5. EXAMPLE OF A DEGREE DAY MODEL: HINTEREISFERNER, AUSTRIA

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1 Glacial Gelgy 5. Example f a Degree Day Mdel 5. EXAMPLE OF A DEGREE DAY MODEL: HINTEREISFERNER, AUSTRIA = 55 Pints Objective: t create a psitive degree-day mdel f glacial mass balance. Yu shuld be able t: Calculate the psitive degree days fr the ablatin seasn frm actual climate data; Use Excel t determine the relatinship between mass balance and psitive degree-days; and, Assess the general accuracy f yur mdel fr predicting glacial mass balance. Resurces: Davies, B Degree day mdels: Mdelling glacier melt. AntarcticGlaciers.rg. Reading assignment. Fischer, A Glaciers and climate change: Interpretatin f 50 years f direct mass balance f Hintereisferner. Glbal and Planetary Change: 71: di: /j.glplacha Reading assignment and inspiratin fr this activity. Pangaea. Data Publisher fr Earth and Envirnmental Science. Institute f Meterlgy and Gephysics (2013): Climate Data Vent, Ötztal Alps, University f Innsbruck, di: /pangaea Surce f temperature data. Wrld Glacier Mnitring Service, Fluctuatins f Glaciers. Surce f mass balance data. Intrductin Scientists have develped varius mdels t estimate change in glacier mass balance withut actually taking field measurements f mass balance, althugh field measurements are necessary t assess the accuracy f the mdels. These mdels allw scientists t estimate mass balance fr glaciers that are difficult r dangerus t access, t estimate mass balance fr glaciers that have n field data due t lack f funds t cllect the data, and t imprve ur understanding f the relatinship between glaciers and climate. Simple mdels use a single climate factr, such as temperature, t estimate mass balance while cmplex mdels use multiple factrs such as temperature, humidity, clud cver, and wind speed. One example f a simple mdel used t estimate the amunt f glacier melting is the psitive degree-day mdel a psitive degree-day is any day with an average temperature abve 0 C (the melting pint). The degree-day mdel assumes that fr each 1 C ver 0 C, a specific amunt f ice r snw, measured in millimeters f water equivalent (mm w.e.) will melt (Davies 2015; Fischer 2010). The degree-day mdel sums the ttal psitive degree-days during the ablatin seasn t estimate the energy available t melt snw and ice thrugh the ablatin seasn (Fischer 2010), and frm that energy estimate, calculates the ttal mass lst. We are ging t apply the degree-day mdel t Hintereisferner Glacier in Austria t see hw well air temperature predicts glacier mass balance. T d this, we need t sum the psitive degree-days (sum all the average daily temperatures abve 0 C) ver the curse f the ablatin seasn fr each year f recrd. Accrding t Fischer (2010), the ablatin seasn fr Hintereisferner runs frm May 1 September 30, s we want t sum all the psitive temperatures ver that time interval fr each year. We then want t crrelate the sum f the psitive-degree days with the glacier mass balance t see hw well air temperature predicts glacial mass balance. Prcedure Open the Excel file called degree_day_data. The file cntains the fllwing data: Clumn A: Year; Clumn B: Mnth; 1 12 repeating Clumn C: Day; 1 28, 29, 30 r 31 depending n the mnth Clumn D: Average daily temperature measured at 2 m abve grund in degrees C K.A. Lemke UWSP 31

