Hydrology and Surveying

Hydrological and measurement instruments

Hydrological instruments are easiest to learn by the quantity each one senses, so the working method is to name the quantity first and then recall its sensor. A tensiometer measures soil-water tension, the suction with which water is held in the soil, and therefore indicates soil moisture status. A piezometer measures the hydrostatic pressure head in a liquid or in a porous medium, usually through a standpipe installed in a well or aquifer. A lysimeter is a soil tank, often planted, that isolates a soil-plant column. Its water balance gives the consumptive use (evapotranspiration), and the water collected at its base gives the deep percolation. Evaporation alone is measured by pans or atmometers, and infiltration is not a direct output of a lysimeter.

Flow-measuring devices rely on energy relations. A pitot tube uses Bernoulli's equation to relate stagnation pressure and static pressure, while an elbow meter infers volumetric flow rate from the pressure difference across a bend.

• Current meter — measures stream flow velocity; used in open channels.
• Anemometer — measures wind speed.
• Hygrometer — measures atmospheric humidity.
• Symon's gauge — a non-recording rain gauge that measures rainfall.
• Pyranometer — measures global solar radiation.

When a question gives a list of instruments and quantities, match each pair by its defining measurement and then check that every item is used exactly once.

S-curve hydrograph and phi-index

An S-curve hydrograph is the response of a catchment to a continuous rainfall excess of unit intensity. Its ordinates rise from zero and approach an equilibrium discharge. The ordinate of a t_r-hour unit hydrograph for 1 cm of excess rainfall is obtained by subtracting from each S-curve ordinate the ordinate t_r hours earlier. Hence the unit hydrograph ordinates are U(t)=S(t)-S(t-t_r) for 1 cm of excess, and a larger excess depth scales the ordinates proportionally.

The \phi-index is the constant rate of loss that is assumed to act whenever rainfall intensity exceeds it. Rainfall in an interval counts as direct runoff only in the amount by which intensity exceeds \phi, so the excess in an interval is (i-\phi)\Delta t when i>\phi and zero otherwise. The direct runoff volume is the sum of these excesses multiplied by the catchment area.

The W-index is the average rate of loss over the storm. It is computed as W=(P-R-I_a)/t, where P is total rainfall, R is direct runoff depth, I_a is initial abstraction and t is storm duration. Runoff depth is obtained by dividing runoff volume by area.

• \phi-index — a constant loss rate, not a cumulative loss. Only intensity above \phi becomes runoff.
• W-index — a mean loss rate, so W is a loss, not an excess rainfall intensity.
• S-curve — accumulated response to continuous unit excess; its difference over one unit-hydrograph duration yields the unit hydrograph.

Reservoir routing methods

Hydrologic routing is based on the continuity of mass alone: inflow minus outflow equals the rate of change of storage, I-O=\dfrac{dS}{dt}. It is combined with a storage-outflow relation, and it does not solve the momentum equation of the flow. Hydraulic routing, by contrast, also uses the momentum (Saint-Venant) equations.

The Muskingum method is a hydrologic channel-routing technique with two parameters, the storage time constant K and the weighting factor X. With routing interval \Delta t, the constants are D=K(1-X)+0.5\Delta t, C_0=(-KX+0.5\Delta t)/D, C_1=(KX+0.5\Delta t)/D and C_2=(K(1-X)-0.5\Delta t)/D. The three constants sum to unity. The outflow is then O_{j+1}=C_0I_{j+1}+C_1I_j+C_2O_j.

• Continuity-only routing — hydrologic methods; they need inflow, storage and outflow relations, not momentum.
• Muskingum K — storage time constant, roughly the travel time of the flood wave through the reach.
• Muskingum X — weighting factor between 0 and 0.5; it controls how strongly storage depends on inflow versus outflow.

Traverse closure and area computation

In a closed traverse each line has latitude L\cos\theta and departure D\sin\theta. The algebraic sums of latitudes and departures should both be zero for a perfect closure. When they are not, the closing error is the distance between the origin and the computed final point, e=\sqrt{(\sum L)^2+(\sum D)^2}. The relative precision is the closing error divided by the perimeter, and the Bowditch rule distributes the error to each line in proportion to its length.

