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National Operational Hydrologic
Remote Sensing Center

Robert W. Maxson1, Milan W. Allen1, Timothy L. Szeliga1

theta - Image Classification by Comparison of Angles Created Between Multi-Channel Vectors and an Empirically Selected Reference Vector


NOHRSC captures AVHRR data directly from the NOAA polar orbiting satellites. The 10-bit integer transmission format is translated to 16-bit as each individual band is extracted. These data are geometrically corrected, calibrated to percent albedo and temperature, navigated to find earth location and reprojected to a 30 arc-second latitude/longitude grid. Full resolution GOES (GVAR formatted) data are purchased from a private vendor and acquired over a dedicated T1 line. These data are navigated, calibrated, and converted to McIDAS area format. GOES-East and GOES-West visible data are calibrated to albedo. Infrared data are calibrated to temperature and resampled to match the 1 KM resolution of the visible data using an inverse distance function. All bands are reprojected to a 30 arc-second latitude/longitude grid.

Multi-spectral scanners, such as those carried on board the NOAA polar orbiting (AVHRR) and geostationary (GOES) satellites, image the earth's surface by sampling multiple electromagnetic spectrum frequency bands. Depending upon the reflective or emissive properties of the imaged surface, each band may contain information which allows analysts to classify sub-portions of the image into spectrally distinct categories. For example, snow may be distinguished from cloud or snow-free land may be distinguished from water.

It is often impossible to separate one feature from another when only a single band of the electromagnetic spectrum is examined. For example, in the visible portion of the spectrum, both snow and clouds are essentially white. However, when multiple bands are examined simultaneously, the discriminating power increases dramatically.

Differences emerge which are a consequence of variations in the spectral response across the spectrum exhibited by different materials. For example, while snow and clouds appear to be white in the visible spectrum, the clouds may appear much warmer in the thermal portion of the spectrum.

The ability to discriminate one feature from another depends on two factors:

  1. being able to distinguish differences between each feature on the basis of spectral response
  2. being able to image these features with sensors sensitive to those portions of the spectrum in which they are spectrally distinct.

NOHRSC Window 1: AVHRR Channel 2 - 8 February 1995

The National Operational Hydrologic Remote Sensing Center (NOHRSC) in the Office of Hydrology of the National Weather Service has developed an operational multi-band snow classification algorithm theta designed to separate snow from cloud, land and water in North America. The theta algorithm has been used successfully to map the areal extent of snow using both AVHRR and full-resolution GVAR imagery.

theta assumes multi-band imagery as input. Each row-column coordinate location in the image is treated by theta as a vector quantity. The components of each vector are calibrated digital counts for the combination of bands selected for the analysis. Presently, NOHRSC uses two classification routines and then differences the two resultant images to map the areal extent of snow. The first routine uses channels 3, 4, and 5 in AVHRR data or channels 2, 4, and 5 in GOES data to form a three-space vector algorithm to identify clouds. The second routine uses channels 1, 2, 3, and 4 in AVHRR data and channels 1, 2, 4, and 5 in GOES data to form a four-space vector algorithm to identify the areal extent of both snow and clouds.


Three-Space Theta Schematic

To simplify illustration, the discussion of theta will be limited to a three-band AVHRR example. However, the theta algorithm is extensible to n-space for any multispectral sensor. In either case, we treat the n-band image as a ROW x COL collection of order n vectors.

At a particular row and column location r,c in the co-registered image,
we define vector Fr,c as (a i + b j + c k) and vector Gr,c as (d i + e j + f k).
Vector F is formed by using the values for location r,c for AVHRR channels 3, 4, and 5 (degrees Kelvin, real values) as the vector components. Again, the non-zero reference vector G may be positioned anywhere in space depending on the aim of the classification analysis.
The magnitudes of vectors F and G are (a2 + b2 + c2)1/2 and (d2 + e2 + f2)1/2 respectively.
We express the angle between these two non-zero vectors with the following equation:

cos(theta) = (F dot G)/(||F|| ||G||) ááááááááááááááááááááááááááááááááááááááááááááááá[1]

where the cosine of the angle between the image vector F and the reference vector G is expressed as the dot (or scalar) product of the two vectors divided by the product of their magnitudes. The dot product of F and G is also a scalar and is equal to ((a * d) + (b * e) + (c * f)). ááátheta is then found by calculating the arc cosine of the right hand side of the equation and converting from radians to degrees.

Members of the same classification category in the image tend to have similar values in each band and by extension similar theta angles. Members of different classes will exhibit theta angles which should align on opposite sides of a threshold angle. The theta algorithm offers an opportunity to exploit clustering tendencies using an efficient algorithm for excising a targeted class. The algorithm is fast enough that near-real time configuration and execution is feasible.

