Myths of Nature: a reflection on our tacit assumptions about the global environment

It is not difficult to realize that the world around us is vast and complex, its dynamics happening somewhere in the fascinating but paradoxical realm of complexity, beyond the easier to envision extremes of complete order or chaos. It is no coincidence then that one of the pleasures of life is to grasp some aspect of the Universe, be it through the cosmovision of some philosophy, or using advanced computer models.

In everyday life, though, we are not roaming the world hooked to a supercomputer with just the appropriate set of equations to understand every single situation happening around us. No matter how rational and knowledgeable we believe we are, we rely heavily on intuitive mental models imprinted in us by our education and culture to make sense of life and express our impressions about it. We are forced to face, evaluate and act on situations based on more or less blurry ideas and incomplete information. These mental models almost always have some irrationality built into them, something that is particularly true when we assess multidimensional and multiscale problems such as those related to the global environment and sustainability.

These issues are nicely highlighted in “Panarchy. Understanding Transformations in Human and Natural Systems” [1], a thoughtful book edited by Lance H. Gunderson and C.S Holling, and published in 2002. In chapter 1 of this book “In quest of a theory of adaptive change“, Gunderson, Holling and Ludwig present a clever picture of some common mental models used to understand Nature and justify our approach to it. These authors call these five models “caricatures”, or “myths”, simple metaphors expressed as balls rolling in different landscapes and encapsulating our ideas about the dynamic behaviour of the world. None of these models is completely right, nor completely wrong, but all are just incomplete visions of one and the same world.


The first model, “Nature random” or “Nature flat“, expresses the worldview of a nature without feedbacks or nonlinearities. Changes in this world are driven by random causes, natural or political, and it is therefore a world where humans can do as they please to satisfy their needs, with no risk of triggering undesirable irreversible processes. The ball just moves in a flat terrain, going wherever some random force pushes it.

Nature stable” expresses the view that the world is largely in balance, a static world dominated by negative feedbacks equilibrating any change. This is the worldview that emphasizes maximum yields and believes that we can optimize things to navigate turbulent times.

Nature unstable” or “Nature anarchic” is the opposite worldview, where Nature is in delicate balance and extreme caution is needed not to push it away from it. This is the mental model where the precautionary principle dominates, and is the worldview held by many environmental organizations.

Nature Resilient” is a step forward in our understanding of Nature, it encompasses the previous three views. It relies on the lessons from the science of complex systems and dynamical systems applied to natural science. A resilient world is one in which several dynamical basins of attraction exist, where a given part of nature might be forced to move to a neighbouring basin by human intervention, or by changes in variables at larger scales of time and space. In this worldview, a perturbed ecosystem can flip to a different state, a move that is fully admissible under the prevailing environmental conditions, although possibly unknown and probably not pleasant for its inhabitants and users. An example of this is the often observed transition of grasslands to dry unproductive woodlands due to heavy overgrazing (Australia, Patagonia).

Nature evolving” is presented as the last frontier. It depicts an even more complex world than “Nature resilient”, one where the dynamical landscape changes with time. Gunderson, Holling and Ludwig convincingly argue that this view might be the one closest to our real world, and the one that we should be exploring to assess our impacts on Nature and evaluate our chances for long term survival in the Anthropocene. It is also the one worldview that is least accessible to our intellects, and the one that poses the greatest challenges to understanding and prediction. The one where intuition might not longer serve us so well.

I believe that these models capture wonderfully well some of the most common sets of beliefs that people use to summarize and classify the behaviour complex natural systems. Even if they are just overly simple caricatures, they also show us that having a look at our often unchallenged mental models can be a worthy exercise that can teach us lessons on how to approach Nature in more harmonious ways.

Now, read again the myths. Which is the one closest to your beliefs?

[1] Gunderson, Lance H., and C. S. Holling, eds. Panarchy: Understanding Transformations in Human and Natural Systems. Island Press, 2002.

GIS in a nutshell

I have recently accepted the challenge of giving a very short introduction to GIS to an eclectic audience composed of highly educated professionals, albeit not versed in geospatial technologies or geography. This has prompted me to rethink the question “what is GIS?“, modifying accordingly my approach to teaching it from scratch. The idea was to achieve that beautiful blend of minimal elementary theory, for meaningfulness, and minimal sufficient practical knowledge, for usefulness, that enable any person to start producing simple results and delving further in the matter, if desired.