2 5. Example f a Degree Day Mdel Glacial Gelgy Nte: in Excel, capital letters represent clumns while numbers represent rws. If we want t refer t a specific cell in the spreadsheet, we must use the clumn letter and the rw number. Fr example, E10 represents the cell in clumn E, rw 10. Step 1: identify all the days with temperatures abve 0 C. We will put these psitive temperatures in a new clumn (clumn E) t make summing the psitive-degree days fr each year easier. Clumn E: Temps>0; we will fill in this clumn with: zers fr days when the average daily temperature is less than r equal t zer, r with the actual average daily temperature fr days when the average daily temperature is greater than zer. a) T add values t clumn E, Temps>0, we will use the IF functin in Excel, which allws us t set a cnditin. If the cnditin is met, the cell will cntain ne value and if the cnditin is nt met, the cell will cntain anther value. Since psitive-degree days are temperatures abve zer, ur cnditin will be Temps>0: IF average daily temp > 0 value in clumn E cell = average daily temp (frm clumn D) IF average daily temp is nt >0 b) Excel functins: All Excel functins must start with an equal sign. value in clumn E cell = zer All functin names are fllwed by infrmatin cntained in parentheses. c) The syntax fr the IF functin is: =IF(lgical test, value if true, value if false). Althugh spaces exist between the infrmatin cntained in parentheses, this is just fr ease f reading. When yu type the frmula in Excel, there shuld be n spaces separating the infrmatin inside the parentheses, just cmmas. Our lgical test will be D<0; Excel will test each cell in clumn D t determine whether the cell value is less than zer r nt. Our value if true will be 0; fr each cell in clumn D that cntains a negative value, that value will be represented by a 0 in clumn E. Our value if false will be D; fr each cell in clumn D that cntains a nn-negative value, that nn-negative value in clumn D will be entered in clumn E. Our final functin, which we will enter int cell E2, will be: =IF(D2<0,0,D2) Instead f repeatedly typing this functin fr every day f recrd, we can simply cpy the functin and paste it int all the remaining cells in clumn E that have a crrespnding value in clumn D. Each time we paste this functin int a new cell, the rw number will change. Type this frmula int cell E2 and then cpy and paste it int all the cells dwn t rw 21,913. d) When yu are finished, clumn E shuld cntain zers and psitive values. [1] Step 2: add up all the psitive degree-days fr each ablatin seasn (btain a sum fr each year f recrd). Befre summing the psitive degree-days fr each ablatin seasn, let us lk at the rest f the data in ur spreadsheet in rder t figure ut an efficient way t calculate these yearly sums. Clumn H: Year: a list f years frm , the years fr which mass balance data are available. Clumn I: Psitive Degree Days; we will fill in this clumn with the ttal accumulated psitive temperatures thrughut the ablatin seasn, which accrding t Fischer (2010) runs frm May 1 September 30. Clumn J: Mass balance; the verall mass balance fr the glacier in millimeters f water equivalent (mm w.e.). This number represents the millimeters f water we wuld get if we melted the ice/firn/snw lst frm the glacier. It des nt represent actual thickness f ice lst. 32 K.A. Lemke UWSP

3 Glacial Gelgy 5. Example f a Degree Day Mdel a) T calculate the psitive degree-days fr each year, we must add all the temperatures in clumn E frm May 1 Sep 30 fr each year. Remember, clumn E cntains zers and psitive values nly. T d this we will use the SUMIFS functin. This functin adds up (sums) the values in a specified range f cells prvided they meet a set f cnditins. The cnditins that must be met fr cells t be summed are: Cnditin 1: the year fr the cells we sum must equal the year in clumn H. Cnditin 2: the mnth fr the cells we sum must be greater than 4 (April) and less than 10 (Octber). As with the IF functin, ur functin must start with an equal sign and the functin name must be fllwed by infrmatin cntained in parentheses. b) The syntax f the SUMIFS functin is: =SUMIFS(sum_range,criteria_range_1,criteria_1,criteria_range_2,criteria2,criteria_range_3,criteria_3,etc.) Our sum_range will be E2:E21913, the cells in clumn E cntain the temperatures we want t sum up Our criteria_range_1 will be A2:A21913, the cells cntaining the years Our criteria_1 will be H2, H3, H4, etc., the cells cntaining the year fr which we want the psitive degree-days. Our criteria_range_2 will be B2:B21913, the cells cntaining the mnths Our criteria_2 will be >4 ; we nly want mnths with values greater than 4. The >4 must be put in qute marks. Our criteria_range_3 will be B2:B21913, the cells cntaining the mnths Our criteria_3 will be <10 ; we nly want mnths with values less than 10. The <10 must be put in qute marks. As with the IF functin, in additin t listing the clumns in this functin, we als need t list rw numbers in which case ur functin wuld lk like this: =SUMIFS(E2:E21913,A2:A21913,H2,B2:B21913, >4,B2:B21913, <10 ) c) As with the IF functin, we can cpy and paste the SUMIFS functin int the cells belw instead f typing it ver and ver, but this time we have a prblem: each time we paste the functin int a new cell, all the rw numbers change t match the number f the rw int which we paste the functin. We d want the rw numbers fr the year (criteria_1) t change each time we paste the functin int a new cell. We d nt want the rw numbers fr criteria_2 t change. We d nt want the rw numbers fr criteria_3 t change. We d nt want the rw numbers fr the sum_range t change. T prevent the cell ranges frm changing when we paste the functin int a new cell, we need t add dllar signs befre and after the letter representing the rw. Fr example, $A$2 tells Excel t always g back t cell A2 n matter where the functin is pasted. Our final functin shuld lk like this: =SUMIFS($E$2:$E$21913,$A$2:$A$21913,H2,$B$2:$B$21913, >4,$B$2:$B$21913 <10 ) d) Type this functin int cell I2. Then cpy and paste it int the cells belw dwn t cell I58. When yu are finished, clumn I shuld cntain the sum f the psitive degree-days fr the ablatin seasn fr each year f recrd. [4] Step 3: test the relatinship between mass balance and psitive degree-days. We will d this several ways. a) Create a scatter plt with psitive-degree days n the x-axis and mass balance n the y-axis. [6] When making scatter plts, the usual standard is t put the independent variable n the x-axis and the dependent variable n the y-axis. Since we want t use temperature t predict mass balance, temperature is the independent variable and mass balance is the dependent variable. K.A. Lemke UWSP 33

