Module 7
Caribbean Regional Node

To learn how to carry out "depth-invariant" processing in order to compensate for the effect of light attenuation in the water column (i.e. water depth) on bottom reflectance using a CASI airborne image.
 
 

This lesson relates to material covered in Chapter 8 of the Remote Sensing Handbook for Tropical Coastal Management and readers are recommended to consult this for further details of the techniques involved. Some familiarity with Excel spreadsheets is needed to complete the lesson in full.

Familiarity with Bilko for Windows 2.0 is required to carry out this lesson. In particular, you will need experience of using Formula documents to carry out mathematical manipulations of images.

A Canadian airborne multispectral digital imager called the Compact Airborne Spectrographic Imager (CASI) was mounted on a locally-owned Cessna 172N aircraft using a specially designed door with mounting brackets and streamlined cowling. An incident light sensor (ILS) was fixed to the fuselage so that simultaneous measurements of irradiance could be made. A Differential Global Positioning System (DGPS) was mounted to provide a record of the aircraft's flight path. Data were collected at a spatial resolution of 1 m² in 8 wavebands (Table 5.1) during flights over the Cockburn Harbour and Nigger Cay areas of South Caicos, Turks and Caicos Islands (21^o 30' N, 71^o 30' W) in July 1995. Further details are given in Clark et al. (1997).

Water column correction (depth-invariant processing) will be carried out on a CASI image of the area to the south of the island of South Caicos. The CASI image was acquired at approximately 10 a.m. local time on 16 July 1995. For the purposes of this lesson Bands 3 and 4 (green) of the CASI image (CASI3C16.DAT and CASI4C16.DAT) will be processed to create a depth-invariant image. The CASI image has undergone geometric correction and radiometric correction to apparent reflectance. Full atmospheric correction to surface reflectance would require use of the 6S radiative transfer code but we will carry out a crude correction using the dark pixel subtraction (DPS) method. The images are stored as 16-bit integer data (2 bytes per pixel) although the underlying sensor data is 12-bit (allowing a radiometric resolution of 4096 which is 16 times as sensitive as Landsat TM and SPOT XS).

When light penetrates water its intensity decreases exponentially with increasing depth. This process is known as attenuation and it exerts a profound effect on remotely sensed data collected over water. The severity of attenuation differs with the wavelength of electromagnetic radiation. The red part of the visible spectrum attenuates more rapidly than shorter wavelength blue light and infra-red light hardly penetrates water at all. Thus as depth increases, longer wavelengths are progressively absorbed and the spectra of habitats as seen at the water surface change. The spectrum of sand at a depth of 2 m will be very different to that at 20 m - yet the substratum is the same. In fact, the spectral signature of sand at 20 m may be similar to that of seagrass at (say) 3 m. The spectral radiances recorded by a sensor are therefore dependent both on the reflectance of the substrata and on depth. The influence of depth on the signal will create considerable confusion when attempting to use visual inspection or multispectral classification to map habitats. Since most marine habitat mapping exercises are only concerned with mapping benthic features, it is useful to remove the confounding influence of variable water depth. This lesson describes a fairly straightforward means of compensating for variable depth which is applicable to clear waters such as those surrounding coral reef environments.

Jerlov (1951) formally classified oceanic water types according to their optical attenuation properties. Type I waters were represented by extremely clear oceanic waters. Most clear coastal waters were classified as Type II because attenuation tends to be greater than that for oceanic waters of low productivity. Most reefal waters fall into categories I or II. Type III waters are fairly turbid and some regions of coastal upwelling are so turbid that they are unclassified.

Lyzenga (1978, 1981) described a simple technique for removing the influence of depth from spectral data for Type I and II waters. This is thus suitable for the clear reefal waters of the Caicos Bank and is the method which will be used in this lesson.

The method involves four steps.

The first step is a crude atmospheric correction based on the "dark pixel subtraction" method. If a full atmospheric correction (as done in Lesson 3) had already been carried out this step would not be needed and the surface reflectance values could be used directly. The CASI image we are using here has not been atmospherically corrected so that you should carry out this step. This involves selecting a large number of pixels from "deep water" and calculating their average apparent (at sensor) reflectance (and its standard deviation). This value minus two standard deviations is then subtracted from all other pixels in each band respectively to give a crudely corrected reflectance.

Atmospherically corrected reflectance = L_i - L_si   Equation 5.1

where L_i is the apparent pixel reflectance in band i and L_si is the average apparent reflectance for deep water in band i minus two standard deviations.

[Note: In practice, full atmospheric correction as carried out in Lesson 3 would be preferred to the cruder, dark pixel subtraction method we are using here, but this method is useful if you do not have access to atmospheric models such as the 5S or 6S radiative transfer codes].