After some pondering, and many sketches, I finally came to this:

GIS in a nutshell

GIS in a nutshell

This is of course nothing else that the classical “a system to store, manipulate and visualize geographical data” stripped to its bare minimum.

In this scheme of things, we may start describing the two basic representations of geographical reality: vectors, for discrete features, and rasters, for continuous fields, for which different formats are used (in the most simple cases just ShapeFiles and GeoTIFFs). Here, I find convenient to introduce the concept of data as layers, which is essential for both processing and visualization in GIS.

We may then continue explaining that, for each layer, we usually have associated data, that are allocated in database tables called attributes. These can be used to store information, to choose a particular subset (spatial queries) or make operations on the layers. Vectors and raster data models have associated sets of spatial operations, and also operations for converting one type into the other. It is at this level that we most often use GIS to explore relationships in spatial data.

As all this makes sense only in the context of a particular coordinate system, a Spatial reference System (SRS), we may continue by introducing some essential cartography. I found that it is sufficient, and more convenient, to limit myself to the practical basics of a projection of interest for the audience (such as UTM), introduced merely as a coordinate system. I prefer this approach to the more classical lecture on geodesy, with the gory details of the geoid, ellipsoid and the myriad of projections and reference systems, which more often than not are just confusing for beginners.

Finally, I found relevant to say some closing words about where our data might come from: models, field measurements, regional surveys, GPS, satellite images, on-screen digitizing; and where our data might go to: maps, statistics, datasets as input for other studies, etc. This gives context and adds that tiny dose of practical meaningfulness that most people (that is, not GIS nerds) appreciate.

This was my brand of “GIS in a nutshell“, as of today. Which is yours?

Elementary atmospheric correction of Landsat images: haze correction via histogram minimum

Sensors operating in the visible and mid-infrared portions of the electromagnetic spectrum measure the fraction of the incoming solar radiation reflected at the surface. If the sensors are mounted on board satellites in orbit, the radiation coming up to the sensor does not entirely arises from the reflection at a particular location on Earth, but also partly due to the interaction of light with the atmosphere. Apart from the obvious case of a cloudy sky, absorption and dispersion of light by the atmosphere can be very important, even in conditions that we could consider clear sky.

The basic types of scattering are Rayleigh and Mie. Rayleigh scattering arises from the interaction between the light and the air molecules. it affects more the shorter wavelengths and is responsible for the blue color of the sky. Mie scattering arises from the interaction of light with particles, more importantly water drops, dust and pollution. Their combined effect on a satellite image is that, apart from the radiation reflected at the surface, we also get radiation coming from Sun light scattered in the atmosphere along the view path (path irradiance), and from Sun light scattered in the atmosphere in the direction of the surface element and reflected to the sensor (sky irradiance). Eventually we also get light scattered from the neighboring surface (neighbor irradiance).

Depending on our purpose these effects might be negligible and we can simply disregard them, as when measuring the extent of a feature. In other cases they may severely degrade the quality of an image, for example when we need to use the value of the surface reflectance. We might then need to apply atmospheric correction. In essence, this is about modeling the amount of radiation arising from the above described processes and subtracting them from the measured values. Atmospheric correction models, such as 6s, used in GRASS GIS, are based on radiative transfer theory and depend on information about current atmospheric conditions at the moment of image acquisition, be them measured (rarely) or assumed (most often). We here show a simple application of the haze correction method, a simple yet effective method that relies on image values and simple assumptions about the effects of the atmosphere for visible and near infrared Landsat bands. In the tradition of this blog, we focus on the fundamental aspects, leaving sophistication aside. As an example, we use a subset of a Landsat 8 image of the Bay of Naples, Italy, subject of previous posts.

The central assumption of the method is that all values in an image are shifted upwards due to the additive effects of scattering, and that the effects are uniformly distributed across the region covered by the image. Thus, if we could find the value of this upward shift in each band, we could simply subtract it from each pixel and correct the image. To find these values we plot histograms of the bands and inspect them to find the minimum significant values in each band. We begin with the reflectance in bands 2 (blue), 3 (green), 4 (red) and 5 (near infrared):

Histograms of reflectance in bands 2, 3, 4 and 5, of the Landsat 8 image path/row 189/32.

