Global Warming? Effects on Glaciers

Transcription

1 This project has been funded with support from the European Commission. This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein. Global Warming? Effects on Glaciers Menntaskólinn í Kópavogi, Iceland HAK Grazbachgasse, Graz, Austria Description of the Project Open ê Close Print Description Openê Close This project is about analyzing the melting of glaciers in Austria and Iceland. The most important question this project should answer is: Site: Desktop 1

2 "Do Icelandic glaciers behave the same way as Austrian ones concerning the retreat due to global warming?" Finding the proper data was quite difficult but in the end the underlying data is received from various institutions: KF Uni Graz, UNI Innsbruck, ZAMG, Alpenverein. A comparison of the collected data of the Icelandic glacier "Hyrningsjökull" and the Austrian glacier "Pasterze" is carried out by using the program Mathematica especially the application M@th Desktop. Therefore various mathematical models for the year-temperature, year-treatment, temperature-retreat are created. The project gives information about the forecasted retreat for both glaciers in the year Correlation of Glacier Movement, Hyrningsjökull, Iceland - Pasterze, Austria Open ê Close Print Reading the Excel files of Hyrningsjökull, Iceland (IS) Openê Close Here is a picture from the Snaefellsjökull near Reykjavik. But we work only with the data of Hyrningsjökull. This is only the lowest part of the big glacier, an icefall. Site: M@th Desktop 2

3 Source: The picture below shows you how the whole glacier looses its thickness between the years Site: Desktop 3

4 Regressionmodels Years - Temperature (IS) Openê Close We have saved all informations under the CSV MD-DOS format. We read in the temperature data with our palette, button Import. The data ware taken in Stykkishólmur. The period of time is All data were provided from private communications of our teacher Guðrún Angantýsdóttir with the glacier experts on Iceland. Site: M@th Desktop 4

5 path = "D:\\C1 2008\\Meeting, , Reykjavik 2\\Hyrningsjokull Year Temperature.csv"; strdata = Import@path, "Words"D; data = HstrPos = StringPosition@, ","D@@1, 1DD; 8ToExpression@ StringTake@, 81, strpos 1<D, ToExpression@ StringTake@, 8strPos + 1, StringLength@ <D<L &ê@ strdata 8retr, temp< = Transpose@ data; , 3.8<, 81932, 4.6<, 81933, 5.1<, 81934, 4.1<, 81935, 4<, 81936, 3.9<, 81937, 4.1<, 81938, 4.2<, 81939, 5.1<, 81940, 4<, 81941, 5.2<, 81942, 4.5<, 81943, 3.3<, 81944, 3.7<, 81945, 4.9<, 81946, 5.1<, 81947, 4<, 81948, 4.1<, 81949, 3.2<, 81950, 3.9<, 81951, 3.4<, 81952, 3.6<, 81953, 4.4<, 81954, 3.9<, 81955, 3.6<, 81956, 4.4<, 81957, 4.2<, 81958, 3.9<, 81959, 4.2<, 81960, 4.8<, 81961, 4.2<, 81962, 3.8<, 81963, 3.7<, 81964, 4.8<, 81965, 3.9<, 81966, 3<, 81967, 3<, 81968, 3.2<, 81969, 2.6<, 81970, 3<, 81971, 3.1<, 81972, 4.3<, 81973, 3.1<, 81974, 4.3<, 81975, 3.3<, 81976, 4<, 81977, 3.6<, 81978, 3.8<, 81979, 2.3<, 81980, 3.8<, 81981, 2.5<, 81982, 3.2<, 81983, 2.6<, 81984, 3.6<, 81985, 3.8<, 81986, 3.5<, 81987, 4.7<, 81988, 3.3<, 81989, 3.4<, 81990, 3.9<, 81991, 4.5<, 81992, 3.9<, 81993, 3.8<, 81994, 3.4<, 81995, 3<, 81996, 4.4<, 81997, 4.1<, 81998, 3.7<, 81999, 3.8<, 82000, 4.1<, 82001, 4.4<, 82002, 4.4<, 82003, 5.4<, 82004, 4.9<, 82005, 4.8<< We are going to visualize the data. MDPlotData@data, PlotStyle 8Red, PointSize@.02D<, FrameLabel 8"Years", "Temperature H CL "<, PlotJoined False D 75 Data Points 5 T emperature H CL Years Site: M@th Desktop 5

6 Graphics Answer: The data are wide spread. The temperature range goes from 2.3 C to 5.5 C between 1931 to It is not quite clear, which trend can be seen in the data. So we try different models. A quadratic model could be our first guess. However, we try first a linear model. A linear model will probably not work. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit1@x_d = MDNonlinearFit@data, a x + b, H model L 8x<, 8a, b< D êê ChopA, E &; H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit1@xD<, 8x, start, stop<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x 75 Data Points 5.0 Temperature H±CL Years Answer: The fit does not seem to reflect the trend. The dots are too much scattered Site: M@th Desktop 6

