GIS Term Paper 2002

By Meghan Strand

CE 394K

The objective of this term paper is to create depth-duration frequency maps for the state of Texas using analysis available in the Depth-Duration Frequency of Precipitation for Texas, Water-Resources Investigations Report 98-4044, published by the USGS in 1998 by William Asquith. These maps have numerous applications including the design of hydraulic structures that control and route runoff. Accurate DDF maps are important for cost-effective structural designs and for developing flood prediction models.

Depth-duration frequency (DDF) is an estimate of the depth of precipitation for a specified duration and frequency of a storm event. The purpose of the 98-4044 report was to present procedures that determine DDF for any location in Texas. The procedures were based on data for 173 fifteen-minute, 274 hourly, and 865 daily National Weather Service precipitation stations with at least 10 years of record in Texas. Precipitation durations of 15, 30, and 60 minutes; 1, 2, 3, 6, 12, and 24 hours; and 1, 2, 3, 5, and 7 days were analyzed. To clarify what might appear as redundant, durations such as 60 minutes and 1 hour, the 60-minute duration is associated with four consecutive 15-minute data values as opposed to the 1-hour data from a single value. In the same respect, the 24-hour duration is associated with 24 consecutive 1-hour data values as opposed to the 1-day data from a single value. A correction factor was applied to the durations and accounted for the different measurement scales in the 98-4044 report. For this term paper, the 100-year storm frequency will be investigated along with all the above referenced durations.

The 98-4044 report used L-moment statistics (Hosking, 1990) of the precipitation annual maxima for each duration and for each station using unbiased L-moment estimators. The location, scale, and shape parameters of the distributions for each duration and each station were calculated from the L-moments. The parameters were contoured using spatial interpolation to produce 37 maps that depict the spatial variation and magnitude of each parameter. The only parameter that was not mapped and contoured was the shape parameter for 1-day and greater durations. This was due to the fact that maps for 1-day to 7-day durations were not produced with errors appreciably smaller than the statewide standard deviation. Therefore, a single statewide mean shape parameter was used for each duration. Using the parameter maps created for the 98-4044 report and the following distribution equations, the precipitation depth for a specified frequency and duration for any location in Texas can be determined. This term paper expands on this idea and involves combining the parameter maps developed previously using ArcGIS (ESRI, 2002) to create depth-duration frequency of precipitation maps for Texas.

Two distribution equations were selected in the 98-4044 report to represent the rainfall data. The Generalized Logistic distribution (GLO) was selected for Texas for the durations of 15 minutes to 24 hours. The Generalized Extreme-Value distribution (GEV) was selected for Texas for 1-day and greater durations. Computation of the precipitation depth for a given frequency, X_d(F), from the GLO distribution is as follows:

Computation of X_d(F) from the GEV distribution is as follows:

where

ξ, α, and κ = location, scale, and shape parameters and

F = annual nonexceedance probability, 1 – exceedance probability, or [1 – (1/T)] where T is the recurrence interval.

The 98-4044 report provided maps for all of the parameters in the two distribution equations. The parameter maps combined with the above equations produce maps of depth-duration frequency of precipitation for Texas. For example, the location, scale, and shape parameter maps from the 98-4044 report for the 3-hour duration are shown in Figures 1-3, respectively.

Figure 1 - ξ parameter map for 3-hour precipitation duration in Texas (adapted from Asquith, 1998)

Figure 2 - α parameter map for 3-hour precipitation duration in Texas (adapted from Asquith, 1998)

Figure 3 - κ parameter map for 3-hour precipitation duration in Texas (adapted from Asquith, 1998)

The procedure for creating the depth-duration frequency maps begins with analysis of the parameter maps and determining the usability of the data in ArcGIS. Unpublished coverages and grids had been created associated with the parameter maps in the 98-4044 report. The grids were in a text format and conversion to raster format was necessary. The Arc Macro Language (AML) was used for the conversion by utilizing the “asciigrid” command followed by importation into ArcMap. The conversion is also possible using the ArcToolbox “asciigrid” command. The command also allowed for the specification of the data storage type of floating point (decimals) or integers. A floating value was preferable to acquire better resolution of the DDF maps. Finally, the datasets were projected in the Clarke 1866 Albers Equal Area.

