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The area is simply the width (W) of the river times the depth (Y). Because depth and velocity change through any cross section (its faster and deeper in the center of the river), several measurements must be made. Typically, the stream gauger splits the stream into several cells and measures W, Y, and V in each cell, calculates Q in each cell, then adds up all the cells to get the total Q. The United States Geological Survey recommends 20 cells in a cross section.
The stream gauger moves across the river measuring the distance from the bank to keep track of width, and takes a depth measurement at a station. A salmon swimming upstream could tell you that the stream velocity is slower at the bottom of the river than at the top. How then do we report the velocity in a cell if it changes? Fortunately, the relationships between depth and velocity are fairly well known. The average velocity typically occurs at 60% of the depth measured from the top. So we measure the depth, then place a current meter at 0.6 times the depth to get the average velocity for that station. If the stream is shallow, the gauger will wade. More elaborate suspension systems, boats or bridges are required in deep and fast flow. Here is an example of calculating discharge from field data.
The above techniques give us the discharge of a stream only at the time of measurement. Does this mean that every time the river rises from a rainfall that we have to re-measure the streamflow? Yes and no. We can construct a continuous record of streamflow, called a hydrograph, from what is called a stage-discharge relationship. Stage is simply the height of the water surface relative to the height of a reference marker that doesnt change. Stage is measured with a staff gage. The simplest staff gage is essentially a meter stick placed in the stream. When a stream gauger measures discharge, he/she records the stage. Stage is continuously monitored by charting the history of a float in a stilling well. The float moves up and down with the water level and rotates a drum on which a sheet of paper is attached. At the same time, a pen moves across the drum at a known rate. The result is a line describing the stage history. Fortunately, for any given stage in a cross section that is not altered by erosion, the discharge is always the same. Discharge is measured at several stages, then a relationship is established between stage and discharge through a mathematical equation. If stage is continuously monitored, continuous discharge can be calculated by the stage-discharge relationship.
A hydrograph is a plot of discharge against time. The time scale can be in hours, days, months, or even years. The rise and fall of discharge is the response of the watershed to inputs of precipitation. A simple storm hydrograph contains 4 components: baseflow, the rising limb, the crest, and the falling limb. The arrangement of these 4 components produce a unique shape of a hydrograph that results from a combination of properties of the storm and the watershed. For example, a hard rain on pavement will result in a very steep rising limb, and a watershed containing wetlands will usually produce hydrographs with long falling limbs as the wetlands slowly release water.
Precipitation is reported as a depth of water in inches, and includes rain and snow. The depth does not include the actual depth of snow, but is the depth of water that would result from melting the snow. Rainfall is measured in rain gauges which are essentially buckets that collect water. If you record the change in depth in the bucket with time you get a rainfall rate or intensity. A bar-chart of precipitation that falls in a given unit of time is called a hyetograph. Comparing a hyetograph to a hydrograph allows you to investigate how quickly streams respond to precipitation.