Histograms of reflectance in bands 2, 3, 4 and 5, of the Landsat 8 image path/row 189/32.

We clearly see that band 2, blue, has the largest shift in the histogram, the signature of Rayleigh scattering. Detailed inspection of the data for each histogram enables us to figure out the thresholds for the significant values:

band 2: 0.15
band 3: 0.11
band 4: 0.07
band 5: 0.04

We then subtract these threshold values from each band. We do this using grdmath, the NetCDF calculator of the Generic Mapping Tools (GMT) toolbox:

grdmath vesuvius_b2_refTOA.grd 0.15 SUB = vesuvius_b2_refTOA_atcorr.grd
grdmath vesuvius_b3_refTOA.grd 0.11 SUB = vesuvius_b3_refTOA_atcorr.grd
grdmath vesuvius_b4_refTOA.grd 0.07 SUB = vesuvius_b4_refTOA_atcorr.grd
grdmath vesuvius_b5_refTOA.grd 0.04 SUB = vesuvius_b5_refTOA_atcorr.grd

Note that, to do this, we first have to convert our GeoTIFF files to NetCDF using the GDAL utility gdal_translate. See previous posts. A comparison of our results, in band 2, with the uncorrected image readily shows the effects of “haze removal”:

Band 2 (blue), before (left) and after (right) removing the 'haze'

Band 2 (blue), before (left) and after (right) removing the ‘haze’

A comparison of false color composites, using bands 5, 4 and 3 (RGB), shows the same but in color:

False color composite (bands 5, 4 and 3 RGB), before (left) and after (right) haze removal.

False color composite (bands 5, 4 and 3 RGB), before (left) and after (right) haze removal.

Finally, we might calculate the Normalized Difference Vegetation Index (NDVI):

NDVI = \frac{\rho_{ir} - \rho_{r}}{\rho_{ir} + \rho_{r}}

using our corrected images, and compare with the results with the uncorrected ones:

Normalized Difference Vegetation Index (NDVI) calculated before (left) and after (right) haze removal.

Normalized Difference Vegetation Index (NDVI) calculated before (left) and after (right) haze removal.

Although there is no big visual difference, the image to the right is arguably better because the range of values in the corrected bands is larger, thus allowing us to reveal vegetation patterns more clearly.

This method is closely related to the Dark Object Subtraction (DOS) method, which is based on the identification of dark surfaces, clean lakes or asphalt, and assuming that their reflectances are only due to atmospheric scattering, subtracting their values from the image. The difference here is that we do not have to rely on the existence of such surfaces within our region. In a sense, haze removal by histogram minimum is technically equivalent to a linear stretch, where we cut a fraction of the lowest values, except that in this case we have a reasonable geophysical principle to rely on. Although not perfect, the haze removal by histogram minimum is a robust and simple method that is very reliable as a first approximation. In many cases it might be the only atmospheric correction we need.

Hillshading, useful fun with digital elevation models

Hillshading is the art and science of making a graphical representation of an illuminated landscape. It has a long history that includes names as Leonardo da Vinci (with the chiaroscuro technique) and many famous and not so famous Swiss and Japanese cartographers (have a look at some examples of mastery on manual hillshading by Eduard Imhof). A very complete account can be found in the excellent and very enjoyable paper “Hill shading and the reflectance map”, by Berthold Horn [1].

Shading has an intuitive appeal because it conveys information on shape in a straightforward manner. This is best envisaged as the difference in perception that a person experiences during a cloudy or a sunny day. In a cloudy day light is diffuse and does not produce shadows, giving the impression of flatness, whereas in a sunny day shadows allow us to make an immediate mental model of the shapes and enables better estimation of distances. This is very relevant in many practical situations, in particular “… when the interpreter’s time is limited, as in aviation, for users that are not trained cartographers, and for small scale maps, where contours degenerate into messy tangles of lines.” [1]

Most importantly, the techniques involved in computing shaded maps are fundamental for applying topographic correction to satellite images. They can also become important when it is relevant to convey information about topography as context to something else, but when a exact quantitative specification is not necessary nor desirable, as in tourist maps or as background information in geoscientific studies, as shown below.