7 from the regression line. A quadratic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, 8x<, 8a, b, c < E êê ChopA, E &; H model L H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD<, 8x, start, stop<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x 2 75 Data Points 5.0 Temperature H±CL Years Site: M@th Desktop 7

8 Answer: The sum of squared errors was reduced from about 32 (linear model) to about 25. It also seems to display the trend graphically better than the linear model. What attracts attention is that the average temperature per year fell from 1930, 4.8 C to 3.6 in 1971, then it increased to 4.4 C in It was getting colder in the seventieth on Iceland. We try to make a forecast for the year 2020 if the weather conditions remain more or less the same. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, 8x<, 8a, b, c < E êê ChopA, E &; H model L H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD, fit1@xd<, 8x, start, 2025<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 2020, fit2@2020d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<, PlotRange 8 2.3, 6 <D Iy i ŷ i M 2 Sum of Squared Error : x x 2 Site: M@th Desktop 8

9 Data Points 5.5 Temperature H±CL Years in comparison to the linear model fit1[x] Answer: If the temperature data in the years to come will not change considerably then according to our quadratic model, the average temperature in Stykkishólmur is predicted with 5.3 C. On the plot we can compare both models. The linear model predicts an average temperature of 3.6. But there is no reason to moti vate such a drop in temperature. So we skip the linear model. Note: Sometimes you do not see the blue point, because it is outside the plot range. In this case use the PlotRange button from the Plot Helper palette. A cubic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit@x_d = MDNonlinearFitAdata, a x 3 + b x 2 + c x + d, H model 3rd order L 8x<, 8a, b, c, d < H parameters L E êê ChopA, E &; Site: M@th Desktop 9

10 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit@xD<, 8x, start, stop<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x x 3 75 Data Points 5.0 Temperature H±CL Years Answer: With Mathematica 7 one gets a good model of third order for the increase in temperature. Now a forecast with this model: H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit3@x_d = MDNonlinearFitAdata, a x 3 + b x 2 + c x + d, H model 3rd order L 8x<, 8a, b, c, d < H parameters L E êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit3@xD<, 8x, start, 2025<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 2020, fit3@2020d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<, PlotRange 8 2.3, 10 <D Site: M@th Desktop 10

11 Iy i ŷ i M 2 Sum of Squared Error : x x x Data Points 9 T emperature H CL Graphics fit3@2020d Years Answer: This model is more likely to fail. There are no hints that the temperature will rise so quickly. The predicted value is 7.81 C in Answer: To describe the year- temperature data for the Hyrningsjökull we have chosen the quadratic model, fit2[x] = x x 2. x in years, fit2[x] in C. The average temperature in Stykkishólmur is predicted with 5.3 C. Regressionmodels Years - Glacier Movement (IS) Openê Close A linear model The data for the movement of the glacier in meters are provided from All data were provided from private communications of our teacher Guðrún Angantýsdóttir with the glacier experts on Iceland. path = "D:\\C1 2008\\Meeting, , Reykjavik 2\\Hyrningsjokull Year Movement.csv"; Site: M@th Desktop 11

12 strdata = Import@path, "Words"D; data = HstrPos = StringPosition@, ","D@@1, 1DD; 8ToExpression@ StringTake@, 81, strpos 1<D, ToExpression@ StringTake@, 8strPos + 1, StringLength@ <D<L &ê@ strdata 8retr, temp< = Transpose@ data; , 3<, 81932, 22<, 81933, 70<, 81934, 52<, 81935, 32<, 81936, 103<, 81937, 30<, 81938, 40<, 81939, 21<, 81940, 35<, 81941, 22<, 81942, 17<, 81943, 18<, 81944, 37<, 81945, 38<, 81946, 24<, 81947, 23<, 81948, 30<, 81949, 20<, 81950, 5<, 81951, 15<, 81952, 9<, 81953, 3<, 81954, 48<, 81955, 26<, 81956, 14<, 81957, 18<, 81958, 19<, 81959, 40<, 81960, 35<, 81961, 15<, 81962, 50<, 81963, 40<, 81964, 10<, 81965, 37<, 81966, 13<, 81967, 25<, 81968, 22<, 81969, 42<, 81970, 26<, 81971, 25<, 81972, 9<, 81973, 14<, 81974, 12<, 81975, 31<, 81976, 2<, 81977, 7<, 81978, 37<, 81979, 1<, 81980, 14<, 81981, 24<, 81982, 37<, 81983, 27<, 81984, 40<, 81985, 31<, 81986, 51<, 81987, 35<, 81988, 30<, 81989, 2<, 81990, 9<, 81991, 2<, 81992, 13<, 81993, 12<, 81994, 3<, 81995, 13<, 81996, 3<, 81997, 10<, 81998, 30<, 81999, 20<, 82000, 24<, 82001, 41<, 82002, 36<, 82003, 92<, 82004, 87<, 82005, 28<, 82006, 8<, 82007, 45<, 82008, 21<, 82009, 12<< First have a look at the data. MDPlotData@data, PlotStyle 8Red, PointSize@.02D<, FrameLabel 8"Years", "Movement HmL"<, PlotJoined False D Data Points Movement HmL Years Site: M@th Desktop 12