A total of 37 rasters were created and represent the ξ, α, and κ (location, scale, and shape parameters) for each of the 14 durations. These raster datasets were then combined using the equations for the GLO and GEV distributions to create a new raster dataset representing the depth of precipitation. This was accomplished in two different ways. The first involved using the spatial analyst raster calculator to combine the parameter rasters and the second involved using AML to accomplish the same task. The benefits of using AML were that repetitive operations could be automated. The GLO and GEV distributions required decomposition into elementary steps for the program to work. The following program illustrates how the equations were decomposed:

OUT1 = POW ((-ln(1-1/T)), ) note: is a grid until the 1 day duration where it becomes a constant.

OR Depending on Duration or the distribution equation used (GLO or GEV)

OUT1 = POW (((1 – (1 – 1/T))/(1-1/T)), ) note: is a grid until the 1 day duration where it becomes a constant.

OUT2 = 1 – OUT1

OUT3 = OUT2 * GRID

OUT4 = OUT3 / note: is a grid until the 1 day duration where it becomes a constant

FINAL MAP GRID = OUT 4 + GRID

The program, the parameter rasters are being combined in a way that solves the distribution precipitation equations (GLO and GEV) to produce a new raster representing the depth of precipitation for a given storm duration for any location in Texas. The following figures are the 14 depth-duration frequency of precipitation maps for the 100-year storm frequency created using the steps outlined above:

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Figure 12.

Figure 13.

Figure 14.

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Figure 17.

Once the depth-duration frequency maps were created, analysis was made to determine the reliability of the estimated depths. The first analysis task involved choosing two locations in Texas, one in El Paso County (the most western location in Texas) and the other in Newton County (the most eastern location in Texas). Newton County is expected to greatly exceed El Paso County in the amount of rainfall each county receives for a particular duration. Also, as the durations for each location increase, so should the depth of precipitation. Based on the depth-duration frequency maps, Newton County does in fact receive more precipitation for a given duration than does El Paso County. This is the logical result of the test and is shown in the figure below. The Newton County depth of precipitation is increasing monotonically with duration, which is the expected result. However, precipitation depths in El Paso County are only mostly increasing with duration. Between the 3 hour and the 6 hour duration for El Paso County there is a decrease in precipitation. This leads to an inconsistency Depth-Duration Frequency relation. This is attributable to minor errors in the parameter maps from the 98-4044 report. The potential for inconsistent DDF curves is discussed in the report.

Figure 18.

Inconsistency

The second for analysis task was to take one of the DDF rasters created and subtract a higher duration from a lower duration. This is done using the raster calculator in the spatial analyst. When this is done, if the precipitation depths are monotonically increasing as duration increases, there should be no negative values in the new raster. As is shown below, there are in fact negative values for some of the durations. This reconfirms the notion that inconsistencies might have occurred when analyzing the initial data.

Figure 19.

Further investigation will be required to determine the source of the inconsistencies in the depth-duration frequency of precipitation maps. There are a number of considerations that could possibly cause the inconsistencies to occur. The first of these is the fact that there may be error involved in the data itself. Some of the National Weather Service precipitation station readings may have been inaccurate from the beginning. Another source of inconsistency comes from the fact that there are different numbers of data points for each gauging station. A station that has been taking data for thirty years has a greater reliability than one that has been taking data for ten years. Also, there were numerous precipitation stations located around the bigger cities in Texas but much fewer in west Texas and the lesser populated areas. This could lead to a skew of the data because the more precipitation stations, the greater the reliability. Another source of possible error was once the data was obtained for the 98-4044 report, it was plotted on a map and then contoured using a computer. These precipitation contours were then manually smoothed. The hand smoothing increases the possibility of incomplete interpolations of data contours. These sources contribute to the inconsistencies observed in the DDF maps. Future investigators of the DDF maps should be aware of the inconsistency issue and mitigation steps taken. The 98-4044 report describes some mitigating steps.

William H. Asquith (United States Geological Survey)

Natalie Houston (United States Geological Survey)

David R. Maidment, PhD (University of Texas at Austin)

Asquith, W.H., 1998, Depth-Duration Frequency of Precipitation for Texas: Austin, Texas., Water-Resources Investigations Report 98-4044

ESRI, 2002, ArcGIS 8.2: ESRI GIS & Mapping Software; Redlands, California

Hosking, J.R.M., 1990, L-Moments-Analysis and estimation of distributions using linear combinations of order statistics: Journal Royal Statistical Society B, v. 52, no. 1, p. 105-124.