Palaeo-ice-stream onsets in south-central Baffin Island, Amadjuak Lake region. Section of geological map of Canada (GSC 1860A*; Wheeler et al., 1997) draped over shaded relief with superimposed landforms. Map 1860A is produced, compiled and published by the Geological Survey of Canada (GSC). This section is reproduced here according to the permissions granted by the GSC for non-commercial applications. Modified from De Angelis and Kleman (2008) [2].

Palaeo-ice-stream onsets in south-central Baffin Island, Amadjuak Lake region. Section of geological map of Canada (GSC1860A*; Wheeler et al., 1997) draped over shaded relief with superimposed landforms. Map 1860A is produced, compiled and published by the Geological Survey of Canada (GSC). This section is reproduced here according to the permissions granted by the GSC for non-commercial applications. Modified from De Angelis and Kleman (2008) [2].

Every modern GIS package provides a convenient way to compile a shaded relief map, often including very involved corrections. We here focus on the fundamental aspects, leaving refinements aside, and will confine ourselves to the important and sufficiently general case of Lambertian surfaces, using as an example the DEM we compiled in previous posts of the region around Vesuvius volcano, in Italy.

The central concept here is to understand that the irradiance, the amount of light coming upon a surface element, depends on the position of the light source relative to the surface element. The radiance, the amount of light being reflected by the surface element, depends on the irradiance but also on the reflectance properties of the surface. As we mentioned, we here focus on Lambertian surfaces, reflecting diffusively and equally on all directions, in which case we can safely assume that radiance will be proportional to irradiance. For natural surfaces illuminated by the Sun the problem of finding the irradiance, or solar incidence, reduces to an application of the spherical law of cosines:

cos \gamma_{i} = cos \theta_{p} cos \theta_{z} + sin \theta_{p} sin \theta_{z} cos (\phi_{a} - \phi_{o})

where \gamma_{i} is the solar incidence, \theta_{p} is the slope of the surface, \theta_{z} is the zenith angle, or the angle between the Sun position and the vertical, \phi_{a} is the the Sun azimuth from the North, and \phi_{o} is the surface aspect, that is, the direction in which the surface is sloping.

We begin by calculating the slope and aspect from the DEM:

1. slope:

\theta_{p} = atan ( \sqrt{ (\frac{dz}{dx})^{2} + (\frac{dz}{dy})^{2} })

we can easily apply this manipulation using grdmath, the GMT reverse polish notation calculator, on our DEM grid, here denoted dem.grd:

grdmath dem.grd DDX dem.grd DDY R2 SQRT ATAN R2D = slope.grd

Note that the function R2 is equal to the sum of the squares of the first two terms, and R2D is the Radian To Degree converter.

2. aspect:

\phi_{o} = atan (\frac{dx}{dy})

and similarly:

grdmath dem.grd DDX dem.grd DDY ATAN2 R2D 180 ADD = aspect.grd

Note that we used the function ATAN2, which specifically accounts for the signs of the numerator and denominator and thus gives the angles in the right quadrant. We add 180 degrees to get angles from 0 to 360, instead in the range -180 – +180.

Our partial results look like this:

Slope (left) and aspect (right) of the Vesuvius DEM.

Slope (left) and aspect (right) of the Vesuvius DEM.

We can now turn into computing cosine of the irradiance. As Sun azimuth and elevation we choose 315° and 45°, that is: illumination from the northwest at half sky, following the cartographic convention for shaded maps:

We do:

grdmath slope.grd D2R COS 45 D2R COS MUL slope.grd D2R SIN 45 D2R SIN MUL 315 D2R aspect.grd D2R SUB COS MUL ADD = cosgammai_315.grd

(this can also be accomplished using GMT’s grdgradient, which also has a function for specifying the properties of the reflective surface)

Our results look like this:


Presto! This is an image upon which we could drape a photograph or satellite image, or even draw the contours. This is also a fundamental piece of information needed for topographical correction of images, a topic of a previous post.

[1] Horn, B. K. (1981). Hill shading and the reflectance map. Proceedings of the IEEE, 69(1), 14-47. Find it in ResearchGate

[2] De Angelis, H. & Kleman, J. (2008). Palaeo‐ice‐stream onsets: examples from the north‐eastern Laurentide Ice Sheet. Earth Surface Processes and Landforms, 33(4), 560-572. Find it in ResearchGate.

