G E O 3 2 8 A s s i g n m e n t # 3 Page 1 University of Florida Department of Geography GEO 328 Assignment 3 Modeling Precipitation and Elevation Solar Radiation Precipitation Evapo- Transpiration Vegetation Surface Sub- Surface Interception Effective Precipitation Evapo-transpiration Excess Precipitation Storage Infiltration Drainage Basin Direct Runoff Sub-surface Runoff Runoff INTRODUCTION: The changing seasonal role of topography in controlling precipitation within the Tiribí drainage basin is investigated and represented. Different precipitation generating processes operate with varying intensities during the course of the year (seasonality). The purpose of this assignment is to illustrate the role of geographic variables, such as elevation and aspect, in determining the nature of this changing relationship, and to find a way to estimate the basic precipitation input to our basin. A simple mathematical model is introduced and applied annually and seasonally. The changing parameters (coefficients) are interpreted physically. The important concepts of calibrating and validating a mathematical representation are introduced, as well as a very simple test of the reliability of the representation. The parameters of a mathematical model may also vary from year to year. This property is illustrated by investigating the relationship between elevation and mean annual precipitation in La Niña and El Niño years.
G E O 3 2 8 A s s i g n m e n t # 3 Page 2 What you should take away from Assignment #3. Some introductory ideas about the nature of mathematical models and their application. In particular, the notion that the same model can be used during various seasons, but that the coefficients within the model will change with each season and that these changes can, and should be interpreted physically. We are specifically interested in the seasonally changing role of elevation as it influences the quantity and spatial pattern of precipitation input into the basin. The importance of testing the results from your model not just against the data that you used to formulate the model (calibration data set), but against a set of data which have not formerly been used (validation data set). The concept of a quantifiable measure of your confidence that the mathematical representation employed actually provide a reasonable approximation to the observed data and the degree to which it can be used as a reasonable forecast tool. Further evidence of how our small regional basin may be influenced by variability in atmosphere-ocean circulation patterns at a much larger scale, and how we can graphically portray those changes over space.
G E O 3 2 8 A s s i g n m e n t # 3 Page 3 1. Does a Relationship Exist Between Topography and Precipitation? Using the information provided in the file ANSTATS: A) Produce a graph (scattergram, XY dots only) of station elevation (x-axis) versus the mean annual precipitation (y-axis) observed at all fifteen stations. (3 Marks) B) On the graph, identify those stations lying in the Caribbean watershed and those in the Pacific watershed, as defined in Table 2.1 of the previous asignment and Figure 3.2. Comment on any differences you observe in the relationship between annual precipitation and elevation, between these two sets (Pacific vs. Caribbean) of stations. Provide a physical explanation for your observation. (3 Marks) Building on what you know. Refer to the west-east cross section created in Assignment 1 and the discussion of dominant wind directions over Costa Rica at different times of the year from Assignment 2. C) Using only the 11 meteorological stations within the Pacific drainage, estimate the straight line relationship between station elevation and mean annual precipitation by means of the REGRESSION option in your software package, of the following form: PRECIP (mm) = b + m.elev (meters) Retain and report values of the coefficients for the INTERCEPT (a) (to one decimal place), X-VARIABLE (slope or b) (to 3 decimal places), and R-SQUARED (to 3 decimal places). (1 Mark) See EXCEL How-To #25 Hints: Representations of the graphical meaning of the regression coefficients of intercept (b), slope (m) and r-squared are shown in Figures 3.1. R-squared can be viewed as a measure of the variability in the observed data (the dots) which is explained by the fitted straight line. A perfect fit (1% of variability explained) yields an r-squared of 1., while a line which explains none of the observed variability (%) returns a value of..
G E O 3 2 8 A s s i g n m e n t # 3 Page 4 1 Annual Precipitation, P (mm) 8 6 4 2 Observed data Fitted Regression Line P(mm) = m.e(m) + b r 2 = 1. Intercept = 2mm Slope =.2 mm.m -1 5 1 15 2 25 3 35 Station Elevation, E (m) Annual Precipitation, P(mm) 1 8 6 4 2 1 8 6 4 2 m = 189.6 mm b =.25 mm.m -1 r 2 =.936 5 1 15 2 25 3 35 m = 994.5 mm b = -.26 mm.m -1 r 2 =.769 5 1 15 2 25 3 35 1 8 6 4 2 1 8 6 4 2 Elevation, E (m) m = 698. mm b = -.1 mm.m -1 r 2 =.2 5 1 15 2 25 3 35 m = 1. mm b = -.2 mm.m -1 r 2 = 1. 5 1 15 2 25 3 35 Figure 3.1. Examples of changing values of the regression coefficients of intercept (m) and slope (b), and the measure of goodness of fit of the straight line to the data, r-squared.
