The assumptions commonly used in operational shallow water modeling are the following.

• Simplified RTE: shallow water modeling is only possible because the very complex radiative transfer equation of sun and sky light through atmosphere and water has been extensively simplified for sake of operationality.

• Homogeneity: the optical properties of the atmosphere column and of the water column are assumed to be horizontally homogeneous.

• Operational K: the various specific diffuse attenuation coefficients for upwelling and downwelling irradiances, K _up and K _down, which compound into the depth-dependent diffuse attenuation coefficients for irradiance K _d are conveniently rounded up into an effective two-way attenuation coefficient for near-nadir radiances, denoted K, for sake of operationality.

• Multiple scattering in the water column is neglected, as is multiple reflection at the water-air interface.

• Dark target assumption: in 4SM, the radiance measured in the imagery at the longer wavelength available (possibly a NIR -or Red- band) over an optically deep water area is assumed to be representative of the radiance of the atmospheric path at that wavelength.

For all these reasons,
bathymetry modeling is best conducted using images,
or part of them,
which exhibit a very high radiometric quality standard,
and it should be considered a safe precaution
that certain images or part of them
may not be used for this specific purpose.
see Image Worthiness

• This has been presented at the ERIM Marine RS conference in 1998 (reprints still available).

• 4SM only relies on the information contained in thebareland and marine parts of the spectral imagery in order to derive scene-dependent modeling parameters.

• Seed value: then 4SM uses

  • the table of diffuse attenuation coefficients for downwelling irradiance in marine/coastal Case I waters worldwide

  • published in 1976 by Jerlov in his reference book "Marine Optics" (Elsevier),

  • in order to derive a seed value for the operational attenuation coefficient for the water column under study which applies to the central(?) wavelength of each of the spectral waveband.

• Spectral K: then, using that seed value, spectral operational K values are derived from K _i/K _j ratios observed for all possible pairs of bands i and j of the image itself.

• Therefore, 4SM does not require, neither does it uses, any preliminary field data for the estimation of any of the modeling parameters.

• As a consequence, the only thing that is needed is the imagery itself and the specifications of its band setting.

But all computed depths still need to be multiplied by a final depth correction factor to be derived from sea truth evidence when available.

• NIR : the availability of a Near InfraRed waveband is highly desirable in bathymetry modeling.

  • A NIR waveband hardly contributes to the results because sunlight only penetrates down to a few decimeters in water.

  • Nonetheless, a NIR waveband is extremely useful

    • in the preliminary search for adequate modeling parameters (dark pixel assumption, deglinting),

    • for providing an alternate seed value for spectral K,

    • and of course for precise delineation of the waterline.

• Visible : besides the NIR waveband(s), if any, the multi/hyperspectral imagery is comprised of N wavebands in the visible part of the solar spectrum, i. e. from 400 to 700 nanometers.

• Separation of colors:

  • Strictly speaking, bathymetry modeling refers to the dual estimation of the bottom's depth and spectral reflectance at one particular shallow water pixel.

  • Bathymetry modeling is only possible if the pixel considered exhibits bottom detection in at least one pair of wavebands i and j which happens to show a value of the ratio K _i/K _jless than ~0.8: this is the concept of separation of colors.

• Three cases : therefore, in bathymetry modeling, three cases are to be distinguished:

  • The N-BANDS case is where modeling is possible for at least three pairs of wavebands, like when using the Blue/Green/Red bands of Landsat imagery.

    • this yields a spectral bottom reflectance

    • and a depth which may be slightly biased to the extent that the actual bottom signature is contrasted.

  • The 2-BANDS case is where modeling is possible for only one pair of wavebands, like when using the Green/Red bands of SPOT XS imagery.

    • this yields an average bottom reflectance

    • and a depth which may be severely biased to the extent that the actual bottom signature is contrasted.

  • The 1-BAND case is where modeling may not be performed in spite of the fact that the bottom is detected by at least one waveband i.

    • if the attenuation coefficient K_i is known,

    • if the bottom reflectance LB_i is known over homogeneous bottoms,

    • then a good a depth may be derived.

• Multispectral : the imagery needs to be multispectral.

  • Two bands is the minimum.

  • More bands is better.

• Red to Green : the red to green (600 to 500 nm) part of the visible solar spectrum is where the majority of bands should be placed, with only a few bands outside this range.

• A Panchromatic band may be included, provided it has been acquired concurrently with the spectral bands:

  • aPanchro band is a very wide band which may not be used like normal multispectral bands.

  • we have successfully used the SPOT PAN waveband, in addition to the normal SPOT XS bands, to extend the modeling depth range from 4-6 m down to 10-13 m in the clear waters of Bora Bora.

• Near vertical viewing at high altitude on a "calm and clear" slightly hazy day is best.

• Bathymetry modeling has been tested successfully using

  • SPOT XS, Landsat TM and ETM, IKONOS

  • scanned color aerial photography,

  • airborne spectrographic imager data (CASI).

• QUICKBIRD, FORMOSAT2, THEOS, PLEIADES should all prove to be suitable.

• Digital multispectral video (DMSV) and digital aerial photography should prove to be suitable.

• Spaceborne hyperspectral imagery should prove to be the ultimate source of data for bathymetry modeling in the years to come, subject to ground resolution though.