To convert a reduced bearing to a whole circle bearing, first note the quadrant from the signs of departure and latitude. For the south-east quadrant, the whole circle bearing equals 180^\circ minus the reduced bearing, where the reduced bearing is \tan^{-1}(|D|/|L|) measured from the south. Similarly the north-east quadrant gives WCB equal to the reduced bearing, the south-west gives 180^\circ+RB, and the north-west gives 360^\circ-RB.

• Latitude — north-south component, L\cos\theta; north is positive.
• Departure — east-west component, D\sin\theta; east is positive.
• Area by coordinates — A=\tfrac12\sum(x_iy_{i+1}-x_{i+1}y_i) once the traverse is closed.

SCS curve number runoff estimation

The SCS curve number method gives direct runoff depth Q=\dfrac{(P-I_a)^2}{P-I_a+S} for P>I_a, where P is rainfall, I_a is initial abstraction and S is potential maximum retention. In SI units S=25400/CN-254 mm. Runoff is zero when P\le I_a. A common convention is I_a=0.2S, but some problems specify another fraction such as 0.25S, so always read the definition in the question.

A lower S (higher CN) produces more runoff, so changing a forest cover to cultivated land with a smaller S increases the runoff for the same storm. The percentage change is measured relative to the earlier runoff depth.

For a watershed with mixed land uses, the composite CN is the area-weighted mean of the individual curve numbers, and the composite value is then used in the runoff equation. Average (normal) antecedent moisture conditions are assumed unless stated otherwise.

• Potential maximum retention S — depends on CN only.
• Initial abstraction I_a — interception, depression storage and early infiltration before runoff starts.
• Composite CN — area-weighted average, not a simple average.

Chain survey area correction

A chain that is shorter than its nominal length makes every measured distance too large, so the computed lengths and areas are overstated. If the chain is short by \delta, its true length is L_{nom}-\delta, and a measured distance L_m corresponds to the true distance L_m(L_{nom}-\delta)/L_{nom}. For a rectangular field, both sides scale by this ratio, so the area scales by its square.

Tape corrections follow a fixed sign rule. Sag correction is always subtracted, because the tape measures the longer chord between supports rather than the horizontal distance. Slope correction is also subtracted, and wrong alignment is subtracted because the measured path is longer than the straight line. Temperature correction is additive when the field temperature exceeds the standard temperature and subtractive when it is below.

• Short chain — overestimates length and area; multiply measured values by the ratio of true to nominal length.
• Long tape — underestimates length; the correction is additive.
• Sag correction — always negative (subtractive).

Rational method peak discharge

The rational formula estimates peak discharge as Q=CiA, where C is the runoff coefficient, i is rainfall intensity for a duration equal to the time of concentration, and A is the catchment area. The method therefore needs the time of concentration to select the intensity, and this is the essential input. In SI units with i in mm/h and A in km^2, Q in m^3/s equals CiA/3.6.

The Kirpich formula gives the time of concentration as t_c=0.0195\,L^{0.77}S^{-0.385} minutes, with L in metres and S in m/m. Intensity at t_c is read from the intensity-duration-frequency data, and the depth for a duration that falls between tabulated values is found by linear interpolation.

When a catchment has several land uses, the composite runoff coefficient is the area-weighted average of the individual values.

• Time of concentration — time for runoff from the most remote point to reach the outlet.
• Runoff coefficient C — fraction of rainfall that becomes runoff; it is not an infiltration rate.
• Return period — the design frequency that selects the rainfall depth-duration data.

Aridity index and drought classification

An aridity index compares water supply with water demand. In the convention used in this exam, the index is the fractional moisture deficit AI=(PET-P)/PET, where P is rainfall and PET is potential evapotranspiration. A positive value means that rainfall is less than the atmospheric demand, so a deficit exists, and larger positive values correspond to more severe aridity. A negative value means that rainfall exceeds PET, so no moisture deficit exists and the area is classified as non-arid.

Some texts define the aridity index as P/PET, in which case the sign and class limits are reversed. The definition given in the question always governs the classification.

• Non-arid — rainfall meets or exceeds demand; the deficit index is zero or negative.
• Mildly, moderately, severely arid — classes of increasing positive deficit values.
• PET — the evaporative demand of the atmosphere under non-limiting water supply.