G E O 3 2 8 A s s i g n m e n t # 3 Page 5 Figure 3.2. Disposition of meteorological stations listed in ANSTATS with respect to the divide between Pacific and Caribbean drainage. D) On the basis of the coefficients you derive and the definitions provided in Figure 3.1, answer the following questions: i) What is your estimate of mean annual precipitation at sea level on the Pacific coast? ii) What is the expected increase, or decrease, of mean annual precipitation (in mm) with each additional meter of elevation within the Pacific drainage? iii) What percentage of the variation in the mean annual rainfalls observed at these stations can be explained by the differences in their elevations? (1.5 Marks) 2. How can we Generate Estimates of Annual Precipitation from the Digital Elevation Model? Fields, or maps, of annual and monthly precipitation for the basin are to be generated, based solely upon the elevation information contained in the DEM and the mathematical representation, or model, that you have just derived. From the previous discussion, we know that elevation is not the only control on precipitation, but it is an important one.
G E O 3 2 8 A s s i g n m e n t # 3 Page 6 A. Using the coefficients derived the previous question (using the number of decimal places specified in question 1C), estimate mean annual precipitation at each cell within the D.E.M. (TIRIBELEV). Express answers to one decimal place. Produce a 3-D surface of mean annual precipitation across the basin - not outside of it. Use BASINLIMITS to achieve this. (5 Marks) Settings: Use a perspective of 315, a minimum vertical axis value of 19 mm and a maximum vertical axis value of 35 mm.! Warning: As annual precipitation is calculated as a simple linear function of elevation, this figure should look very similar in slope and orientation to the 3-D graph of elevation in Assignment 1. 3. Can Differences or Changes in Hydrologic Fields be Mapped? The phase of ENSO (El Niño/La Niña) has considerable effect upon rainfall in the region. Annual rainfall totals from the period 1941-1989 from the 11 stations in the Pacific drainage have been extracted. These records are then sub-divided according to the COAPS classification scheme, and mean annual precipitation estimated at each station under El Niño (Warm Phase of ENSO) and La Niña (Cold Phase of ENSO). Table 3.1 contains a listing of the coefficients estimated for the best fit regression line relating station elevation and mean annual precipitation observed under each ENSO phases. ENSO Condition Intercept Slope r-squared La Niña (Cold phase) 178.9.959.617 El Niño (Warm phase) 129.9.721.627 Table 3.1 Coefficients of estimated regression relationships between station elevation and mean annual precipitation under La Niña and El Niño conditions.
G E O 3 2 8 A s s i g n m e n t # 3 Page 7 A. Answer the following short questions, based on the information provided in Table 3.1: i) What is the expected difference in mean annual precipitation between Cold and Warm Phases of ENSO at a station located at sea level in the Pacific drainage? ii) Under which ENSO condition does a unit change in elevation (a meter) have a greater influence on the anticipated change in mean precipitation? How many more millimeters of precipitation would you expect to arise from the same increase in station elevation of 1m in a La Niña year than an El Niño year? (Note that this requires that you compare two stations separated by 1m) iii) Is there any evidence that the linear regression model is more appropriate (fits better) under one of the two ENSO conditions? Support your answer. (3 Marks) B. Use the D.E.M. data (TIRIBIELEV), BASINLIMITS and the coefficients listed in Table 3.1 to generate an anticipated mean annual precipitation field in the basin under a) El Niño conditions, and b) La Niña conditions. i) Create a new spreadsheet which represents the field of differences (in millimeters) between mean annual precipitation under La Niña and El Niño conditions (La Niña value minus El Niño value) within the basin. Produce and submit a 3-D view of this field. (3 Marks) Settings: A rotation of 315, a minimum vertical axis value of 2mm and a maximum vertical axis value of 7mm. ii) Using these two surfaces: Estimate and submit the mean annual precipitation input into the basin under these two ENSO phases. Express and submit the increase in mean annual precipitation between El Niño and La Niña as a percentage of the El Niño mean. (2 Marks) Building on what you know. You have already calculated mean basin precipitation via several different methods, and have been given observations at stations. Your computed values here will be different, but should be in the same ball park!
