down to the north and south-east from a belt of high ground in the south- west of the area. This surfacc fits the data points very well, accounting for nearly go per cent of their variability. It is concluded that the summit surface of the Alston Block represents a warped surface above which there rise several monadnocks. It could be argued that, instead of there being one warped surface, there is really a

2W series of closely spaced levels graded to a number of base-levels. If this were the situation, then it would be expected that the residuals would form belts of alternating positive and negative anomalies. This pattern does not appear, and therefore the second alternative is not so plausible as the first. In the Askrigg Block the summit erosion surface is best described by the cubic trend surface. The surface in this area is also warped and the degree of warping is rather more complex than in the Alston Block. The more 370 380 390 4~ 410 420 complex cubic surface accounts for FIGURE9-The residuals from the cubic trend surface in the rather less of the variability than the Askrigg lock. Contours are at zoo feet (60 m) intervals. quadratic surface in the Alston Block. The cubic surface in the Askrigg Block has a closed high area in the central western part, while the rest is dominated by the easterly slope of the surface, apart from the extreme south where the influence of the Craven faults is felt. The more complex pattern of the surface in the Askrigg Block may reflect its less complete structural isolation compared with the Alston Block, which is cut off by recent faults or structural warps along three of its margins. The Askrigg Block is joined westward to the Howgill and across the Dent fault. This fault, which forms the north- western margin of the area, has not moved since the Mesozoic at the latest. Uplift during the Tertiary has taken place across the Dent fault and not along it. The downthrow along the Dent fault is on the east, unlike the other faults that have moved more recently, the downthrow along which is to the west. This structural pattern may help to account for the greater com- plexity of the summit surface of the Askrigg Block. The northern part of the block has also been considerably faulted, thus adding to the structural complexity. The faulting helps to account for the complexity of the summit surfacc in this area where lithological control of the landscape is strong. Reconstrt~ction.Trend-surface analysis provides a useful method of reconstructing an erosion surface much of which has been subsequently removed by erosion. The isolated summit heights can be linked to form a continuous surface. Points which are anomalously higher or PENNINE EROSION SURFACES 5 7 lower than the general trend do not disturb the general trend pattern seriously, and can be recognized by their high residual values. The trend-surface maps provide a good picture of the erosion surfaces after they had been warped and uplifted, but before they had been dis- sected by subsequent erosion to any marked degree. Testit~ga hypothesis. The earlier work in the Alston Block by Trotter (1929) suggested that there was an uplifted and warped erosion surface in the area, above which monadnocks rose as isolated high points. This hypothesis can be tested by trend-surface analysis. The high proportion of the variability explained by the quadratic surface in the Alston Block supports and strengthens the evidence put forward by Trotter. The form of this surface closely follows his views of the warping of the surface in this area. The presence of monadnocks is also con- firmed by the pattern of the positive residuals in the western part of the area. In the Askrigg Block, the hypothesis of an easterly tilted erosion surface is also confirmed by the analysis, which shows how much of the explanation of the variability of the data is accounted for by the x variable. This variable defines the tilt in an easterly direction. The view that the surface in this area has been warped in a fairly complex manner is also confirmed by the trend-surface analysis, because the cubic surface is needed to raise the percentage variability explained to just below 80 per cent. The presence of monadnocks is also confirmed by the pattern of positive residuals in the Askrigg Block. The residuals do not support the view that the summit surface was cut at different times with successively lower base-levels. Thus the model of a warped erosion surface is supported by the analysis. Search procedure. The residuals are the most useful part of the analysis from the point of view of search procedure, since they can reveal patterns that were not obvious from other work. In this area the pattern of the negative residuals is particularly valuable. The closeness' with which the main lines of drainage follow the pattern of the trend-surface contours has been noted. This agreement suggests that the streams were initiated on the warped surface, or at least adjusted to it. The negative residuals give some clue as to the reason for the exact position of the streams on the surface. It has been suggested that the linear belts of negative residuals follow minor downwarps across the general trend of the warped surface. These minor features are confirmed by the heights of the outcrops of particular beds of limestone in the area. These beds are lower in the areas where the negative residuals form linear belts along some of the major valleys. This applies particularly to Wensleydale, which is the largest valley in the Askrigg Block.