A reflection on learning in technology dominated fields

This post is a short reflection on some aspects of the influence of technology in learning, and is based on my experience as teacher of remote sensing and GIS in higher education settings. I nevertheless think that many aspects of this reflection apply to other fields as well.

A comfortable and amusing side of modernity is that we have at our disposal a plethora of fancy software that will, at a mouse click, create beautiful results without us having to bother about the boring details. This is particularly evident in computing, where we can rapidly get results from data, but more strikingly in visualization problems, where shiny colorful plots can be created in a second. This is all good and fine, but there are dangerous side effects of this.

The heavy price we pay for this handiness is that we very often forget, or even neglect to adequately learn, the essential ideas behind the methods and, in extreme cases, even their purpose. In this way, we unnecessarily expose ourselves to the risk of ignoring the potentials and limitations of our tools. I think that this is most clearly evident when it comes to visualization, a complex field in which many software packages today enable us to create shockingly colorful charts in an eye cast. In this sort of double-sided sword, the tools invite us to produce results rapidly thereby opening the door to forget fundamental aspects of visualization like scientific purpose and human perception.

Conceptual model of scientific visualization, with the three outstanding dimensions and their caveats.

Conceptual model of scientific visualization, with the three outstanding dimensions and their caveats.

Another trap lurking behind is that we may be caught unaware by the fact that we usually become addicted to specific software tools, particularly of the GUI variety, and might end up associating a given procedure with a sequence of clicks on virtual buttons, instead of ideas and equations that can be done in every computer language or even paper and pencil (albeit not recommendable!).

As a teacher in remote sensing and GIS I have seen this unfortunate situation in many higher-education settings, where for a sizable part of students learning goes as to learn some loose basics and then jump to click on programs and produce results. This is not learning, at least not for me. The seriousness of the problem is manifested by the typical question ‘ok, now, how do I do this or that?‘, meaning: ‘which button do I click on now?‘ I assume that this approach will be then transported to other settings, industrial or academic, as the now experts, pressed to produce rapid results but acquainted with performing series of clicks in a particular software package, become fragile to changes in software packages, tool configurations and new problems.

I believe that all this is avoidable. Focusing on the essential core of principles in a discipline can help people to construct their knowledge, upon which long-term academic excellence and professional expertise can be built. In this sense, I agree with the late mathematician Richard Hamming in that developing an attitude of learning to derive results from principles is essential. Learning the principles behind the methods is, as much as learning to learn, a cornerstone of lifelong learning. I believe that stressing on this approach is a way of helping our students become more adaptable and resilient in a changing world.

Getting started with SRTM elevation data: downloading, processing, clipping and visualizing.

The Shuttle Radar Topography Mission (SRTM) is a tremendous resource because of its high accuracy and its consistency as a “snapshot” of Earth’s surface, acquired over just 10 days during February 2000. It is also freely available for anyone! SRTM was a specially designed mission of the Endeavour space shuttle, with a specially built sensor antenna, and aimed to map the Earth’s land topography using radar interferometry. As a result, during the 10 days of the mission, the land surface topography between latitudes 60°N and 56°S, or about 80% of Earth’s surface, was mapped with a final cell size of 1 arc-second (about 30 m) and absolute elevation errors under 10 m. More details about the mission can be found in the paper by Farr and collaborators [1] (which is open access and can be downloaded here).

SRTM data have been publicly available relatively soon after the data was processed. Three versions exist: version 1, the original sensor raw data; version 2, a processed version with data voids (arising from weak radar returns or other anomalies caused by, for example, deep valleys); and version 3, the void-filled version, compiled with the help of ASTER DEM. During many years, though, only data over the United States was available as V3 at the maximum resolution of 1 arc-second, whereas data for the rest of the world was offered as V2, 3 arc-second, or about 90 m pixel size. Fortunately, a recent happy decision by the US government has finally made available all data at full resolution for the whole globe. The process is gradual and is expected to be finished by next year.