13 Answer: We can see from the data that in 1936 the biggest decrease of the glacier was about 102 meters and the biggest increase was in 1986 with 52 meters. The decrease of the glacier slowed down until about Then it started to grow until Then it decreased again but at a big rate per year. We are quite sure that the linear model will fail. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit1@x_d = MDNonlinearFit@data, a x + b, H model L 8x<, 8a, b< D êê ChopA, E &; H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit1@xD<, 8x, start, stop<, FrameLabel 8"Years", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x Data Points Movement HmL Years A quadratic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, H model L Site: M@th Desktop 13

14 8x<, 8a, b, c < E êê ChopA, E &; H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD<, 8x, start, stop<, FrameLabel 8"Years", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x Data Points Movement HmL Years Answer: We have 3 arguments to believe that quadratic model fits very good in reality. 1. the sum of squared errors drops from around (linear model) to (quadratic model) 2. Graphically the quadratic model fits the data very well. 3. From the year temperature analysis we know that average temperature in the 1970s was quite low. So it makes sense that the glacier started to grow in the 1970s. We try to do a forecast for H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, H model L 8x<, 8a, b, c < H parameters L E êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; Site: M@th Desktop 14

15 8x, start, 2025<, FrameLabel 8"Years", "Movement HmL"<, Epilog 8Red, Point data, Blue, D, 2020, <D<, PlotStyle 88DarkGreen, Iy i ŷ i M 2 Sum of Squared Error : x x Data Points M ovement HmL Years Graphics fit2@2020d Answer: According to our quadratic model the glacier will retreat with about 60 meters in The assumption of our model is that the data between 2005 and 2020 will follow the trend we have detected in the temperature data. How ever it will be interesting to get a data up to 2010 to increase the quality of our forcast. A cubic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit3@x_d = MDNonlinearFitAdata, a x 3 + b x 2 + c x + d, H model 3rd order L 8x<, 8a, b, c, d < H parameters L E êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; Site: M@th Desktop 15

16 8x, start, 2023<, FrameLabel 8"Years", "Movement HmL"<, Epilog 8Red, Point data, Blue, D, 2020, <D<, PlotStyle 88DarkGreen, PlotRange 8 55, 250 <D Iy i ŷ i M 2 Sum of Squared Error : x x x Data Points 0 M ovement HmL Years Graphics fit3@2020d Answer: This model seems to be unrealistic, the retreat of the glacier is -142 m much to high depending on the years before. If we have chosen the 3rd model of the temperature this would make sense. However we see from our modeling that very good model for the temperature is absolutely necessary. Answer: To describe the year- movement data for the Hyrningsjökull we have chosen the quadratic model, fit2[x] = x x 2. x in years, fit2[x] in meters. The predicted retreat in 2020 is about 60 m. Regressionmodels Temperature - Glacier Site: M@th Desktop 16

17 Movement (IS) Openê Close A linear model We are going to read in the data. path = "D:\\C1 2008\\Meeting, , Reykjavik 2\\Hyrningsjokull Temperature Movement.csv"; strdata = Import@path, "Words"D; data = HstrPos = StringPosition@, ","D@@1, 1DD; 8ToExpression@ StringTake@, 81, strpos 1<D, ToExpression@ StringTake@, 8strPos + 1, StringLength@ <D<L &ê@ strdata 8temp, retr< = Transpose@ data; 883.8, 3<, 84.6, 22<, 85.1, 70<, 84.1, 52<, 84, 32<, 83.9, 103<, 84.1, 30<, 84.2, 40<, 85.1, 21<, 84, 35<, 85.2, 22<, 84.5, 17<, 83.3, 18<, 83.7, 37<, 84.9, 38<, 85.1, 24<, 84, 23<, 84.1, 30<, 83.2, 20<, 83.9, 5<, 83.4, 15<, 83.6, 9<, 84.4, 3<, 83.9, 48<, 83.6, 26<, 84.4, 14<, 84.2, 18<, 83.9, 19<, 84.2, 40<, 84.8, 35<, 84.2, 15<, 83.8, 50<, 83.7, 40<, 84.8, 10<, 83.9, 37<, 83, 13<, 83, 25<, 83.2, 22<, 82.6, 42<, 83, 26<, 83.1, 25<, 84.3, 9<, 83.1, 14<, 84.3, 12<, 83.3, 31<, 84, 2<, 83.6, 7<, 83.8, 37<, 82.3, 1<, 83.8, 14<, 82.5, 24<, 83.2, 37<, 82.6, 27<, 83.6, 40<, 83.8, 31<, 83.5, 51<, 84.7, 35<, 83.3, 30<, 83.4, 2<, 83.9, 9<, 84.5, 2<, 83.9, 13<, 83.8, 12<, 83.4, 3<, 83, 13<, 84.4, 3<, 84.1, 10<, 83.7, 30<, 83.8, 20<, 84.1, 24<, 84.4, 41<, 84.4, 36<, 85.4, 92<, 84.9, 87<, 84.8, 28<< We will have a look at the data. MDPlotData@data, PlotStyle 8Red, PointSize@.02D<, FrameLabel 8"Temperature H CL", "Movement HmL"<, PlotJoined False D Site: M@th Desktop 17