G E O 3 2 8 A s s i g n m e n t # 3 Page 8 iii) Create a new spreadsheet which represents the field of percentage differences between mean annual precipitation under La Niña and El Niño conditions ({La Niña value minus El Niño value}/el Niño value*1) within the basin. Produce and submit a 3-D view of this field. (3 Marks) Settings: A rotation of 315, setting the lower limit of the vertical axis to 15%. 4. How are the Parameters of the Precipitation/Elevation Model Estimated (Calibrated) for Monthly Data? We use mathematical models or representations to provide physical insights and to allow us to make forecasts. In this series of questions, we are investigating whether the information provided by our simple mathematical model makes sense in terms of what we know to about rainfall generating processes within the basin at different seasons. The best fit linear regressions between station elevation and mean monthly precipitation have already been estimated for all months and are recorded in the file MONTHLY REGRESSION. Bearing in mind the seasonal definitions used early (pre-veranillos, dry, Veranillos etc.), answer the following: A) Create a total of three line and symbol graphs, each of which shows the twelve monthly values of the coefficients (listing provided in file MONTHLY REGRESSION), with month along the x-axis and the value of the coefficients - r- squared, intercept, and slope - on the y-axis. (3 Marks) B) Write brief answers (only a line or two) to the following questions. Reference to Figure 3.1 should help you: i) In which months/seasons does the elevation of a station provide the greatest explanation of the observed variations in monthly precipitation between stations? On what statistical basis do you make this claim? ii) In which months/seasons does the elevation of a station provide the least explanation of the observed variations in monthly precipitation between stations? On what statistical basis do you make this claim? iii) In which month/season does the elevation of a location have the greatest effect upon the seasonal precipitation total at that station? What provides the statistical basis for this claim? iv) In which month/season does elevation have the least effect upon the seasonal precipitation total at a station? On what statistical basis do you make this claim?
G E O 3 2 8 A s s i g n m e n t # 3 Page 9 (2 Marks) Building on what you know D) Provide a physical interpretation (no more than one page) of the seasonal changes in the monthly numerical relationships between elevation and precipitation, bearing in mind the nature of the seasonally changing climatological setting of the region (refer to the previous assignments, such as Assignment 2 question 1, the article dealing with the seasonality of precipitation and dominant wind directions, and the topographic cross-section from Assignment 1). The seasonal divisions used in Assignment 2 are used throughout the manual and provide a structure for your discussion. Completion of Table 3.2 may help guide your thoughts. (5 Marks) JFMA Wind Direction Basin Windward or Leeward Intercept (mm) Slope (mm. m -1 ) r-squared MJ JA SO ND Table 3.2. Possible framework for developing a physical interpretation of the observed monthly/seasonal patterns of regression terms. 5. How Can the Reliability of the Model be Tested? One way in which the reliability of this representation (model) of the relationship between elevation and monthly precipitation can be tested is to estimate the mean monthly precipitation at the 6 recording stations, or calibration stations, that fall within the basin, from each of our twelve monthly surfaces. This has already been completed and graphical comparisons of the estimated and observed means, and ranges of historic data are shown in Figures 3.4, 3.5 and 3.6.