Luckily, for our Landsat 8 example, the Bay of Naples is already available with 1 arc-second as of November 2014. We proceed then to download the data and prepare them. We start by opening the Earth Explorer site ( and choose the area of interest, either with the path/row or, as shown below, specifying a coordinate area with the map in the menu to the left:


we then choose the SRTM dataset in the menu to the left, under the tab ‘Data Sets’ by navigating the menu as follows: DigitalElevation -> SRTM -> SRTM 1 Arc-Second Global:


Clicking in “Results” will show us the 1° x 1° tiles in which SRTM data are compiled. We simply download them, using the icon with a green arrow next to each result item in the results list. I here chose to get the four westernmost tiles:


Once downloaded we get the four files:


These are GeoTIFF files. It is also possible to get the raw format grids, “hgt” files, but these require a bit more involved processing, so we here proceed with the GeoTIFFs. The name of the files refer to the coordinates of the lower left corner of each tile. Each of the files contains 3601 x 3601 elements, in 16 bit integer data type, with missing values signaled as -32767. The file coordinates are given as longitude and latitude, referred to the WGS84 ellipsoid, with elevation expressed in meters referred to the mean sea level (EGM96 geoid). As is tradition in this blog, we use the free and open source tools GDAL and Generic Mapping Tools (GMT) from the command line:

> gdalinfo n40_e013_1arc_v3.tif
Driver: GTiff/GeoTIFF
Files: n40_e013_1arc_v3.tif
Size is 3601, 3601
Coordinate System is:
SPHEROID["WGS 84",6378137,298.257223563,
Origin = (12.999861111111111,41.000138888888884)
Pixel Size = (0.000277777777778,-0.000277777777778)
DTED_UniqueRef_DSI=F02 011
DTED_UniqueRef_UHL=F02 011
Image Structure Metadata:
Corner Coordinates:
Upper Left ( 12.9998611, 41.0001389) ( 12d59'59.50"E, 41d 0' 0.50"N)
Lower Left ( 12.9998611, 39.9998611) ( 12d59'59.50"E, 39d59'59.50"N)
Upper Right ( 14.0001389, 41.0001389) ( 14d 0' 0.50"E, 41d 0' 0.50"N)
Lower Right ( 14.0001389, 39.9998611) ( 14d 0' 0.50"E, 39d59'59.50"N)
Center ( 13.5000000, 40.5000000) ( 13d30' 0.00"E, 40d30' 0.00"N)
Band 1 Block=3601x1 Type=Int16, ColorInterp=Gray
NoData Value=-32767
Unit Type: m

We now proceed to compile a mosaic with our four files:

> -o bayofnaples.tif -n -32767 *.tif
0...10...20...30...40...50...60...70...80...90...100 - done.

Note that we have specified the no-data value, with the option ‘-n’. We now have a file with the following characteristics:

> gdalinfo bayofnaples.tif
Driver: GTiff/GeoTIFF
Files: bayofnaples.tif
Size is 7201, 7201
Coordinate System is:
SPHEROID["WGS 84",6378137,298.257223563,
Origin = (12.999861111111111,42.000138888888884)
Pixel Size = (0.000277777777778,-0.000277777777778)
Image Structure Metadata:
Corner Coordinates:
Upper Left ( 12.9998611, 42.0001389) ( 12d59'59.50"E, 42d 0' 0.50"N)
Lower Left ( 12.9998611, 39.9998611) ( 12d59'59.50"E, 39d59'59.50"N)
Upper Right ( 15.0001389, 42.0001389) ( 15d 0' 0.50"E, 42d 0' 0.50"N)
Lower Right ( 15.0001389, 39.9998611) ( 15d 0' 0.50"E, 39d59'59.50"N)
Center ( 14.0000000, 41.0000000) ( 14d 0' 0.00"E, 41d 0' 0.00"N)
Band 1 Block=7201x1 Type=Int16, ColorInterp=Gray

Since we plan to use this elevation model with our image from the Vesuvius volcano (this one), we proceed to clip the DEM to the same area as the image, reprojecting it at the same time to the UTM33N projection, and taking care to specify a 30 m pixel size, so there will be a one-to-one correspondence between the cells in the image and DEM. We accomplish this using gdalwarp:

> gdalwarp -t_srs 'EPSG:32633' -te 430995 4489005 470985 4539015 -tr 30 30 bayofnaples.tif bayofnaples_UTM33N.tif
Creating output file that is 1333P x 1667L.
Processing input file bayofnaples.tif.
0...10...20...30...40...50...60...70...80...90...100 - done.

our new file has the following properties:

> gdalinfo bayofnaples_UTM33N.tif
Driver: GTiff/GeoTIFF
Files: bayofnaples_UTM33N.tif
Size is 1333, 1667
Coordinate System is:
PROJCS["WGS 84 / UTM zone 33N",
SPHEROID["WGS 84",6378137,298.257223563,
Origin = (430995.000000000000000,4539015.000000000000000)
Pixel Size = (30.000000000000000,-30.000000000000000)
Image Structure Metadata:
Corner Coordinates:
Upper Left ( 430995.000, 4539015.000) ( 14d10'46.25"E, 40d59'57.85"N)
Lower Left ( 430995.000, 4489005.000) ( 14d11' 6.14"E, 40d32'56.15"N)
Upper Right ( 470985.000, 4539015.000) ( 14d39'17.97"E, 41d 0' 6.51"N)
Lower Right ( 470985.000, 4489005.000) ( 14d39'26.33"E, 40d33' 4.67"N)
Center ( 450990.000, 4514010.000) ( 14d25' 9.19"E, 40d46'32.18"N)
Band 1 Block=1333x3 Type=Int16, ColorInterp=Gray

Finally, we can display a nice color plot using GMT. We begin by translating our GeoTIFF to a NetCDF grid:

> gdal_translate -of NetCDF bayofnaples_UTM33N.tif bayofnaples_UTM33N.grd
Input file size is 1333, 1667
0...10...20...30...40...50...60...70...80...90...100 - done.

We then clip the values below 0, setting them to zero, to facilitate colour coding:

> grdclip bayofnaples_UTM33N.grd -Gnew.grd -Sb0/0

The new grid, new.grd, has the following properties:

> grdinfo new.grd
new.grd: Title: GDAL Band Number 1
new.grd: Command: grdclip bayofnaples_UTM33N.grd -Gnew.grd -Sb0/0
new.grd: Remark:
new.grd: Gridline node registration used [Cartesian grid]
new.grd: Grid file format: nf = GMT netCDF format (32-bit float), COARDS, CF-1.5
new.grd: x_min: 431010 x_max: 470970 x_inc: 30 name: x coordinate of projection [m] nx: 1333
new.grd: y_min: 4489020 y_max: 4539000 y_inc: 30 name: y coordinate of projection [m] ny: 1667
new.grd: z_min: 0 z_max: 1442 name: GDAL Band Number 1
new.grd: scale_factor: 1 add_offset: 0
new.grd: format: netCDF-4 chunk_size: 134,129 shuffle: on deflation_level: 3

We then proceed to make a nice color table, based on the master table dem2, using makecpt:

> makecpt -Cdem2 -T0/1500/10 > color.cpt

For visualization, it might be nicer if we indicate that the lowest values, the sea level, have to be blue. We do this by opening a text editor and adding a first line that encompasses the values from 0 to 0.1, with blue color (0/0/255 stands for 0 red, 0 green, 255 blue):

0 0/0/255 0.1 0/0/255
0.1 5.2267/105.17/63.16 10 5.2267/105.17/63.16
10 15.68/121.5/47.48 20 15.68/121.5/47.48

We can then plot the grid using grdimage:

> grdimage new.grd -JX15 -Rnew.grd -Ccolor.cpt -X1 -Y1 -P >

and export the postscript file as a raster, and clipping the original page to the area of the map:

> ps2raster -A

Digital elevation model covering the same area that the Vesuvius image, and with the same pixel size.

Digital elevation model covering the same area that the Vesuvius image, and with the same pixel size.

Presto! In the next post we will compute grids of aspect and slope from this DEM.

[1] Farr, T.G. et al. 2007. The shuttle radar topography mission. Reviews of geophysics 45.2.

Working with Landsat 8, a tutorial. 3. conversion to radiance and brightness temperature at top-of-atmosphere

In this third post of the Landsat 8 series we will implement, step by step, the procedures explained in the Landsat web site for converting the digital numbers in the satellite images to geophysical measurements of spectral radiance, as measured at the sensor and without atmospheric correction, called ‘top-of-atmosphere’ or simply TOA.

The images that we download as TIF files are coded into 16-bit unsigned integer images. These are referred to as digital numbers. For some simple purposes, as for example, measuring the extent of features identifiable by eye, these digital numbers are perfectly sufficient. However, when we need to assess the geophysical properties of the terrain or make studies of land cover change over time, we might find convenient to convert these raw digital numbers to their original values of radiance, mostly to be able to compute reflectance, and compare surfaces on a solid geophysical basis.

