Tobler, Steno and Geologic Maps

Waldo Tobler is a geographer at my alma mater, UC Santa Barbara. He is known for Tobler’s Law or the “first law of geography” which states “Everything is related to everything else, but near things are more related to each other than distant things.”

My classmate Sunil Bharuchi recently published a discussion of Tobler’s Law on his blog, GIS 295 Web GIS. He included this image, which explains spatial auto correlation.

Spatial autocorrelation measures how well a set of spatial features and their values are clustered together in space. A spatial feature is a point, line or polygon that identifies the geographic location of a real world object; this object could be a building, a forest, a rock unit or a lake.

According to Tobler’s law, spatial features will be clustered next to more similar spatial features – this is illustrated in the first image above. But, is this always true? Sunil’s post got me thinking.

Here is a geologic map of Yosemite National Park. Which of the images above does it look like?

Map of Yosemite National Park.svg
Map of Yosemite National Park” by General_geologic_map_of_Yosemite_area.png: en:United States Geological Survey derivative work: Grandiose – This file was derived from  General geologic map of Yosemite area.png: . Licensed under CC BY-SA 3.0 via Commons.

I’ve spent years looking at geologic maps, so I told Sunil “image three looks more like geology.” But, does that mean Tobler is wrong?

Not at all.

Nicolas Steno (Niels Stensen, 1638-1686) was a Danish scientist and bishop who made important contributions to the fields of anatomy, paleontology, crystallography and geology. Steno’s principles of statigraphy explain the formation of sedimentary rock and are still used by geologists to determine the history of a rock unit. There are three principles:

  1. The Principle of Superposition: When sediments are deposited, the sediment that is deposited first is at the bottom while sediment that is deposited later is at the top. Therefore, the lower sediments are older.
  2. The Principle of Original Horizontality: Sediment is originally deposited in horizontal layers.
  3. The Principle of Original Continuity: Sediment is deposited in continuous sheets that only stop when they meet an obstacle or taper off because of distance from the source.

Doesn’t the Principle of Original Horizontality sound a lot like Tobler’s Law? Then why don’t geological maps look like the first picture on Sunil’s image?

First of all, sedimentary rock isn’t the only type of rock on Earth.Steno’s principles do not apply to igneous and metamorphic rock.

Second, the Earth is an active planet. Plate tectonics causes sedimentary layers to bend, break and even overturn. Igneous rocks intrude into existing rock from below the Earth’s surface or erupt from above. These processes mean that geologic units are often very complex and the resulting spatial patterns reflect that complexity.

Yosemite USA.JPG
Yosemite USA” by GuyFrancisOwn work. Licensed under CC BY-SA 3.0 via Commons.

James Hutton (1726-1997) was a Scottish physician and geologist who is known as the founder of modern geology. He was the first to suggest that the Earth is continually being formed and that based on the rates of geologic processes, the Earth must be much,much older than the accepted estimate of a few thousand years. He is also known for the Law of Cross-cutting Relationships.

Law of Cross-cutting Relationships: If a fault or other body of rock cuts through another body of rock, then that intrusion must be younger in age than the rock that it cuts or displaces.

It is this Law of Cross-cutting Relationships that helps us interpret geological units and create geological maps.

Can you figure out the temporal relationships in this cross section?

From Earth: Portrait of a Planet, 4th Edition (2011) by Stephen Marshak.

So, how does Tobler’s Law fit in? It depends on scale. If you are standing on an outcrop of sandstone, chances are good that the rock surrounding you will also be sandstone – especially if you are in the tectonically quieter center of a continent. But, If you are mapping Yosemite park using one kilometer pixels, you will find a lot more variation in neighboring areas.



New York: Why the Height of a Skyscraper Depends on Location

Take a look at the New York City skyline. It’s unique and recognizable because of its skyscrapers. New York City is home to some of the tallest buildings in the world.

Suppose you are interested the height of skyscrapers in New York City. You could make a list of building heights, like this list from Wikipedia.

  1. The Freedom Tower, One World Trade Center (1,776 ft.)
  2. 423 Park Avenue (1,400 ft.)
  3. Empire State Building (1,250 ft.)
  4. Bank of America Tower (1,200 ft.)
  5. Chrysler Building (1,046 ft.)
  6. The New York Times Building (1,046 ft.)
  7. One57 (1,005 ft.)
  8. Four World Trade Center (978 ft.)
  9. 70 Pine Street (952 ft.)
  10. The Trump Building, 4o Wall Street (927 ft.)

This list tells me that the Empire State Building is the third tallest building in the City. It tells me that Freedom Tower is about 750 feet taller than the Chrysler Building. But, what does that look like?

I’d get a better idea of what this means with a bar graph. Or. I could use an image like this (also from Wikipedia):


This gives me a much better idea of how building heights compare. But, this information is still limited.

What if I want to know where these buildings are? What if I care about their locations? I will need a map.

I used the Building Footprints shapefile from NY OpenData to create this map of buildings with a roof height over 500 feet tall in NYC. Only 177 out of 1,082,433 buildings in NYC are over 500 feet tall. Those buildings are indicated in red.



The most interesting thing about this map is that all these very tall buildings are clumped in two locations: Midtown and the Financial District. You can see these clumps in this photograph:


Here’s a closer look:TallBuilding2.What is going on? Did New York City specifically zone these locations to have tall buildings? Is this meant to preserve the skyline? Or, is it intended to show the importance of the Financial District?

The answers can also be found in a map. The location of New York’s skyscrapers is all about geology. As you can see in my hideously ugly geological map (colors courtesy of USGS’s New York Geological Map downloaded as a shapefile), the island of Manhattan has a different type of surface rock than the surrounding area. This bedrock is a metamorphic rock called the Manhattan schist (in pale lavender).


The Manhattan schist formed more than 400 million years ago when a volcanic island arc (similar to today’s Japan) crashed into the eastern side of the continent of Laurentia forming a huge mountain range known as the Taconic Orogeny. The high temperatures and pressures associated with mountain building caused the clay minerals in the mud that accumulated of the coast of Laurentia to transform to more resistant minerals such as biotite, muscovite and quartz.

manhattan schist
Manhattan Schist

Throughout most of Manhattan, this erosion-resistant bedrock is covered with large amounts of unconsolidated sediments. But,  this exceptionally hard rock lies very close to the surface in Downtown and Midtown Manhattan. Because this rock is so strong, it makes the perfect foundation for a skyscraper.

What about other cities? There are only two structures over 500 feet tall in Washington DC: the Hughes Memorial radio tower (761 ft) and the Washington Monument (555 ft). Why doesn’t Washington DC have tall skyscrapers?

While there are some strong metamorphic rocks in Northwest DC, most of DC is built on much softer sedimentary rocks. These rocks cannot support a skyscraper.

So, maps can help us understand where things are, but they can also help us understand why they are where they are. In New York, the height of a building depends on location and location depends on geology.

(Many of the other tallest buildings in the U.S. are located in Chicago. Chicago is weird. You can learn more about the challenges of building skyscrapers in a swamp here. )

This post was inspired by Episode 1 of Making North America on PBS.