We are often interested in two different types of rates:

- rate of movement (length per unit time).
- rate of occurrence (frequency, # of events per unit time or recurrence interval).

What geologic rates might be of environmental interest?

- rate of plate motions.
- rate of erosion.
- rate of deposition.
- rate of fault movement.
- rate of uplift.
- frequency of earthquake activity.
- frequency of volcanic eruptions.
- rate that sea level is increasing.
- rate of groundwater movement.
- frequency of flooding.
- rate of surface water movement.
- rate of natural gas production.
- rate of planktonic growth.
- many additional ones.

If you measure rates over the last year, over
the last 10 years, and over the last 1000 years for erosion in
a given area you often get consistently different numbers. __A
general rule is that the longer the period over which the rate
is being measured, the lower the rate__, but the opposite can
occur. This represents a __scaling relationship__, where the
rate is a function of the time interval over which the erosion
has been measured. What might be causing this behavior?

Think of a dam filling up with sediment. This can control the useful life span of the dam. You would like to know how flong this may take up. Knowing the erosion rate for the drainage basin that feeds into the reservoir can help make an estimate. However, if that rate was measured over one year or ten years or one hundred years can make a significant difference in the estimate of how fast the dam basin may fill up.

If you know the scaling relationship, you can correct a rate determined over one time interval so that it can apply to a longer or shorter time interval.

** Recurrence
interval**: is a measure of frequency
of occurrence of an event. The event is often defined by its size.
For example what is the recurrence interval of a flood that is
10 feet above the flood stage, or that has a certain discharge
(the volume of water that moves through a portion of the channel
in a unit of time). More often, for floods, the inverse is described.
What is the size of the flood that can be expected during a 100
year interval - the 100 year flood. It is important to realize
that this terminology does not mean that such a flood occurs at
regular 100 year intervals. Instead it means that over a long
time the average time between floods is 100 years. Think of rolling
dice at regular time intervals. You wouldn't expect to roll a
six every sixth roll, but the recurrence interval for rolling
a six, should be six rolls. Recurrence intervals can be estimated
for floods, earthquakes, volcanic eruptions, hurricanes, meteorite
impacts and more.

How can one estimate recurrence intervals for natural hazards?

__Analysis of historical record__:

*The basic idea is to see if you can establish
a mathematical relationship between the size of a flood (discharge)
and how often that size occurred. If you can you have some basis
for estimating recurrence intervals over longer time spans.*

**Methodology of estimation of flood recurrence
intervals (can be easily done in Excel, or a variety of other
spread sheets):**

**Obtain historical discharge data - source (United States Geological Survey) of maximum discharge (biggest flood) for each year.****Rank the discharges from 1 the largest on down to n the smallest discharge per year, where n is the number of years of record you have. Note that the rank can also be described as the number of years where the discharge was equal to or greater than the specific year the rank is being calculated for.****Calculate recurrence interval for each : RI = (n+1) / rank. This is nothing but asking**__how often did a discharge of this size or bigger occur in the historic time span__.**Plot the log of the RI versus the discharge. Usually a line will fit the data fairly well. Compute the best fit mathematical relationship for the data.****Extrapolate mathematically from this history to longer time period of interest - 100 year flood or 1000 year flood.****example for Elkhorn River near Omaha, from Geodata analysis course****.**

**What is the basic assumption in this type
of analysis?**

**How may this assumption be violated?**

**drainage basin characteristics change (e.g. logging, urbanization, farming practices).****climate change.**

**Is there a better way?**

You can extend your historical approach by looking at the geologic history if the events of interest leave some type of record that can be deciphered. Consider a volcano. One can compile historical accounts of past eruptions and that provides some insight. However, the recurrence interval may be such that only a few eruptions have been sampled during human history. One can date the individual lava flows, and significantly extend that history backwards.

You can also try a theoretical approach, but if system shows complexity, this can be quite difficult, and in certain cases not presently possible.

- Fractals are
__scale invariant__. A smaller part of the pattern looks like a larger part. The viewer can't tell what the scale is by looking at the pattern, they need to be informed as to the scale. - Plotting the log of the size of an element versus the log of the frequency with which it occurs produces a linear relationship where the slope is the fractal dimension.
- Some geologic phenomena have a fractal relationship for the relationship between the size of the event and the frequency with which it occurs.
- They are non-integer in terms of dimensions - for example between 2 and 3 dimensions.

Link to a number of examples of fractal geometries.

What is the length of the line of a section of the coastway of Norway? Instead of converging on a value as you look closer and closer, the length just keeps getting longer because there are always smaller bumps on the larger bumps that come to light as you look closer.

Limits to fractal relationships.

One example of how fractal relationships can be useful - RI of earthquakes.

What is the upper limit to the fractal relationship for earthquakes - or what determines the largest earthquake that can occur in an area?

Link to Yale course material on earthquakes and fractals.