18 50 75 Data Points 25 M ovement HmL Temperature H CL Graphics Answer: According to the data we expect that with rising temperature the retreat of the glacier will grow. We are trying a linear model. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit1@x_d = MDNonlinearFit@data, a x + b, H model L 8x<, 8a, b< H parameters L D êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit1@xD<, 8x, start, stop<, FrameLabel 8"Temperature H CL", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x Site: M@th Desktop 18

19 75 Data Points Movement HmL Temperature H±CL Answer: The linear model confirms that our idea from the raw data is correct Answer: We found out by cursor measurement that on an average temperature of about 3.3 C the glacier is neither increasing or decreas ing. We are making a prediction for the year 2020, entering the 5.3 C from our temperature model. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit1@x_d = MDNonlinearFit@data, a x + b, H model L 8x<, 8a, b< D êê ChopA, E &; H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit1@xD<, 8x, start, stop<, FrameLabel 8"Temperature H CL", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 5.3, fit1@5.3d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : Site: M@th Desktop 19

20 x 75 Data Points Movement HmL Temperature H±CL fit1@5.3d Answer: According to the linear model we expect that at an average temperature of 5.3 C gives a retreat of approximately 40 meters. A quadratic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, H model L 8x<, 8a, b, c < H parameters L E êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD, fit1@xd<, 8x, start, stop<, FrameLabel 8"Temperature H CL", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 5.3, fit2@5.3d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x 2 Site: M@th Desktop 20

21 40 75 Data Points 20 Movement HmL Temperature H±CL Answer: The quadratic model forcasts about 51 meters retreat of the glacier at an average temperature of 5.3 C. We have done a double check with our models. The year-movement model suggests about 60 meters retreat in the year The temperature/movement model suggests about 51 meters retreat in the year Although the difference between 60 m and 51m is not neglectible one can say that our models for the forecast show a certain consistence. Answer: To describe the temperature - movement data for the Hyrningsjökull we have chosen the quadratic model, fit2[x] = x x 2. x in C, fit2[x] in meter. The average temperature in Stykkishólmur is predicted with 5.3 C and will show a retreat of about 5.1. Summary: According to the data our models confirm that the Hyrningsjökull retreat is about 51 meters according to the quadratic model until 2020 if the weather conditions do not change. We found out that the linear model is too simple. Site: M@th Desktop 21

22 Quadratic models gave a reasonable result. They tell us that in 2020 the avarage temperature will be around 5.3 C and the retreat ar ound 51 meters. The average of both models would forecast a retreat of around 55m Reading the Excel files of Pasterze, Austria (AT) Openê Close This picture shows the glacier Pasterze in different years. Source: pasterzenzunge We have saved all informations under the CSV MD-DOS format. We read in the data with our palette. We would like to thank Jakob Abermann (University of Innsbruck) who interpolated the temperature data for us for an altitude of 2500 meters. Original data are from: ECMWF, ERA40 - reanalysis data, Site: M@th Desktop 22

23 He also sent us the retreat data of "Pasterze" according to the annual report of the ÖAV (Oesterreichischen Alpenverein). Regressionmodels Years - Temperature (AT) Openê Close A linear model Note: We got the data from the altitude of the Kaiser Franz Josef Höhe, which is about 2500 m above sea level. So our average temperature data are different than those of Island, they are all negative because of the big height above sealevel. path = "D:\\C1 2008\\Meeting, , Reykjavik 2\\Pasterze Year Temperature.csv"; strdata = Import@path, "Words"D; data = HstrPos = StringPosition@, ","D@@1, 1DD; 8ToExpression@ StringTake@, 81, strpos 1<D, ToExpression@ StringTake@, 8strPos + 1, StringLength@ <D<L &ê@ strdata 8year, temp< = Transpose@ data; , <, 81971, <, 81972, <, 81973, <, 81974, <, 81975, <, 81976, <, 81977, <, 81978, <, 81979, <, 81980, <, 81981, <, 81982, <, 81983, <, 81984, <, 81985, <, 81986, <, 81987, <, 81988, <, 81989, <, 81990, <, 81991, <, 81992, <, 81993, <, 81994, <, 81995, <, 81996, <, 81997, <, 81998, <, 81999, <, 82000, <, 82001, <, 82002, <, 82003, <, 82004, <, 82005, <, 82006, <, 82007, <, 82008, 4.9<, 82009, << We are going to visualize the data. MDPlotData@data, PlotStyle 8Red, PointSize@.02D<, FrameLabel 8"Years", "Temperature H CL "<, PlotJoined False D Site: M@th Desktop 23