ACKNOWLEDGEMENTS The computer program which was used to calculate the trend-surface data was kindly made available by Dr. J. T. Andrews, formerly of the Geographical Branch, Department of Energy, Mines and Resources, Ottawa, Canada, where the program was run. Grateful acknowledgement is made to the Univcrsity of Nottinghanl for .I grmt towards the cost of illustrations.

REFERENCES

BOTT,M. H. P. (1967) 'Geophysical investigations of the Northern Pennine Basement rocks', Proc. Yorks. geol. Soc. 36, 139-68 CHORLEY,R. J. and P. HAGGETT(1965) 'Trend-surf~cemapping in geographical research', Trar~s.Illsf. Br. Geogr. 37, 4747 KING,C. A. M. (1963) 'Some problenis concerning marine planation and the formation of erosion surfaces', Trarls. 111sf. Br. Geogr. 33, 29-43 KRUMBEIN,W. C. and F. A. GRAYBILL(1965) Art ir~trodrictiorrto statisfical models irl geology (New York) TROTTER,F. M. (1929) 'The Tertiary uplift and resultant drainage of the Alston Block and adjacent areas', Proc. Yorkr. geol. Soc. 22, 161-80 RBsuMB-L'atlalyse des tendattces des surfaces d'aplanissemerzt vers le centre de la chaine des Penttitter. La planitk des surfaces d'aplanissement somniitales des massifs d'Alston et d'Askrigg, tousles deux sitkes vers le centre de la chaine des Pennines, a kt6 ktudike en vue d'identifier la direction best-fit des surfaces. Siparks par le synclinal du Stainniore, les deux massifs forment deux blocs definis par des failles iniportantes au nord, B l'ouest et au sud. On a trouvk qu'une surface quadratique explique 89 pour cent de la variabilitk des donnks de la surface d'Alston, tandis que c'est une surface cubique qui explique 78,s pour cent de la variabilitk de la surface sonimitale d'Askrigg. Sur le premier massif la pente de la surface best-fit commence dans une zone &leveeen voisinage des failles et plonge vers le nord et vers le sud-est; la pente plus inportante du massif d'Askrigg plonge ver l'est en partant d'un point Clevk dans l'ouest. Les deux surfaces se rkunissent au fond du fossk qui marque, approximativenient, la ligne du synclinal du Stainmore. Au dessus de la surface sommitale dkformke ressortent des residuels qui reprksentent des monadnocks (residuels positifs). Des combes, qui sont entammkes dans la surface (residuels nkgatifs) et qui representent peut-ttre des petits sillons situks pour la plupart B c6tk des plus grandes vallkes, auraient jouk un r61e dans l'ktablissement d'un rkseau hydrographique. Les cours des ruisseax les plus importants sont perpendiculiires aux courbes de niveaux de la surface best-fit; il se peut que ces cours fussent influences par la forme de la surface sommitale exhausske et dkformke. Par cette mkthode d'ktudier la tendance d'une surface (trend surface analysis) on peut dkcrire prkciskment en utilisant les mathkmatiques les trois dinlensions d'une surface d'aplanissement. Ainsi on peut rkconstituer une surface aujourd'hui fragmentke. Cette mkthode servirait B mettre B l'kpreuve les hypothkses et servirait aussi comme prockde de recherche. Nous traiterons tous ces aspects de trend surface analysis au cours de la discussion.