Converting to radiance is extremely easy because the images are coded using a linear relationship between radiance measurement and digital numbers. For all bands, we retrieve these values by calculating:

L_{\gamma} = M_{L} Qcal + A_{L}

where L_{\gamma} is the spectral, or band-wise, TOA radiance, M_{L} is a multiplicative term, the slope of the line in the linear relationship, Qcal is the digital number, and A_{L} is the additive linear term, the value at the origin. This will give us measurements in:

\frac{W}{m^{2} ~ srad ~ \mu m}

or Watts per square meter (surface) per steradian (solid angle) per micrometer (wavelength), which is the actual radiometric information that sensors collect.

We can accomplish this for each band by searching the metainformation text file *_MTL.txt and extracting for each band the values of RADIANCE_MULT_BAND_X and RADIANCE_ADD_BAND_X, corresponding to the parameters M_{L} and A_{L} in the above equation, respectively. I perform these operations via a Perl script, extracting the parameters using regular expressions, storing them in variables ($RADMULBAND and $RADADDBAND below), and then using the powerful NetCDF calculator grdmath as a raster calculator, in Generic Mapping Tools, and GDAL tools:

gdal_translate -of NetCDF $band.TIF $band.grd
grdmath $band.grd $RADMULBAND MUL $RADADDBAND ADD = $band-radTOA.grd
gdal_translate -ot Float32 $band-radTOA.grd $band-radTOA.tif

In this snippet, I first convert the band file to NetCDF using gdal_translate, then perform the operation with grdmath, and finally export to GeoTIFF using gdal_translate. The results, for band 8, the panchromatic, in the area around the Vesuvius volcano looks like this:

Subset of a Landsat 8, band 8 (panchromatic) image, depicting radiance at top-of-atmosphere of the Vesuvius area, on 29 July 2013.

Subset of a Landsat 8, band 8 (panchromatic) image, depicting radiance at top-of-atmosphere of the Vesuvius area, on 29 July 2013.

The image looks very similar to the uncorrected one because what we made is just a linear transformation, so the relationship between nearby pixels has not been changed. However, the image now is in 32-bit floating point format, and has double the size in memory.

For the particular case of the thermal bands (bands 10 and 11), we can proceed to calculate the brightness temperature, again at TOA, without correcting for the atmospheric distortions, by applying:

T = \frac{K_{2}}{ln(\frac{K_1}{L_{\gamma}} + 1)}

where K1 and K2 are constants called K1_CONSTANT_BAND_x and K2_CONSTANT_BAND_x, respectively, in the metainformation file. As before, we extract these using a regular expression within a script, and proceed to generate a corrected TIF using:

grdmath $K1 $band-radTOA.grd DIV 1 ADD LOG INV $K2 MUL = $band-temp.grd
gdal_translate -ot Float32 $band-temp.grd $band-temp.tif

The results are in degrees Kelvin [K], which we can easily transform to Celsius by subtracting the value at 0°C, 273.15 °K. The brightness temperature in band 10, together with the difference between band 10 and band 11 look like this:

Brightness temperatures (°C) calculated for the Vesuvius area on 29 July 2013, at satellite pass time, using band 10 (left), and difference with temperature in band 11 (right).

Brightness temperatures (°C) calculated for the Vesuvius area on 29 July 2013, at satellite pass time, using band 10 (left), and difference with temperature in band 11 (right).

As you see, the difference is very small, with mean -1.79 °C, and outliers on the top of the volcano and on some urban areas. Note how the northern slopes are slightly colder than southern ones. Pay also attention to the fact that these are uncorrected for atmospheric effects and might therefore depart from the real temperatures measured on surfaces at the same time. The above figures where drawn using R, which can also be successfully used to perform the calculations described in this page.

Finally, the Landsat use page also points to an illumination correction by which one can calculate reflectance. The procedure is the same as above, applying a linear transformation using the digital numbers and two parameters (REFLECTANCE_MULT_BAND_X, REFLECTANCE_ADD_BAND_X). Landsat 8 data comes in L1T processing level, meaning that they have already been corrected for topographic effects. According to the Landsat website this is done using a combination of SRTM, ASTER DEM, etc. These conversions work fairly well in most cases, but might not be entirely correct for polar areas, for example, where the existing DEM might be too coarse.