Geography 416 - Study Guide 2

Our primary concern this week is with statistical analysis of hydrologic data. Our immediate concern arises from the fact that hydrologic processes exhibit considerable variation over time. Some of that variation is cyclical (e.g. patterns driven by seasonal climate changes), and some of it is random or stochastic (occurrence of droughts or storms). Often we are concerned about the consequences of wet or dry conditions and we would like to have the ability to make predictions. When we look at a plot of some hydrologic variable against time, however, it quickly becomes apparent that prediction of what will happen on a given day or during a given week or month is for the most part impossible. When we plot a sequence of observations in time, we often can learn something useful and important from the pattern displayed. However, in order to make any predictions concerning the likelihood or probability of certain events taking place, we have to transform our analysis into the frequency domain.

To do this we need to understand what random variables are (both discrete and continuous) and we need also to understand frequency distributions and histograms. Histograms are bar charts that plot frequencies with which different ranges of values are observed for some variable or parameter, and they are closely related to probability distributions and to their probability density functions; to understand this more clearly we need to know the difference between empirical and theoretical frequency distributions. We also need to understand the cumulative frequency distribution. Flow duration curves, illustrated in chapter 2, are cumulative frequency distributions for specific data sets. One of the exercises in your next homework asks you to calculate the cumulative frequency distribution and plot the flow duration curve for a 10-year record of mean daily streamflow values. I will provide a template showing how to do it in one spreadsheet file, and a listing of the data you should use in completing your own analysis will be in another spreadsheet file.

In order to facilitate this work you should have several 3-1/2 inch diskettes at your disposal and I would recommend bringing at least one to class with you in case you need to copy something onto it. Any files that I make available to you in class will also be available electronically, which means they can be accessed in any computing lab on campus or from home via modem. I will provide instructions on how to do this if you need them.

Part of dealing with time-series data also involves knowing how to analyze and plot a series of instantaneous values in order to see the temporal patterns of rainfall and runoff and also calculate the cumulative volumes of these two variables. In another exercise I will provide data to allow you to reconstruct the hydrograph of a recent flood and to calculate and compare cumulative volumes of rainfall and runoff.

If we are going to make any predictions about likelihood of certain events taking place, we must rely on some statistical techniques together with a set of assumptions about the underlying processes and associated probabilities. Therefore we need to be able to calculate sample statistics, including the moments of a distribution, and we have to understand the characteristics of at least a couple of the probability distributions that are most important in hydrology (the most common being the "normal" distribution). Use of the sample statistics, combined with assumptions made about the distribution or distributions, allows us to make estimates of the probabilities of various future scenarios (such as the probability of the 200-year flood occurring two years in a row). We can also attempt to estimate the reliability of our results by using what we know about the properties of the most common probability distributions.

If we get all the way through this set of issues before the end of the week, we will also talk briefly about some topics in chapter 3. Much of what is in this chapter involves topics you have been exposed to before in Physical Geography or Weather and Climate. Rather than cover this material in great detail, I will probably just review the major precipitation-forming mechanisms and the spatial patterns of precipitation, evapotranspiration , and runoff that have been mapped at a global scale.

Geography 416 - Hydrology

Problem Set 2

Due Monday, Feb. 9

1.	Open the spreadsheet file wdf77_87 . wk1 using Lotus, Quattro, or Excel. In this file you will find a listing of the mean daily streamflow records for Watts Branch in Rockville, Maryland over a ten-year period between Oct. 1, 1977 and Sept. 30, 1987. I have sorted these records and placed them in rank order from smallest to largest, and I have calculated cumulative frequencies and exceedence probabilities for each day of record. The formulas necessary to do this are built into the spreadsheet. In addition I have also used some of the statistical functions in Quattro Pro version 5 to calculate sample statistics for this ten-year record. I have also used a procedure for calculating cumulative frequencies associated with a subdivisions of the data more or less following the guidelines on p. 11-13 in the textbook. I have used another graphics package to plot the resulting flow duration curve (discharge or streamflow on the y-axis, exceedence probability on the x-axis). I am handing out two versions of this plot. Both have probability scales on the x-axis; one has a linear y-scale and the other has a logarithmic y-scale.

Your job is to perform the same kind of analysis for another ten-year period from the same stream gage: from Oct. 1, 1957 to Sept. 30, 1967. This record is contained in the spreadsheet file wdf57_67 . wk1 . You should calculate the mean, median, standard deviation, coefficient of variation and skewness of this record and compare it with the values calculated for the period 1977-1987. (If you don't know how to do this, follow the procedure in Box C-2, p. 496.) Once you have completed the analysis necessary for generating a flow duration curve, plot the results in pencil on the same graph I have provided for you. In your opinion are the two records substantially the same or are they different in significant ways? Compare the flows associated with specific quantiles , e.g. exceedence probabilities of 5, 10, 20, 50, 70, 80, 90, 95 percent. What might this comparison tell you about processes going on in the Watts Branch watershed?

You may choose to hand in your spreadsheet file as part of your homework - if you wish to do so, let me know in advance.

(20 points)

2.	A third spreadsheet file, patsav96 . wk1 , contains a record of rainfall and streamflow for the Little Patuxent River at Savage, MD (about 15 minutes from campus) for a flood that occurred about 2-1/2 weeks ago. You need to plot both rainfall and runoff on the same plot (if you don't have a version of a spreadsheet that can do this, see me and we'll make an arrangement). Also you need to convert the instantaneous discharge record into a series of flow volumes for each time period, and to convert those into equivalent water depths over the watershed drainage area. Finally you need to plot cumulative runoff vs. time and cumulative rainfall vs. time in inches. How does the total depth of rainfall compare with the cumulative equivalent depth of runoff generated? How can you explain the observed result?

If you don't know how to perform these analyses , don't worry. The purpose of the exercise is to teach you how to do this kind of work with real-world examples. We will be discussing the details before you actually have to complete the assignment. There is also a specific point to each analysis that has to do with the actual hydrologic processes involved, and as you go along I want you to think about the meaning of the assignment and discuss it with others once the actual computations are complete.

(20 points)