FIG. I-Carte de la chaine des Pennines, partie centrale, niontrant les massifs d'Alston et d'Askrigg avec les complexes des failles les plus importantes FIG. 2-La surface alignke du massif d'Alston, avec les grandes failles FIG. 3-La surface quadratique du massif d'Alston. Les courbes de niveau (feet) sur le

Zus~~~~~~~ssu~~-Trer~dobe~ac~~eltana~yseauf die GipfelerosiorrsoberjacIte irr dern mittleren Peirrtirregebirge. Trendober- flachenanalyse wird auf die GipfelerosionsoberflPche der in den1 mittleren Penninegebirge befindlichen Alston- und Askrigg-Blocke angewandt. Die Blocke bilden klar umrissene, durch die Stainmore-Syncline getrennte Struktureinheiten. die in1 Norden, im Westen und im Suden durch eine Reihe grosser Verwerfungsspalten begrenzt sind. Durch die quad- ratische OberAache wird die Variabilitat der Angaben in. dem Alston-Block zu 89 prozent wie durch die kubische Oberflache die Variabilitat in dem Askrigg-Block zu 78,s prozent erklart. Die Trendoberflache in den1 Alston-Block fa11t von einem hochliegenden, den Verwerfungsspalten angrenzenden Gebeit nach Norden und Sudosten ab, wahrend der Hauptabhang der kubischen OberAache des Askrigg-Blocks von einem isolierten, im Westen befindlichen, hoch- liegenden Gebiet nach Osten geht. Die beiden Oberflichen verbinden sich in einer tiefen Neiderung, die ungefallr so situiert ist wie-die Stainmore-Vertiefung. Die positiven Residualien stellen Monadnocks dar, die uber die krumme Gipfeloberflache hinaufragen. Die negativen Residualien liegen meistens in den Haupttalern und diirfen kleine Abwarts- kriimmungen darstellen, die beim Bau des Abwasserungsmusters eine Rolle spielten. Die Hauptgewasser fliessen beinahe rechtwinklig zu den Trendoberflachenkonturen und durfen durch die Form der hochgehobensn und gekrumnlten GipfeloberflSche beeinflusst worden sein. TrendoberAachenanalyse kann verwendet werden, unl ,cine dreidimensionale Oberflache nlathematisch zu beschreiben; sie kann verwendet werden, urn 2ine teilweise zergliederte Oberflache wider- herzustellen; sie kann auch verwendet wcrden, um eine Hypothesc zu prufcn und nls cine Suchprozcdur. Alle Aspckte werden in der Analyse angewandt, die in den1 Artikel besprochen wird.

ABB. I-Karte des mittleren Penninegebirges mit Darstellung der Alston- und Askrigg-Blocke und dergrossenverwer- fungsspalten ABB. Z-Die lineare Oberflache des Alston-Blocks mit den grossen Verwerfungsspalten ABB.3-Die quadratische Oberflache des Alston-Blocks. Die Hohenlinien (fect) an den1 'grossen Knlkstcin' sind niit Punkten angezeigt. PENNINE EROSION SURFACES 59 ABB.4-Die lineare Oberflache des Askrigg-Blocks mit Darstellung der grossen Verwerfungsspalten und der Regional- grenzen. Hohen sind in 'feet'. ABB. 5-Die quadratische Oberflache des Askrigg-Blocks. Hohen sind in 'feet'. ABB. 6-Die kubische Oberflache des Askrigg-Blocks. Die Hohenlinien (feet) an dem 'Hauptkalkstein' sind mit Punkten angezeigt. ABB. 7-Kombination der quadratischen Oberflache des Alston-Blocks mit der kubischen Oberflache des Askrigg- Blocks. Hohen sind in 'feet'. ABB. 8-Die quadratische Oberflache des Alston-Blocks mit Residualien. Der Abstand zwischen den Hohenlinien betragt zoo Fuss (60 m). ABB.-Die kubische Oberflache den Askrigg-Blocks mit Residualien. Der Abstand zwischen den Hohenlinien betragt zoo Fuss (60 m).