4 5. Example f a Degree Day Mdel Glacial Gelgy Select (highlight) all the cells cntaining mass balance and degree-day values. On the very tp menu bar in Excel, click n the Insert tab. Find the blck with different types f charts and click n the ne fr a scatter plt. Select the ptin with n lines, just dts. Frmat yur chart: Adjust yur chart size t 2.5 high by 6.5 wide. Make sure yur chart has an apprpriate title and apprpriate labels (including units) fr the x- and y-axes. Adjust the x-axis s it appears at the bttm f the chart, nt in the middle. Adjust the minimum value fr the x-axis t smething clse t 1000 t eliminate the large empty space n the left side f the chart. Add a trend line t yur scatter plt. Include the equatin fr the trend line and the R 2 value. Adjust the psitin f the equatin and the R 2 values s they are easy t read. b) Calculate a crrelatin cefficient using the =CORREL functin. [1] The CORREL functin assesses the linear dependence between tw variables, in ur case, mass balance and psitive degree-days. In essence, the functin measures hw far away ur data pints are frm the trend line. Values fr the crrelatin cefficient range frm 1 t +1. A negative crrelatin means that as ne variable increases, the ther decreases. A psitive crrelatin means that as ne variable increases, the ther als increases. A cefficient f 1 indicates a perfect negative crrelatin and a cefficient f +1 indicates a perfect psitive crrelatin. With a perfect crrelatin, all f the data pints fall exactly n the line. As the distance f the data pints frm the trend line increases, the magnitude f the crrelatin cefficient decreases and appraches zer. A cefficient f zer means that n crrelatin exists at all between the tw variables. The syntax fr the =CORREL functin is: =CORREL(array_1, array_2). Array_1 will include all the cells cntaining the psitive degree days fr each year; I2:I58 Array_2 will include all the cells cntaining the measured mass balance; J2:J58 Our final functin is: =CORREL(I2:I58,J2:J58). Type this functin in an empty cell and add a label adjacent t it. c) The equatin fr ur trend line is: MB = ( 3.55 PDD) where MB = mass balance and PDD = psitive degree-days. Use this equatin t predict the mass balance using the psitive-degree days. [1] Excel frmulas, similar t functins, must start with an equal sign. In additin, ur frmula requires parentheses t insure Excel des the math calculatins in the crrect rder. Type the equatin fr the trend line int cell K2. Instead f typing in the value fr PDD, click n the cell cntaining that value. The frmula shuld lk like this: =( 3.55*I2) After typing the frmula int cell K2, cpy the frmula and paste it int the cells belw s yu have a predicted value fr each year f recrd. d) Create a line graph (nt a scatter plt) that shws the measured mass balance and the predicted mass balance. [6] Select (highlight) the cells cntaining the measured mass balance values. On the very tp menu bar in Excel, click n the Insert tab. 34 K.A. Lemke UWSP