G E O 3 2 8 A s s i g n m e n t # 3 Page 1 Validation: Comparisons of estimated values to those of the calibration stations are not the fairest test of the model, as these stations were themselves used in establishing the regression coefficients which form the numerical basis of the model. A more widely accepted test would be to apply the method to stations that had not been used previously in the model calibration procedure. Thus far, only precipitation stations with at least 2 complete years of record have been used in the study. There are several other stations in and near the basin, possessing between 1 and 2 years of record, which were not used in the calibration process. The historic data from these stations provide a useful, independent means of validating the model, comparing the forecast regimes to the observed regimes, because the data at these stations were not used in formulating our numerical model. The locations of 4 validation stations are shown in Figure 3.7. Unfortunately, only one, San Juan, actually falls within the limits of the Tiribí basin, but the other three are nearby and lie within the Pacific drainage. A. Using the monthly regression relationships (expressed to 3 decimal places) and the elevations of each validation station, estimate the mean monthly precipitation during each of the five missing months, at each station. Complete Table 3.3. Express all estimates of mean monthly precipitation to one decimal place only. (5 Marks) Validation Cell Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Elev. Station (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (m) San Joaquin G9 2.9 15.4 61.9 236.2 184.6 328.1 124.9 978 San Antonio I14 2.7 17.4 57.5 261.2 28.8 368.8 172.8 132 San Juan N15 11.6 16.4 59.8 248.3 196.3 347.9 148.2 1144 IPIS V3 18.7 17.2 58. 258.3 26.1 364.2 167.4 1281 int. -47.99 9.517 74.64 164.824 115.114 211.765-11.975 slope.52.6 -.13.73.71.119.14 Table 3.3. Validation station cell addresses and estimated mean monthly precipitation.
G E O 3 2 8 A s s i g n m e n t # 3 Page 11 CALIBRATION STATIONS 7 6 Historic Estimated Curridabat Precipitation Total (mm) 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 11 12 7 6 Historic Estimate Month Desamparados Precipitation Total (mm) 5 4 3 2 1 Percentiles 95th 9th 75th Mean 5th 1 2 3 4 5 6 7 8 9 1 11 12 Month 25th 1th 5th Figure 3.5. Comparison of observed monthly rainfall totals and mean monthly total estimated from the regression analysis for the calibration stations of Curridabat and Desamparados.
G E O 3 2 8 A s s i g n m e n t # 3 Page 12 CALIBRATION STATIONS 7 6 Historic Estimate Rancho Redondo Precipitation Total (mm) 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 11 12 Month 7 6 Historic Estimate San José Precipitation Total (mm) 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 11 12 Month Figure 3.6. Comparison of observed monthly rainfall totals and mean monthly total estimated from the regression analysis for the calibration stations of Rancho Redondo and San José.
CALIBRATION STATIONS G E O 3 2 8 A s s i g n m e n t # 3 Page 13 7 6 Historic Estimated El Alto de Ochomogo Precipitation Total (mm) 5 4 3 2 1 7 6 1 2 3 4 5 6 7 8 9 1 11 12 Historic Estimated Month Hacienda Concepción Precipitation Total (mm) 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 11 12 Month Figure 3.7. Comparison of observed monthly rainfall totals and mean monthly total estimated from the regression analysis for the calibration stations of El Alto de Ochomogo and Hacienda Concepción.
G E O 3 2 8 A s s i g n m e n t # 3 Page 14 B. The available historic observed monthly precipitation data for these validation stations and a possible framework for the analysis are found in the following files: SAN JUAN MONTHLY P SAN JOAQUIN MONTHLY P SAN ANTONIO MONTHLY P IPIS GUADALUPE MONTHLY P i. Open each file in turn. At the base of each column (month) estimate, and save, the following PERCENTILES from each of column (month) - 1 th, 25 th, 5 th, 75 th and 9 th, and the observed mean. There is no need to print these data. See EXCEL How-To #26 ii. Using a Line chart, construct a graph for each validation station showing the month along the x-axis and the values of the monthly percentiles (5 values) and observed mean on the y-axis (monthly precipitation). Each graph will therefore consist of 6 separate line plots (LINE ONLY, no symbol). iii. On the same graph, plot the forecast monthly means, both those provided in Table 3.3 and those you calculated above, as symbols only on the same graphs. Enlarge the default symbol shape, color and size so that it stands out clearly. (1 Marks) See EXCEL How-To #27 Check out time saving hint on next page. Settings: Ensure that the y axes run from to 6mm in each chart. C. Provide a short (half page) discussion of how well the estimated means at the validation stations match the historic data. (4 marks)
G E O 3 2 8 A s s i g n m e n t # 3 Page 15 Figure 3.7. Mean annual precipitation total and the locations of the four validation stations. Codes beneath station names indicate the cell addresses corresponding to these stations.
