### parachute_lab

```Short course on
“Model building, inference and hypothesis testing in hydrology”
21-25 May, 2012
Martyn Clark
Approach
• Stick to very simple (yet robust) numerical methods
– Simpler than those presented in “Numerical Recipes”
– Yet very easy for people to code from scratch, and retrofit existing
hydrologic models
• Learn by doing
– “I never really understand something until I code it up myself”
– Exercises to implement the methods that are introduced
• Encourage discussion
The typical hydrologic modeling recipe
• Perceptual Model
– Understanding of the Earth System
– Not constrained by mathematics
– Different for different people (based on training, experience)
• Conceptual Model
– Synthesis (simplification) of the perceptual model
– Requires specifying
• System boundaries
• Relevant inputs, state variables and outputs
• Physical laws to be obeyed and simplifying assumptions
– Different conceptual models = different hypotheses of system
behavior
• Numerical Model
– Temporal integration of the governing model equations
3
Outline
• Numerical methods for bucket-style hydrologic models
–
–
–
–
The explicit Euler method
The implicit Euler method
Ad-hoc numerical implementations in hydrologic models
• Exploring impacts of unreliable numerical implementation
– Macro-scale and micro-scale discontinuities in model response surfaces
and associated difficulties in model calibration and sensitivity analyses
– Computational cost
– Fragility of unreliable numerical implementations in predictive mode
• Summary and outlook
State equations
dS s
dt
dSc
dt
dSr
dt
dSa
dt
 ps  ms
 pr  ec  Sc   pt  Sc 
 pt  Sc   qix  Sr   qsx  Sr   er  S r   d  S r 
 d  Sr   qb  S a 
5
The exact solution of the ODE system
(and the need for approximations)
•
The ODE system from Clark et al. (2008) can be written as
dS
 g  S, t 
dt
•
The exact solution of the average flux over the interval tn (start of the time
step) to tn+1 (end of the time step) is
g n  n 1 
•
1
t
t n1
  g(S,  ),   d
tn
Given an estimate of the average flux, the model state variables can be
temporally integrated as
S  t n 1   S  t n   tg nn 1
•
The exact solution is computationally expensive, so approximations to the
exact solution are used
•
The approximation controls the stability, accuracy, smoothness, and
efficiency of the solution
The bread and butter of hydrology:
The explicit Euler method
•
The most straightforward approximation to the exact solution is the explicit
Euler method
nn 1
g EE (1)  g (S n )
where the flux at the start of the time step is used to approximate the
average flux over the time step
•
The explicit Euler method linearly extrapolates from the initial condition
along the initial slope of g(S)
Try this…
•
A famous hydrologist (weight = 90 kg) has decided to go skydiving, and has asked you
to determine how fast they will fall.
•
The hydrologist in question thinks that it is acceptable for you to use Newton’s second
law of motion considering only the gravitational force (Fgrav) and the force due to air
resistance (Fres) . That is,
dv
m
 Fgrav  Fres
dt
where m = mass (kg) and v is the fall velocity (m s-1).
•
Further, the hydrologist pursues the paradigm of parsimony, and is quite happy for you
to use an empirical representation of the resistance force. After some discussion, you
agree that the terms on the right-hand-side can be written as
Fgrav  mg
Fres  kv
where k is a proportionality constant (for freefall, assume k = 15 kg s-1), and g is the
gravitational acceleration 9.81 m s-2).
•
Your mission, should you choose to accept it: Based on an initial velocity of zero
(the hydrologist is hopping out of a hovering helicopter), temporally integrate the ODE
for a 3 minute period using the explicit Euler method with 10 second time steps.
The parachute problem (freefall)
• ODE:
dv
m  mg  kv
dt
• Constants:
– m = mass of the hydrologist (90 kg)
– g = gravitational acceleration (9.81 m s-2)
• Hence
dv
 f (v )
dt
• Numerical approximation: explicit Euler
vn1  vn  f (vn )t
• Length of time step: Δt = 10 seconds
• Initial velocity = zero m s-1
– the hydrologist is hopping out of a hovering helicopter
…and, the end result…
Any observations?
Let’s see if we can do better…
SOLVE THE PROBLEM AGAIN,
EXCEPT THIS TIME USE THE
EXPLICIT EULER METHOD WITH A
TIME STEP OF 0.1 SECONDS
Ta-Da!
The parachute problem (landing)
•
Velocity ODE:
m
•
dv
 mg  kv
dt
Position ODE:
dz
 v
dt
•
Initial conditions:
– v = initial velocity (terminal velocity from the freefall example = 58 m s -1)
– z = height to open the parachute (500 m)
•
Use the same constants as in the freefall example, except for k:
– m = mass of the hydrologist (90 kg)
– g = gravitational acceleration (9.81 m s-2)
– k = proportionality constant for the open chute (250 kg s-1)
•
Numerical approximation: explicit Euler with Δt = 10 seconds
Poor numerical implementation can be
downright dangerous!
Once again, short steps save the day…
Terminal velocity
= 3.53 m/s
= 12.71 km/hr
= 7.89 mi/hr
Can we do better
than the explicit
Euler method?
Can we both increase
accuracy and decrease
computing costs?
The implicit Euler method
• Approximates the exact solution of the average flux by
estimating the flux at the end of the time step
nn1
g( IE1)  g(S n 1 )
• The implicit Euler method requires that the computed
flux at the end of the time step is consistent with the
estimated solution:
S n 1  S n  t g  S n 1 
• In general, this requires iteration and can be
computationally expensive.
Newton-Raphson iteration
•
The Newton-Raphson method finds the root of a function by evaluating the
function f (x) and the function derivatives f ́ (x) at arbitrary points of x.
•
The function: In the implicit Euler method f (x) is the residual between a
trial state variable (x) and the solution at the end of the time step estimated
using the trial value x:
f ( x)  x   x n  t g ( x) 
•
The derivative: The derivative f ́ (x) can be computed using finite
differences, as
f ( x) 
•
f ( x)  f ( x   )

The new value of x can then be computed as
x new  x 
f ( x)
f ( x)
and iterate until either the function or the derivative are below a userspecified error tolerance
Let’s give it a go:
Can the implicit Euler method save our famous hydrologist?
• Step 1:
– Provide an initial guess of v (vm)
– Compute the function evaluation (the residual) at vm and vm+ɛ
• requires the derivatives g(vm) and g(vm+ɛ)
f (v m )  v m  v n  t g (v m ) 
• Notation: m=iteration index, and n=time step index; n denotes the start of the
time step, and v0 is the initial guess of v
– Compute the derivative of the function evaluation
f (v ) 
m
f (v m )  f (v m   )

• Step 2: compute a new estimate of v
x new
m
f
(
v
)
m
v 
f (v m )
and go back to step one, replacing the initial guess of v with
the new estimate of v
• Keep going until you are happy!
… and, the result…
Summary and outlook
• In many cases model analysis complexities are
consequences of numerical artifacts in the model
implementation
• Numerical errors of uncontrolled time stepping schemes
routinely dwarf the structural errors of the model
conceptualization
• The robust numerics paradigm is yet to be accepted as the
required standard in conceptual hydrology
• Reliable hydrologic modeling is a major challenge without