In relatively clear water, the intensity of light will decay exponentially with increasing depth (Figure 5.1 - 1). If values of light intensity (radiance) are transformed using natural logarithms (ln), this relationship with depth becomes linear (Figure 5.1 - 2). Transformed reflectance values will therefore decrease linearly with increasing depth. If X_i is the transformed reflectance of a pixel in band i, this step is written as:

The attenuation coefficient k_i describes the severity of light attenuation in water for each spectral band i. It is related to radiance and depth by the following equation where a is a constant, r is the reflectance of the bottom and z is depth.

Theoretically, it would be possible to rearrange the equation and generate an image of bottom type, r (reflectance) which is the measurement we seek. However, this approach is not feasible because there are too many unknown quantities - i.e. the value of the constant a, the attenuation coefficient for each band and the depth of water at each pixel. The method developed by Lyzenga does not require the actual calculation of these parameters but gets around the problem by using information from more than one band. All that is required is the ratio of attenuation coefficients between pairs of spectral bands. Use of ratios cancels out many of the unknowns in Equation 5.3 and the ratios can be determined from the imagery itself.

Two bands are selected and a bi-plot made of (log transformed) reflectances for the same substratum at differing depths (Figure 5.1 - 3). Since the effect of depth on measured radiance has been linearised and the substratum is constant, pixel values for each band will vary linearly according to their depth (i.e. points will fall on this straight line). The slope of the bi-plot represents the relative amounts of attenuation in each band. In fact, the slope represents the ratio of attenuation coefficients between bands. Conceptually, the line represents an axis of reflectance values for an unique bottom type. As one moves along the line, the only change is depth.

If reflectance values for another bottom type were added to the bi-plot (Figure 5.1), a similar line would be obtained - once again, the only change between data points would be depth. However, since the second bottom type will not have the same reflectance as the first, the new line will be displaced either above or below the existing line (e.g. if line 1 was derived from sand which generally has a high reflectance, and line 2 was generated from seagrass with lower reflectance, the latter line would lie below that for sand). The gradient of each line should be identical because the ratio of attenuation coefficients k_i/k_j is only dependent on the wavelength of the bands and clarity of the water.

An index of bottom type can be obtained by noting the y-intercept for each bottom type (Figure 5.1). For example, while pixel values lying on the line for sand show considerable variation in radiance, they all represent the same bottom type and have the same y-intercept. The y-intercept for pixels of seagrass is considerably different. The y-axis therefore becomes an axis (or index) of bottom type with bi-plots for each habitat crossing at a characteristic point.

Of course, not all pixel values for a given bottom type lie along a perfectly straight line. This is because of natural variation in bottom reflectance, patches of turbid water and sensor noise. Nevertheless, each pixel can be assigned an index of bottom type once the ratio of attenuation coefficients has been estimated (k_i/k_j). This is accomplished by "connecting" each pixel on the bi-plot to the y-axis using an imaginary line of gradient k_i/k_j. Pixel values on the bi-plot are then converted to their corresponding positions on the y-axis (index of bottom type). Using this method, each pixel value is converted to a depth-invariant index of bottom type which is (as its name implies) independent of depth. These depth-invariant indices of bottom type lie along a continuum but pixels from similar habitats will have similar indices.

The mathematics of the depth-invariant index are simple. For this lesson we will be working with bands 3 and 4 of the CASI image and making a bi-plot of the transformed and corrected band 3 and band 4 values (respectively X₃ and X₄ from Equation 5.2):

where X₃ is the y-axis variable, ln (L₃ - L_s3), index is the intercept of the regression line with the y-axis (the depth-invariant index of bottom type), k₃/k₄ is the gradient of the regression line and X₄ represents the x-axis variable, ln(L₄ - L_s4). The equation can be rearranged to give the depth-invariant index of bottom type:

The general equation for any pair of bands i and j, written in full is:

Each pair of spectral bands will produce a single depth-invariant band of bottom type. If the imagery has several bands with good water penetration properties (e.g. Landsat TM, CASI), multiple depth-invariant bands can be created. The depth-invariant bands may then be used for image processing or visual inspection instead of the original bands.

Figure 5.1. Processes of water column correction, showing the difference of sand and seagrass.

We will look at one image in the green part of the spectrum (CASI3C16.DAT) and use a transect to quantify the effect of depth on reflectance.

Pairs of spectral bands are selected which have different bottom reflectances but good penetration of water (i.e. visible wavebands). Out of 10 possible depth-invariant index bands which can be produced from bands 1-5 of the CASI in the visible blue to red (Table 5.1), the four CASI bands (2, 3, 4 and 5 in Table 5.1) which produced the most useful depth-invariant index images have been used in this lesson. For display and analysis purposes three of the six depth-invariant index images which can be produced from the four bands were found to be most useful; these were b2_b4, b3_b5 and b3_b4 combinations (where b2_b4 means a depth-invariant index image produced from bands 2 and 4 of the CASI data). For this lesson we will just create the b3_b4 depth-invariant index image from CASI bands 3 and 4 in the green part of the visible spectrum.