24 -4 40 Data Points T emperature H CL Graphics temp êê Sort Years , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 4.9, , , , , , , , , < Answer: The data are not wide spread, temperature range goes from -6.3 C to C. It is not quite clear, which trend can be seen in the data. So we try different models. A linear model could be our first guess. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit1@x_d = MDNonlinearFit@data, a x + b, H model L 8x<, 8a, b< D êê ChopA, E &; H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit1@xD<, 8x, start, stop<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : Site: M@th Desktop 24

25 x Data Points T emperature H CL Graphics Years Answer: The fit seems to reflect the trend. The dots are not so much scattered from the regression line. The average temperature per year rose from 1970, -6 C to -4.6 C in A quadratic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, 8x<, 8a, b, c < E êê ChopA, E &; H model L H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD<, 8x, start, stop<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x 2 Site: M@th Desktop 25

26 -4 40 Data Points T emperature H CL Graphics Years Answer: The sum of squared errors was reduced from about 7.88 (linear model) to about 7.82 It also seems graphically that the squared model displays the trend as good as the linear model. We try to make a forecast for the year 2020, if the weather conditions remain more or less the same. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, 8x<, 8a, b, c < E êê ChopA, E &; H model L H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD, fit1@xd<, 8x, start, 2025<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 2020, fit2@2020d <D, Red, Point@8 2020, fit1@2020d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x 2 Site: M@th Desktop 26

27 40 Data Points -4 T emperature H CL Graphics fit1@2020d fit2@2020d Years Answer: If the temperature data in the years to come will not change considerably then according to our quadratic model, the average temperature at the Sonnblick is predicted with -3.9 C. On the plot we can compare both models. The linear model predicts an average temperature of It is not clear at the moment which model is more reliable. We try another model, a cubic model. Note: Sometimes you do not see the blue point, because it is outside the plot range. In this case use the PlotRange button from the Plot Helper palette. A cubic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit3@x_d = MDNonlinearFitAdata, a x 3 + b x 2 + c x + d, H model 3rd order L 8x<, 8a, b, c, d < H parameters L E êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; Site: M@th Desktop 27

28 8x, start, stop<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x x Data Points T emperature H CL Graphics Years Answer: With Mathematica one gets a good model of third order for the increase in temperature. Now a forecast with this model: H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit3@x_d = MDNonlinearFitAdata, a x 3 + b x 2 + c x + d, H model 3rd order L 8x<, 8a, b, c, d < H parameters L E êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit3@xD, fit1@xd, fit2@xd<, 8x, start, 2025<, FrameLabel 8"Years", "Temperature H CL "<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 2020, fit3@2020d <D, PointSize@ D, Red, Point@8 2020, fit1@2020d <D, PointSize@ D, Green, Point@8 2020, fit2@2020d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Site: M@th Desktop 28

29 Iy i ŷ i M 2 Sum of Squared Error : x x x 3 40 Data Points -4 T emperature H CL Graphics Years 8fit1@2020D, fit2@2020d, fit3@2020d< , , < Answer: This model nearly predict the same temperature as as the linear model of the year - temperature data. So we will stick to the linear model for years-temperature data. Its definition x is simpler than x x x 3 Answer: To describe the year- temperature data for the Pasterze we have chosen the linear model, fit1[x] = x. x in years, fit2[x] in C. The average temperature at the Pasterze is predicted with C. Site: M@th Desktop 29

30 Regressionmodels Years - Glacier Movement (AT) Openê Close We are going to read in the data. path = "D:\\C1 2008\\Meeting, , Reykjavik 2\\Pasterze Year Movement.csv"; strdata = Import@path, "Words"D; data = HstrPos = StringPosition@, ","D@@1, 1DD; 8ToExpression@ StringTake@, 81, strpos 1<D, ToExpression@ StringTake@, 8strPos + 1, StringLength@ <D<L &ê@ strdata 8year, temp< = Transpose@ data; , 11.4<, 81971, 30.8<, 81972, 23.4<, 81973, 29.2<, 81974, 12<, 81975, 4.28<, 81976, 12.1<, 81977, 10.9<, 81978, 14.2<, 81979, 11.2<, 81980, 10.7<, 81981, 20.7<, 81982, 17.7<, 81983, 16.1<, 81984, 9.1<, 81985, 4.3<, 81986, 24.2<, 81987, 7.4<, 81988, 37.4<, 81989, 11.3<, 81990, 8.3<, 81991, 33.5<, 81992, 7.6<, 81993, 17.5<, 81994, 17.1<, 81995, 13.6<, 81996, 14.2<, 81997, 10.2<, 81998, 17.7<, 81999, 25.8<, 82000, 30.6<, 82001, 19.4<, 82002, 7.1<, 82003, 29.6<, 82004, 13.4<, 82005, 23.4<, 82006, 25.8<, 82007, 33.1<, 82008, 12.7<< First have a look at the data. MDPlotData@data, PlotStyle 8Red, PointSize@.02D<, FrameLabel 8"Years", "Movement HmL"<, PlotJoined False D Data Points -10 M ovement HmL Years Site: M@th Desktop 30

