Theory of the Slope-Area Method

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Detailed Description

This video covers the theory behind a slope-area indirect measurement, the most common type of indirect measurement used in the USGS. 


Date Taken:

Length: 00:04:34

Location Taken: Las Vegas, NV, US

Video Credits

Hampton Childres, Mark Smith, Michael Steiner, Todd Geiger, Office of Employee Development


Script for Slope-Area Theory

Hi, this is Megan Poff and I’m the Field Office Chief at the USGS in Las Vegas, Nevada.  In this video, I’ll be discussing the background and theory for the slope-area indirect measurement method.   Don’t click off of this video yet!  It’s important that you understand why you’re doing what you’re doing and knowing a little theory can help with your understanding – so you can collect the best data possible.  Slope-area measurements are the most common type of indirect measurement method used by the USGS. Generally if you hear someone say that they were out “running indirects” they were likely referring to the slope-area method.  You can find more information on this topic in TWRI 3-A2: Measurement of peak discharge by the slope-area method.

What is a slope-area measurement?  A slope-area measurement is a one-dimensional flow model developed using a uniform flow equation that accounts for channel geometry, channel roughness, and the water-surface profile at the peak of the flood.  For the uniform flow equation, the USGS uses a variation of Manning’s equation, which is defined as Q=(1.486/n)AR2/3S1/2, where Q is discharge, n is a roughness coefficient, A is the area of the channel cross section, R is the hydraulic radius, and S is the slope or energy gradient.  Hydraulic radius (R) essentially is the mean depth of flow, defined as area of the channel cross section divided by the wetted perimeter.  Wetted perimeter is simply the perimeter of the cross section that is, well, wet.  Sometimes, Manning’s equation is simplified as Q=KS1/2, and K is referred to as the conveyance.  Think of the conveyance as a measure of channel carrying capacity. In the slope-area method, the slope S is approximated by the water-surface profile defined by high-water marks.

The slope-area method assumes you have conditions close to steady and uniform flow (these conditions are known as “gradually varied flow”).  Steady flow means that your flow isn’t varying by time, and uniform flow means that your flow isn’t varying spatially – so depth, area and velocity are the same from section to section.  Generally, steady flow is a safe assumption at the flood peak because we will have conditions that approximate steady flow just at the end of the rise and before the start of the recession.  However, we also need to choose a measurement reach in which the channel geometry is fairly uniform throughout.

Take a look at a natural channel.  To me, this sure doesn’t look like completely uniform conditions.  See those areas of expansion and contraction?  As long as channel changes in the reach are not too drastic (that is, they change gradually), we can basically assume steady and uniform flow in this reach by placing cross sections at the major hydraulic changes, which we will identify by assessing the flood high-water marks and resulting water-surface profile.  We’ll get more into detail on the water-surface profile in the slope-area surveying video. 

Okay, I gave you a flow equation and I gave you some definitions of steady and uniform flow.  What does all this have to do with a slope-area indirect measurement?  Manning’s equation applied to several cross sections in a stream reach is the basis for a slope-area indirect measurement!  Remember I just mentioned that our simplified version of Manning’s equation is Q=KS1/2?  We can apply the equation to several cross sections by using the geometric mean of the conveyances at adjacent cross sections, because we’re assuming the conveyance varies gradually between sections.

Before you start panicking and I start giving you more equations to account for things like velocity head coefficients, energy losses, and friction losses, keep in mind that nobody does a slope-area computation by hand anymore.  The USGS has developed a helpful computer program called SAC (which stands for Slope-Area Computation) – it calculates discharge using the slope-area method from a user-prepared input file that includes the spacing between cross sections along a longitudinal baseline, the water-surface elevation at each cross section, the X-Y geometry of each cross-section starting from an arbitrary horizontal reference (remember, this is a 1-D model), and user-defined estimates of the roughness at each cross section. There also are Graphical User Interface (or GUI) platforms like iRIC and SAC-GUI that can help you create the input file; the SAC executable program is integrated with these GUIs! You can download a copy of the SAC executable program, along with its user’s guide, from

Before attempting a slope-area indirect measurement, please refer to the video that covers survey requirements for a slope-area.  If you need help in the field, call your supervisor, surface-water specialist, or indirect measurement specialist.