It should be fairly obvious that the success of the algorithm is highly dependent on a number of factors:

  • The selection of bands for that target class. At NOHRSC the selection of bands was based on known physical properties of snow and clouds.
  • The selection of an appropriate reference vector. This too is based largely on known properties of snow and cloulds and refined by multiple iterations of the algorithm varying the reference point between executions.

In this example, the reference vector G was initially defined as vector ( 1, 1, 1 ) and executed on imagery known to contain clouds. After several empirical shifts of the reference vector, it was discovered that the vector formed by ( 1, -1, -1 ) yielded the strongest cloud discrimination.

Three-Space Cloud Identification Algorithm (AVHRR)

Theta Angles: AVHRR Channels 3,4,5

In images created using these techniques, cloud surfaces start at the minimum theta values found in the synthetic image and end within one standard deviation of the theta simple mean. Because clouds are the minimum values in the image it is relatively easy to set the proper threshold after the image is created by using graphical masking subroutines. By creating the theta image as a real image, extremely fine adjustments can be made in setting the single cloud, no-cloud threshold.

Theta Cloud Screen-AVHRR Channels 3,4,5

It is interesting to note that this cloud screen detects the presence of both cirrus and cumulus clouds. Additionally, it is capable of distinguishing cirrus clouds that overlay the snowpack, a feature that has proven to be difficult for simple multi-spectral algorithms to resolve.

Cloud and Snow in Four and Five-Space (AVHRR)

The four and five space algorithms use visible and near-infrared data that must be corrected for variations in solar illumination and sun angle across each scene. Each AVHRR visible and near-infrared pixel and each GOES visible pixel is normalized to solar noon using a simple cosine correction. A scaling factor is calculated for each pixel by computing the cosine between tha actual sun angle and the sun directly overhead.

As well as the differences created in the various theta images by varying the position of the reference vector, channel combinations can be changed to create new images. Further, the technique examined in Case 1, which is geometrically viewed in three-space, can be extended beyond three-space into n-space. Two combinations of AVHRR channels have been investigated and both offer similar results.

Theta Snow and Cloud Screen-AVHRR Channels 2,3,4,5

The first approach uses channels 1, 2, 3, and 4 as a four-space vector and a reference vector (-1,1,1,1). The angle theta is calculated using equation 1.0 which is extended into n-space. Very strong surface separation is noted between snow and cloud and all other surfaces (water, snow-free land). Snow and cloud, in aggregate, are represented by the lowest values of images created using these combinations. As theta increases, the next surface to be separated is snow free land, followed by large bodies of water. The four channel vector combination outlined above can be used to separate snow and cloud from all other surfaces on the image. The three-space vector combination used above can then be applied to separate the cloud from the snow. If desired, large bodies of water can also be segregated by using this four channel vector technique.

Combined Theta Output

An alternate approach is very similar to the four-space procedure only now channels 1,2,3,4, and 5 along with a reference vector of (1,1,1,-1,1) are inserted in to the original equation. When this combination is used, the output is nearly identical to that used with the four channel procedure .

Comments on GOES Imagery

The method of image classification that has been described was originally developed using AVHRR image data. Once it proved to be useful for those data the similarities in the spectral response of AVHRR sensors and those flown on the next-generation GOES satellites (GOES-8 and GOES-9) allowed the technique to be extended for use on full resolution GOES data. The successful implementation of these algorithms with GOES data permits NOHRSC to map on a near daily basis, the extent of cloud cover notwithstanding.

Comparison of the AVHRR channels to GOES channels
Comparison of the AVHRR Channels to GOES channels

Note that these crude comparsions are only applicable for next-generation GOES satellites .
Also, be aware that channel 1 GOES data is approximately one kilometer in resolution, which closely matches all the AVHRR channels in resolution. The remaining GOES thermal channels have an approximate nadir resolution of four kilometers. GOES satellites do not have a near-infrared sensor in the .75 - 1.10 Ám range, corresponding to AVHRR band 2.


The technique described is based on the fact that different physical surfaces will behave differently given the same solar input and will therefore reside in different spatial regions formed in vector space. These spatial regions can be simply differentiated by examining the angle formed between the multiple channel pixel vector and a common unit vector. The classification routine is less effective in the presence of cloud shadows and under estimates the areal extent of snow in heavily forested areas.

The NOHRSC snow-mapping algorithms are not automated and require the analyst to set two thresholds for each image classification. Additionally, the analyst must resolve mixed pixel regions subjectively. Further, this routine is a daylight-only classification technique.


The NOHRSC snow classification routine is a subjective technique that isolates surfaces of interest by examing the angle formed between a multi-channel vector and a unit reference vector. It is easy to code and runs very quickly. Two intermediate images are formed, one that isolates cloud and the other that isolates snow and cloud in combination. The analyst must set thresholds for each of these images. The two intermediate images are then differenced to determine the overall areal extent of snow.

Future plans will integrate this snow mapping technique with other snow classification algorithms to form a more automated, spatially-based snow classification system.


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