The Power Function Studies of stream length and drainage basins determined an empirical relationship : L = 1.4 A 0.6 where L is stream length and A is.

The Power Function

Studies of stream length and drainage basins determined an empirical relationship :

L = 1.4 A0.6

where L is stream length and A is drainage basin

area. (Hack, 1957)

The Power Function

In our lab exercise on sinking forams, you derived the equation for Stokes settling velocity,

Vstokes

= (g/18 d2

Stokes derived this equation from consideration of the driving forces and resisting forces for sinking foraminifera.

The Power Function

Both the empirically defined Hack equation and the analytically derived Law for Stokes velocity are examples of power functions.

A power function is written in general form by

y = axb .

The Power Function

In the case of Hack's Law,

y = Lx = Ab = 0.6(a = 1.4)

L = 1.4 A0.6

y = axb

The Power Function

In the case of Stokes velocity,

y = Vstokes

x = db = 2(a = g/18)

y = axb

Vstokes

= (g/18 d2

The Power Function

y = axb

What is interesting about this equation is what happens when you apply logarithms.

How would you do it ?

log y = log a + b*log x

Does this equation remind you of anything ?

Logarithm of a Power Function


This may be similar to the equation for a straight line where y = b + m*x

b = 3m = 1

y = 3 + 1*x y = 3 + x

What kind of scale would we need to plot the logarithmic equation to simulate a linear equation ?

(at the x intercept when x = 0, y = 3)



If a and x are a set of measurements and y is a column of results

and we take the log of each of these numbers

Then plot these log values on normal graph paper...

We see a straight line.

With b as the slope.

The x intercept (log x = 0)

occurs at log ax

y



If we plot x against y on log-log paper,

We also see a straight line

Again, b is the slope

The line crosses x = 1

Where y = a

x

y

Power Functions in Geology

Log - log plots are common in geology

As a result, power functions often arise in geology

C = CoF(D-1)

As crystals settle out of a magmaelement concentrations, C, in theremaining liquid change accordingto this equation.

Where Co is initial concentration, F is the fraction of liquid remaining, and D is the distribution coefficient.

Linear plot of

C = CoF(D-1)


Log - log plots are common in geology

As a result, power functions often arise in geology

C = CoF(D-1) log-log plot

log C = log Co + (D-1) log F


Stream length (y) and drainage-basin area (x) are measuredand listed in the table above.

The logs of each measurement are listed in column 4 and 5

If we plot columns 4 and 5 and try to “fit” a line to the data

Constant = 0.148761, and slope is 0.53687


Constant = 0.148761, and slope is 0.53687

How can we write this in a linear style equation with logs ?

log y = 0.148761 + 0.53687 log x


log y = 0.148761 + 0.53687 log x

Plot columns x and y (squares)

Test theory, but plotting the line for the log eqn above.

Pretty good fit!


log y = 0.148761 + 0.53687 log x

Remember that if we take the “antilog” of both sides

We get y = 100.148761 x0.53687

Simplifying, y = 1.41 x0.54

Power Functions on a Linear Scale

y = 1.41 x0.54

Data in a power function plotted on a linear-linear scale

The curve continues to increase

But it increases at an ever decreasing slope


y = 1.41 x0.54

To understand the “slopes” of a function, take it's derivative

dy = 0.76 x-0.46

dx The exponent, b is < 1 (negative)

This says the slope will decrease, as x progresses

riserun

=


Taking the derivative in general

dy = (a) xb-1

dx If the exponent, b is > 1 (positive)

Then the slope will increase, as x progresses

What if b = 1 ? Then what ?


y = 1.41 x0.54

To Summarize: For y = axb-1

Plots will be convex- upward if b < 1 (negative exp) Plots will be convex -downward if b > 1 (positive exp) Plots will be a straight line if b = 1.

Power Functions and Exponential Functions

It is easy to confuse power fns with exponential fns

We've already looked at exponential functions

But we have not studied power functions until today.

y = xb y = bx

Exponential functions produce a straight line when plotted on a linear-log scale.

Where as power functions produce a straight line when plotted on a log-log scale

Power Functions and Exponential Functions

y = xb

In a power function, for every increase in x by some factor y increases by some other factor

In an exponential function, for every increase in x by some factor y may increase by an order of magnitude

(assuming b is a whole number) This is where the concept of a half-life comes from.

y = bx

Studies of stream length and drainage basins determined an empirical relationship :

L = 1.4 A0.54

where L is stream length and A is drainage basin

area. (Hack, 1957)

Back to Drainage Basins and Hack's Law

The exponential “b” value here has been debated.

L = 1.4 A0.6

Back to Drainage Basins and Hack's Law

Some say that if b > 0.5

Then the length/area relationship implies that large basins are more elongated.

L = 1.4 A0.54

Shape of Drainage Basins

Understanding length/area ratio

If A = wL

Then, L = 1.4 (wL)0.54

Simplifying.... w/L = 0.53L-0.15

w

L

Notice that the exponent is negative.

How will w/L change as you go downstream (increasing L) ?

Shape of Drainage Basins

Put L on one side: w/L = 0.53L-0.15

w = 0.53 L0.85

Will a plot of L versus w be convex up or down ?

w

L

The Power Function Studies of stream length and drainage basins determined an empirical relationship : L = 1.4 A 0.6 where L is stream length and A is.

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