Surface Runoff

Question 1

What are the common responses of a unit hydrograph

Answer 1

Impulse, step and pulse response function

Question 2

Describe and derive the impulse response function

Answer 2

• Describes the response of a linear system to a unit impulse
• If tau is the moment at which the impulse is applied, the response at time t is described by q(t) = u(t-tau) where t-tau is the time lag since the impulse is applied
• A continuous input can be considered a sum of infinitesimal impulses. If i(t) represent the continuous input rate to the linear system, the amount of input concentrated between the time tau and tau + dtau after the start of the input may be considered as such an infinitesimal impulse. The latter amount equals i(tau)dtau. The response resulting from this impulse is i(tau)u(t-tau)dtau. The response to the continuous input rate may be found by integrating the responses to the pulses that constitute the input. An integral of this type is called a convolution integral:
  integrate i(tau)u(t-tau)dtau between 0 and t

Question 3

Derive and describe the step response function

Answer 3

• Describes the response of the linear system to a unit step input
• The unit step is defined as:
  i(t) = 0 for t<0
  i(t) = 1 for t>0
  where i(t) represents the input rate
• the unit response function is defined as:
  q(t) = s(t) = integrating u(t-tau)dtau between 0 and t
• the step response function at time tau corresponds to the integral of the impulse response function up to that time by substituting l = t - tau in the above equation:
  s(t) = integrating u(l)dl in between 0 and t

Question 4

Derive and describe the pulse response function

Answer 4

• Describes the response of the linear system to unit pulse input
• the unit pulse input is defined as
  i(t) = 0 fot t<0
  i(t) = 1/deltat for 0 <= t <= deltat
  i(t) = 0 for t>deltat
• the response to a pulse input is found by applying the principles of superposition and proportionality to an input consisting of a step input with intensity 1/deltat, starting at time 0 and a step input with intensity 1/deltat, starting at time delta t
• the subtraction of both of the above steps produced an input of p(t) = 1/deltat[s(t)-s(t-deltat)]
• by substituting the step response equation yields p(t) = 1/deltat multiplied by the integral od u(l)dl between t-deltat and t

Question 5

Linear reservoir derivations

Answer 5

Refer to notes