G E O 3 2 8 A s s i g n m e n t # 3 Page 16 6. How can the Model be Applied to Estimate Mean Monthly Precipitation Input to the Basin? Maps such as Figure 3.7 are only approximations of the true distributions of precipitation, or Precipitation Fields, based on simple numerical interpolation between the existing meteorological stations. In areas of rapidly changing topography these mathematical interpolations may be very misleading and erroneous. Monthly precipitation fields for the basin can also be generated, based solely upon the elevation information contained in the DEM. From the previous discussion it is clear that elevation is not the only control on precipitation, and that the degree of control varies seasonally. However such an approach may give a better approximation to the actual precipitation falling into the basin, particularly during the rainy season. A. Using the information provided in MONTHLY REGRESSION, estimate mean monthly precipitation fields for all twelve months across the entire area. Use coefficients to 3 decimal places, and express answers to one decimal place. Save the computed precipitation fields as separate files. For each computed monthly precipitation field, calculate the average (mean) of precipitation entering the basin in that month. Insert these average monthly values in the appropriate cells in Table 3.4. Repeat the process for each month. Submit a completed version of Table 3.4, or the estimated monthly mean basin precipitation, in each of the five months, which are your responsibility. (6 Marks)! Warning: The annual value is merely the sum of the twelve monthly answers. J F M A M J J A S O N D Ann Prec. 27.6 18.2 55.8 27.8 218.2 384.6 191.4 Table 3.4. Mean monthly precipitation within the basin (mm).
G E O 3 2 8 A s s i g n m e n t # 3 Page 17 Hint: In order to set up the above process more efficiently it is recommended that TIRIBIELEV and BASINLIMITS be opened. 1. On a new worksheet, named for example Precip, use the requisite monthly coefficients, e.g. those for January, to calculate estimates of mean January precipitation in each grid square from using elevation information stored in TIRIBIELEV. 2. On a second new worksheet, named perhaps Basin Precip, enter the product of the Precip and BASINLIMITS worksheets. 3. Below this, calculate the sum of all values in the Basin Precip worksheet and divide by the number of falling cells in the basin. 4. Using the SAVE AS command, save this as a file called January. 5. Return to the Precip worksheet, change the regression coefficients to those of the next month (e.g. February), and copy the result to all cells. All subsequent steps should change automatically. 6. Using the SAVE AS command, save this as a file called February. 7. Repeat steps 5 and 6 for all months changing the file name appropriately. Check: The calculations of mean monthly basin precipitation in the basin (step 3) in the 7 months, for which I have already furnished answers, should match those provided in Table 3.4.
Assignment #3 - Submission checklist: G E O 3 2 8 A s s i g n m e n t # 3 Page 18 1A. 1B. 1C. Graph of Elevation vs. Annual Precipitation at 15 stations. Observations of differences in relationship between Pacific and Caribbean stations. Coefficients of regression line fitted to only those stations in the Pacific watershed. 1D. Physical interpretation of coefficients. 2. 3-D surface of annual precipitation across the basin. 3A. Short answers interpreting changes in coefficients between El Niño and La Niña years. 3Bi. 3-D surface of differences in annual precipitation across the basin between La Niña and El Niño years. 3Bii. Mean annual basin input under El Niño and La Niña conditions, and the difference between the two as a percentage of input during El Niño years. 3Biii. 3-D surface of differences in annual precipitation across the basin between La Niña and El Niño years expressed as percentage of cell s precipitation during El Niño years. 3Biv. Observations on the spatial changes in absolute and percentage differences between conditions.
G E O 3 2 8 A s s i g n m e n t # 3 Page 19 4A. Graph of monthly intercepts, slopes and r-squared values for all 12 months. 4B. Short answers showing the general physical interpretation of the regression coefficients. 4C. Interpretation of the seasonally changing values of the regression coefficients as related to the regional precipitation generating processes. 5A. Estimates of mean monthly precipitation at the four (4) validation stations for the 5 missing months. (Table 3.3) 5B. Four (4) graphs showing the historic percentiles and mean vales of monthly precipitation at the four (4) validation stations. Graphs should also display the mean monthly precipitation forecast by the regression of monthly precipitation on elevation. 5C. Discussion of goodness of fit of the fitted precipitation field to the observed regime of mean monthly precipitation at the four (4) validation stations. 6. Completed Table 3.4.