31 Graphics Answer: We can see from the data that in the seventieth it was a bit colder than in the first decade of the second millenium. The data are quite scattered. It will not be easy to find a model reflecting the trend best. A linear model What is the result of the linear fit? H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit1@x_d = MDNonlinearFit@data, a x + b, H model L 8x<, 8a, b< H parameters L D êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit1@xD<, 8x, start, 2022<, FrameLabel 8"Years", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@0.035D, Point@8 2020, fit1@2020d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x Data Points -10 M ovement HmL Graphics fit1@2020d Years Site: M@th Desktop 31

32 Answer: The linear model shows us clearly that the average decrease of the glacier is growing. The forecast tells us that in 2020 the decrease will be around -22 m. A quadratic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, 8x<, 8a, b, c < E êê ChopA, E &; H model L H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD<, 8x, start, stop<, FrameLabel 8"Years", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x Data Points -10 M ovement HmL Graphics Years Answer: It is really difficult to decide on which model describes best the trend. On the one hand the sum of squared error drops from 2886 (lin model) to 2738 (quad model). Site: M@th Desktop 32

33 On the other hand the quadratic model tells us that between 1970 and 1988 the glacier retreat was decreasing. That implies that the average temperature should be lower in that time preventing the glaicer from melting to much. Looking at all models of the temperature data between 1970 and 1988 above, it is clear that the temperature is steadily rising. This is the reason why we reject the quadratic model. We try to do a forcast for 2020 and compare it with the linear model. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, H model L 8x<, 8a, b, c < H parameters L E êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD, fit1@xd<, 8x, start, 2025<, FrameLabel 8"Years", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 2020, fit2@2020d <D, Red, Point@8 2020, fit1@2020d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x Data Points -10 M ovement HmL Years Site: M@th Desktop 33

34 Graphics A cubic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit3@x_d = MDNonlinearFitAdata, a x 3 + b x 2 + c x + d, H model 3rd order L 8x<, 8a, b, c, d < H parameters L E êê ChopA, E &; 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit3@xD<, 8x, start, 2023<, FrameLabel 8"Years", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 2020, fit3@2020d <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x x Data Points -10 M ovement HmL Graphics fit3@2020d Years Answer: This model seems to be unrealistic. The local maximum indicates that the average temperature is decreasing, so it is getting a bit cooler. Site: M@th Desktop 34

35 The local minimum would require a warmer period say between 1995 and 2015 which you cannot be derived from the temperature model. The forecast for 2020 is around - 14 m. Why should it be cooler then. There is no evidence that this will happen. So we reject the cubic model. Answer: To describe the year- movement data for the Pasterze we have chosen the linear model, fit1[x] = x. x in years, fit1[x] in meters. The predicted retreat in 2020 is about 22m m. Regressionmodels Temperature - Glacier Movement (AT) Openê Close We are going to read in the data. path = "D:\\C1 2008\\Meeting, , Reykjavik 2\\Pasterze Temperature Movement.csv"; strdata = Import@path, "Words"D; data = HstrPos = StringPosition@, ","D@@1, 1DD; 8ToExpression@ StringTake@, 81, strpos 1<D, ToExpression@ StringTake@, 8strPos + 1, StringLength@ <D<L &ê@ strdata 8temp, retr< = Transpose@ data; Site: M@th Desktop 35

36 , 11.4<, , 30.8<, , 23.4<, , 29.2<, , 12<, , 4.28<, , 12.1<, , 10.9<, , 14.2<, , 11.2<, , 10.7<, , 20.7<, , 17.7<, , 16.1<, , 9.1<, , 4.3<, , 24.2<, , 7.4<, , 37.4<, , 11.3<, , 8.3<, , 33.5<, , 7.6<, , 17.5<, , 17.1<, , 13.6<, , 14.2<, , 10.2<, , 17.7<, , 25.8<, , 30.6<, , 19.4<, , 7.1<, , 29.6<, , 13.4<, , 23.4<, , 25.8<, , 33.1<, 8 4.9, 12.7<< We will have a look at the data. MDPlotData@data, PlotStyle 8Red, PointSize@.02D<, FrameLabel 8"Temperature H CL", "Movement HmL"<, PlotJoined False D Data Points -10 M ovement HmL Temperature H C L Graphics Answer: According to the data we expect that with rising temperature the retreat of the glacier will grow. A linear model We are trying a linear model. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit1@x_d = MDNonlinearFit@data, a x + b, H model L Site: M@th Desktop 36