Calculation of the parameter L_si, deep water radiance (reflectance) is fairly straightforward. A group of pixels are selected which represent deep water (i.e. greater than 40 m). The pixel data are transferred to a spreadsheet (e.g. Microsoft Excel). The mean and standard deviation of radiance (reflectance) are calculated for each band. Armstrong (1993) recommended subtracting two standard deviations from the mean to account for sensor noise. This (lower) value is then used as L_si in subsequent calculations.
 
 

When water column correction of this CASI image was carried out, nine sand areas of varying depth were selected and bi-plots of the natural logarithms of atmospherically corrected reflectances of all the pixels in each area for each pair of bands were examined. Figure 5.2 shows approximate positions of some of the sandy areas selected. The areas are chosen so that neither of the bands is saturated and both bands penetrate to the seabed. To understand how this is done you will select two 3 x 5 areas of pixels from areas 4 and 7 (asterisked in Table 5.3) in CASI bands 3 and 4 (CASI3C16.DAT and CASI4C16.DAT) and arrange these in a spreadsheet so that for each pixel you have the reflectance in each band. This will allow you to estimate the ratio of the attenuation coefficients (k₃/k₄) for this band pair.

Figure 5.2. CASI image of Cockburn Habour (Turks and Caicos Islands) showing the selection of pixels of sand at variable depth.

Table 5.3. UTM coordinates of the sandy patches selected for bi-plots. For this lesson you will only deal with pixels in two 3 x 5 sand patches, one in the deeper part of the image and one in the shallower part.

The ratio of the attenuation coefficients of CASI Bands 4 (571.9-584.3 nm) and 3 (531.1-543.5 nm) is given by the slope of the bi-plot of the logarithmically transformed atmospherically corrected data. In this case we are using a crude atmospheric correction involving subtraction of the deep water value calculated earlier (Table 5.2). Thus all we have to do is to take natural logarithms of the reflectance of each pixel in the Band 4 column minus the deep water reflectance for Band 4, and then do the same for the Band 3 column.

and (s_i is the variance of band i, s_j is the variance of band j, s_ij is the covariance between bands i and j). In this case i = 3 and j = 4. Calculation of variance (spreadsheet function VAR) and covariance (function COVAR) is relatively straightforward. Let us do this in five steps:

The depth-invariant processing will be removing the variation in reflectance that is due to water depth. To illustrate this you will look at the coefficient of variation (standard deviation/mean) of the 30 pixels in each band and compare these with the coefficient of variation of the same pixels in the depth-invariant bottom index image you will produce later.

Prior to execution of depth-invariant processing all areas of land and cloud should be masked out. It is best to set pixels for these areas to zero. Note that this has already been done on the CASI images you are using. Once the ratios of attenuation coefficients have been calculated for band pairs, the depth invariant algorithm can be implemented. Equation 5.6 can now be used to generate a depth-invariant index image for bands 3 and 4. This is repeated below for your convenience:

where in this case i = CASI band 3 and j = CASI band 4.

Firstly, you will make a false colour composite of the raw CASI data using bands 2 (blue), 4 (green) and 5 (red) and inspect the deep water part of this at the south (bottom end) of the image.

1. The display scale (0-255) reflectance changes from about 35 (approx. average value over first 20 m) to 10 (approx. average value over last 20 m) over the 225 m transect. Expressed as a percentage of the average pixel value over the first 10 m of the transect (about 35) this is 25/35 = 70% change. If you take the highest value of 40 and lowest of 5 and the exact mean of the pixels from 0-10 m along the transect (36), then the change is 35/36 = 97.2% change.

2. The coefficients of variation are 0.6307 (band 3) and 0.7759 (band 4).
3. The pixel values at column and row coordinates 395, 190 are 2104 (band 3) and 1632 (band4). Taking away the respective deepwater reflectances gives 2104 - 344 = 1760 (band 3) and 1632 - 186 = 1446 (band 4). Taking natural logarithms gives 7.4731 (band 3) and 7.2766 (band 4). The attenuation coefficient ratio is 0.73393 so the depth-invariant bottom index value is:

4. The coefficient of variation of the 30 pixels in the CASI3_4.DAT depth-invariant bottom index image is 0.0199. The average coefficient of variation of the same pixels in the two raw bands is 0.7033. 0.0199/0.7033 x 100 = 2.8%. Thus over 95% of the variation in reflectance was due to the effect of water depth.
5. 1) At the bottom (south) of the depth-invariant bottom index image the structure of the reef front in deep water is clearly revealed and no longer does the image get darker as the water depth increases. 2) There is little evidence of sunglint and much less sign of reflection off the sea surface on the right (east) of the bottom-index image.

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