5 Glacial Gelgy 5. Example f a Degree Day Mdel Find the blck with different types f charts and click n the ne fr a line graph. Several ptins fr line graphs appear; select the first ne. Add the predicted values t the chart. On the very tp f the Excel screen yu shuld see Chart Tls and tw tabs belw that labeled Design and Frmat. Click n the Design tab. Tward the right side, yu shuld see a blck labeled Data and within that blck yu shuld see Select Data. Click n Select Data. The menu bx that appears lists Series 1 and this represents the measured mass balance data. Click n the Edit buttn and enter a name fr the series (Measured Mass Balance). Hitting Enter will take yu back t the Data Surce Bx. In the Data Surce Bx click n the Add buttn. Enter a series name (Predicted Mass Balance), click in the Series Values bx and then select the cells cntaining the predicted values in clumn K. Hitting Enter will take yu back t the Data Surce Bx. On the right side yu shuld see Hrizntal (Categry) Axis Labels with the numbers 1, 2, 3, We want t replace these values with the years. Click n Edit. A bx will appear asking fr the Axis Label Range. Select (highlight) the cells cntaining the years. Frmat yur chart. Adjust yur chart size t 2.5 high by 6.5 wide. Add a legend t yur chart. Make sure yur chart has an apprpriate title and apprpriate labels (including units) fr the x- and y-axes. Adjust the x-axis s it appears at the bttm f the chart, nt in the middle. Yu shuld nw have a chart with tw lines, ne representing the measured mass balance values and ne representing the predicted values frm the trend line n the scatter plt. e) Create a table with the fllwing descriptive statistics fr the measured mass balance values and fr the predicted values. [5] Minimum (0 th Percentile) 25 th Percentile Median (50 th Percentile) 75 th Percentile Maximum (100 th Percentile) =MIN(cell range) =Quartile(cell range,1) =Median(cell range) =Quartile(cell range,3) =Max(cell range) f) Create a line graph shwing the measured and predicted mass balance values fr the percentiles yu calculated in part (e). [6] Select (highlight) the cells cntaining the measured mass balance values. On the very tp menu bar in Excel, click n the Insert tab, g t the Chart blck and select the first line graph. Add the predicted values t the chart. Fllw the same steps utlined fr Part (d) abve. Fr the last step, n the right side yu shuld see Hrizntal (Categry) Axis Labels with the numbers 1, 2, 3, We want t replace these values with the percentiles. Click n Edit. A bx will appear asking fr the Axis Label Range. Select (highlight) the cells cntaining the percentiles (Minimum, 25 th Percentile, etc.). Frmat yur chart. Use the same guidelines listed under part (d). Step 4: In Excel, arrange yur three charts, yur crrelatin cefficient and yur table f percentiles s they fit n a single sheet f paper. Print just that ne page and hand it in with the answers t the questins belw. K.A. Lemke UWSP 35

6 5. Example f a Degree Day Mdel Glacial Gelgy Questins 1. Is the crrelatin cefficient fr mass balance and psitive degree-days negative r psitive? Explain in wrds what it means fr ur data when the crrelatin cefficient is psitive/negative. Think abut cause and effect. [2] 2. What is the R 2 value fr yur trendline f temperature and mass balance? [2] What is the value f the crrelatin cefficient? What d yu get if yu square the crrelatin cefficient? Our degree-day mdel is based n the wrk f Fisher (2010) and uses the same data Fisher (2010) used. What is Fisher s reprted R 2 value fr the simple degree-day mdel? What is the R 2 value fr Fisher s (2010) mre cmplex mdel (TS degree-day sum)? Fisher (2010) als calculated an R 2 value fr a mdel using average summer temperature instead f degree-days. What is this R 2 value? 3. What d these R 2 values tell us abut the utility f using temperature fr predicting mass balance? [3] 4. Examine yur scatter plt. Nte the distance separating the data pints frm the trend line. Des there appear t be a difference in the distance f the data pints frm the line when mass balance values are high cmpared t when mass balance values are lw, r des the scatter f pints abve and belw the line appear apprximately cnsistent regardless f mass balance values? [2] 36 K.A. Lemke UWSP

7 Glacial Gelgy 5. Example f a Degree Day Mdel 5. Examine the first line graph yu created shwing the measured and predicted mass balance values by year. What patterns, if any, appear when yu cmpare the predictins t the measured values? Are the tw lines in sync? Are there any patterns t when the predictins appear relatively gd and when they appear relatively bad? [2] 6. Examine the secnd line graph yu created shwing the measured and predicted mass balance values by percentile (minimum, median, etc.). What patterns, if any, appear when yu cmpare the predictins t the measured values? Are the tw lines in sync? Are there any patterns t when the predictins appear relatively gd and when they appear relatively bad? [3] 7. Based n yur answer t questin 6, under what cnditins des the degree day mdel perfrm wrst (i.e. the predictins are the least accurate)? Speculate n sme reasns fr why this might be the case. [3] 8. Glaciers may respnd t sustained negative net mass balance values by thinning, which results in the surface elevatin f the glacier getting lwer, and by the accumulatin area shrinking. Accrding t Fisher s intrductin, hw des the lwering f the glacier surface and the shrinking f the accumulatin area affect the mass balance? [2] K.A. Lemke UWSP 37

8 5. Example f a Degree Day Mdel Glacial Gelgy 9. In the sectin titled TS Mdel f Mass Balance, Fisher (2010) lists specific factrs gverning glacier melting. What are they? [2] 10. Figure 4 in Fisher (2010) lists the actual data used in a mre cmplex mdel f mass balance. What meterlgical data did Fisher use, and what are sme f the reasns prvided fr using these additinal variables (as ppsed t just using temperature t calculate psitive degree-days)? [3] 38 K.A. Lemke UWSP