37 8x<, 8a, b< D êê ChopA, E &; H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit1@xD<, 8x, start, stop<, FrameLabel 8"Temperature H CL", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x Data Points -10 M ovement HmL Temperature H C L Graphics Answer: The linear model confirms that our idea from the raw data is correct. fit1@ 6D fit1@ 4.5D Answer: We found out by cursor measurement that on an average temperature of about -5 C the glacier retreat is about m. We are making a prediction for the year 2020, entering the C from our year - temperature model. H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit1@x_d = MDNonlinearFit@data, a x + b, H model L Site: M@th Desktop 37

38 8x<, 8a, b< D êê ChopA, E &; H parameters L 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit1@xD<, 8x, start, 4.1<, FrameLabel 8"Temperature H CL", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 4.18, fit1@ 4.18D <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x Data Points -10 M ovement HmL Temperature H C L Graphics fit1@ 4.18D Answer: According to the linear model we expect that at an average temperature of C gives a retreat of approximately 22.4 meter s. A quadratic model H Fit data L Clear@x, a, b, c, d, e, f, g, hd; fit2@x_d = MDNonlinearFitAdata, a x 2 + b x + c, H model L 8x<, 8a, b, c < H parameters L E êê ChopA, E &; Site: M@th Desktop 38

39 8start, stop< = 8Min@ D, Max@ D< &@First@ Transpose@ datad; MDPlotFitData@data, 8fit2@xD, fit1@xd<, 8x, start, 4.1<, FrameLabel 8"Temperature H CL", "Movement HmL"<, Epilog 8Red, PointSize@0.02D, Point ê@ data, Blue, PointSize@ D, Point@8 4.18, fit2@ 4.18D <D<, PlotStyle 88DarkGreen, Thickness@0.01D<<D Iy i ŷ i M 2 Sum of Squared Error : x x Data Points -10 M ovement HmL Temperature H CL Graphics fit2@ 4.18D Answer: We have done a double check with our models. The year - temperature model for the Pasterze suggessts in The year - movement model suggests about 22 meters retreat in the year The temperature/movement model suggests about 22 meters for an average temperature of So our modelling was quite consistant with the data. Site: M@th Desktop 39

40 We think the reason for that consistancy is that our average temperature data are taken close to the Pasterze. Answer: The quadratic model forecasts about - 22 m retreat of the glacier at an average temperature of C. So the linear and quadratical model predict pretty much the same. We will stick to the linear model. We have done a double check with our models. The temperature model suggest in The movement model suggests about 22 meters retreat in the year The temperature/movement model suggests about 22 meters for an average temperature of So our modelling was quite consistant with the data. Answer: To describe the temperature - movement data for the Paster we have chosen the quadratic model, fit1[x] = x. x in C, fit1[x] in meter. The average temperature close to the pasterze is predicted with C and will show a retreat of about 22 m in Summary: According to the data our models confirm that the Pasterze glacier retreat is about 22 meters until 2020 if the weather conditions do not change. We found out that the linear models are suited best to describe the data. They tell us that in 2020 the average temperature will be around C and the retreat around 22 meters. In order to compare the trends from Iceland and Austria in one plot on a different temperature scale, the linear fit function of Austria was shifted to the temperature region of Iceland. fit2icelandtempmov@x_d = ` ` x ` x 2 ; fit1austriatempmov@x_d = ` ` x; Site: M@th Desktop 40

41 8tempAT = 4.18, tempice = 5.3<; diff = tempice tempat; xshift = diff; coordat = 8 tempat + xshift, fit1austriatempmov@tempatd <; coordice = 8 tempice, fit2icelandtempmov@tempiced <; MDPlotA8fit1AustriaTempMov@x xshiftd, fit2icelandtempmov@xd<, 8x, 2.3, 5.8<, Frame True, GridLines Automatic, FrameLabel 8"Rising Temperatures in C", "Movement HmL"<, AxesLabel None, FrameTicks 88Automatic, None<, 8None, None<<, Epilog 8DarkGreen, PointSize@ D, Text@Style@"AUSTRIA", FontSize 20D, coordat + 80, 9<D, Point@coordATD, Blue, Point@coordIceD, Text@Style@"ICELAND", FontSize 20D, coordice , 1<D<, PlotRange 8 20, 60 <, PlotLabel "COMPARISON OF PASTERZE, HYRNINGSJÖKULL MOVEMENT \nin THE YEAR 2020" E COMPARISON OF PASTERZE, HYRNINGSJÖKULL MOVEME IN THE YEAR Movement HmL Rising Temperatures ICELAND AUSTRIA Site: M@th Desktop 41

42 A Simple Physical Model to explain Global Warming Open ê Close Print Greenhouse Effect Openê Close GREENHOUSE EFFECT Atmosphere transmits incoming radiation absorbs outgoing radiation Energy absorbed = 1372 Wëm 2 π R 2 Energy reflected = Wëm 2 π R 2 Energy re radiated Wëm 2 4 π R 2 T = 0.31 * * T 4 Temperature = T ê. SolveA1372 == T 4 E@@4DD This is 255 K = -18 C Average Temperature measured on Earth: 15 C Site: M@th Desktop 42

43 33 higher than model Strong correlation between CO 2 concentrations and DT.(Antarctic ice cores) All that in a graphics Source: oregonstate.edu/instruct/ch390/lessons/media/lesson20.pdf Result and Summary Open ê Close Print Result of the Glacier Movements, Summary Openê Close Site: M@th Desktop 43

44 We did only research on two glaciers and not on hundreds of glaciers. However we might assume that the glaciers on Iceland will melt in the near future more than in Austria. Iceland is probably more effected by global warming than Austria. COMPARISON OF PASTERZE, HYRNINGSJÖKULL MOVEME IN THE YEAR Movement HmL Rising Temperatures ICELAND AUSTRIA Iceland Site: M@th Desktop 44

45 75 Data Points M ovement HmL Temperature H CL Summary: According to the data our models confirm that the Hyrningsjökull retreat is about 51 meters according to the quadratic model until 2020 if the weather conditions do not change. We found out that the linear models are too simple. Quadratic models gave a reasonable result. They tell us that in 2020 the avarage temperature will be around 5.3 C and the retreat around 51 meters. Austria Site: M@th Desktop 45

46 -5 39 Data Points -10 M ovement HmL Temperature H CL Summary: According to the data our models confirm that the Pasterze glacier retreat is about 22 meters until 2020 if the weather conditions do not change. We found out that the linear models are suited best to describe the data. They tell us that in 2020 the average temperature will be around C close to the Pasterze and the retreat will be around 22 meters. The Programes and palettes we used Openê Close We used this program to import Excel files. path = "D:\\C1 2008\\Meeting, , Reykjavik 2\\Retreat TemperatureReinhard1.csv"; strdata = Import@path, "Words"D; data = HstrPos = StringPosition@, ","D@@1, 1DD; 8ToExpression@ StringTake@, 81, strpos 1<D, ToExpression@ StringTake@, 8strPos + 1, StringLength@ <D<L &ê@ strdata 8retr, temp< = Transpose@ data; During the EU meeting from 6. February February 2010 in Kopavagur, Iceland, we produced the following palette. Site: M@th Desktop 46

47 First we designed the buttons Linear, Quad, etc. Then we made use of the MD menu command Desktop > Create Own Material > Palette > General Type. All buttons are pasting fitting programs with which we are analyzing the data from the glaciers. The palette is a useful tool for our task. Site: M@th Desktop 47

48 Our Team Open ê Close Print Participants Openê Close This picture was taken in Iceland, during working for the project in February It shows the 2 partners Stefan Scheucher (left) and Jon-Petúr Gunnarsson (right) of our 4-memberteam. This picture shows the other two members Verena Moser (left) and Katja Rieber (right) in Graz, Austria. Site: M@th Desktop 48

49 Site: Desktop 49

50 Our Experience with the Project Openê Close Working on this project was great fun for all of us. Even though the research was hard work, we finally got all facts and figures needed for the project. While creating this notebook, that is about "glacier movement" we were shocked about the hard facts we found out. Both "Pasterze" and "Hyrningsjökull" decrease due to the rising temperatures - and global warming. Stefan and Jon-Peter did a good job, when working on the notebook in Iceland. The calculations of the Icelandic glacier were an important step to being able to compare the figures with the Austrian glacier "Pasterze". But nevertheless it was difficult to compare each other, as the climate conditions are not the same in Iceland and Austria. One aspect which materialized during the work on the project is that an accurate planning and organization is the most important thing if the project shouldn t fail. Furthermore processing things in time or even a little ahead of schedule makes everything easier. Without a good working atmosphere it s very difficult to trust each other and to work in a team. We discussed all big decisions we had to make and at the end we always found a suitable solution for all of us. It is to say, that problems with work sharing have never been on our agenda. In the end we found out that carrying out the whole project was more work than we had expected at the beginning. We think, that working on this project and the presentation in the end will be a big milestone for our futher career. We got a deeper insight how to carry out scientific work and that it is not easy to interpret and compare all results. We would like to thank our teachers accompanying the project: Guðrún Angantýsdóttir in Kópavagur on Iceland and Reinhard Simonovits in Graz, Austra. Site: M@th Desktop 50