For an image to be worthy of shallow water work,
a number of conditions must be met:

• Illumination conditions must be constant through the whole scene

  • ILS correction of airborne flight lines must be applied

    • the main reason is that the incident sun light can vary by several percents from start to end of an airborne flight line as the sun height varies

    • another reason is to correct for transient sun light variations

    • this artifact is wavelength-dependent

  • Removal of limb brightening must be applied

    • the reason is that the path length increases markedly from vertical viewing to oblique viewing across the flight line

  • Atmospheric path radiance must be homogeneous

    • areas affected by haze must be masked out, or "deglinted" if at all possible

    • areas affected by clouds and their shadows must be masked out.

    • atmospheric adjacency effect must be corrected for, if at all possible.

• Sky and Sun glints must be totally removed

  • Best is to discard images or areas which are too badly affected by sun glint

    • like in case of very oblique viewing towards the sun

  • Sun glint and Skyglint can have distinctly different spectral properties

    • sunglint has the spectral properties of direct sun light, i.e. biased towards the yellow region of the solar spectrum

    • skyglint has the spectral properties of the sky light, i.e. usually biased towards the blue region of the solar spectrum for a clear blue sky

  • Best deglinting is achieved when the atmosphere is hazy,

    • as this strengthens the correlations among bands from NIR through BLUE regions (Mie scattering).

• Resampling must be achieved by the nearest neighbor algorithm.

• Optical properties of water bodiesmust be homogeneous across the scene, or this shall show in the output (see the arcachon study case)

Which water type?

This is a question of Which Water and Bottom Types

• in excess of 30 m

  • with a blue-green pair of bands in clear oceanic or spring waters over bright bottom

• even less than 3-4 m

  • with a green-red pair of bands in highly productive green waters over bright bottoms

• even less over dark bottoms

This is a question of How Much Light

• More photons yield deeper bottom detection

  • June is best in the northern hemisphere: waters still fairly clear, sun at its highest

  • December is best in the southern hemisphere: waters still fairly clear, sun at its highest

• Double the amount of photons entails improvement of penetration depth by ~12%

This is a question of How Much Rain

• you don't want rainy season to come into play.

Depends on what?

• The performance in computed depth depends on

  • the particular pairs of wavebands used,

  • the wavelength used to represent each waveband,

  • the radiometric quality of the imagery,

  • the amount of turbidity in the water column,

  • the viewing geometry, illumination conditions and weather conditions,

  • the experience of the practitioner.

• Wavelength dependence:

  • from the blue start of the visible solar spectrum at 400 nm to the red end at 700 nm,

  • the penetration of light decreases from several tens of meters to less than 4 meters

  • in respect of bathymetry modeling in very clear "blue waters",whether marine or inland.

• Waveband pair dependence:

  • bathymetry modeling uses at least one pair of wavebands in order to allow for "un-mixing" of bottom depth versus bottom brightness influences.

  • ratio K _i/K _j of effective attenuation coefficients for that pair must be less than ~0.8.

  • this means that certain pairs are not suitable, depending both on their wavelength and water turbidity.

• Water turbidity dependence:

  • the shallow bottom must be detected through the water column!!

    • shallow water modeling is for "Case 1 waters":

      • in poorly productive clear waters, the penetration of light

        • decreases moderately over the blue-green wavebands,

        • decreases extensively over green-yellow wavebands,

        • and decreases moderately over theyellow-red wavebands.

      • in highly productive clear waters, as a consequence of the buildup of the content of dissolved organic matter (yellow substances),the penetration of light

        • increases extensively over blue-green wavebands,

        • reaches its maximum in green-yellow wavebands,

        • and decreases again over the yellow-red wavebands.

    • bathymetry modeling may not be performed in "Case 2 waters", i.e. where the content of the water column is high in suspended mineral particles.

  • CAUTION: locally high water volume reflectance caused by suspended particles

    • prevents bottom detection

    • mimics bottom detection, very cunningly

• Radiometric noise dependence:

  • Signal/Noise ratio: bathymetry modeling is only possible where the Signal/Noise ratio is significant in the image data.

  • Thresholds: therefore in 4SM a threshold is applied for each waveband, below which that waveband may not be used for modeling.

  • Blue wavebands have a low S/N ratio,

    • mainly because of high turbidity of the atmosphere

    • and also because of poor performances of sensors in the blue domain.

  • "calm and clear" slightly hazy: altogether the imagery to be used is best acquired at high altitude in "calm and clear" slightly hazy weather conditions, in order to minimize difficulties generated by a high level of sky glint or sun glint at the water surface.

• Imaging conditions dependence:

  • near-vertical viewing: the above-mentioned radiometric noise is minimized in near-vertical viewing conditions at high altitude on a calm and clear day.

  • low-altitude airborne: this means that low-altitude airborne imagery collected using a large field of view imaging system

    • may happen to be unusable away from the central part of the image,

    • unless an adequate radiometric pre-processing correction scheme

    • actually succeeds in removing limb brightening and sky/sun glint.

Precision on Computed Depths
HowTo RegressZZ
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Precision on Computed Bottom Reflectance

• Same day: once the image file is ready on disk, one work day is enough for an experienced 4SM practitioner to obtain a reasonable result out of a 4-bands image.

• Longer: finishing can take a while, though, depending on

  • the radiometric quality of the image

    • systemnoise,banding, cross talk, bright target recovery

    • oblique viewing

  • the size and complexity of the scene

    • clouds and shadows

    • floating objects, like boats, seaweed mats, foam, etc

    • wind conditions and breaking waves

    • glints and adjacency effect

    • heterogeous water types, and locally foul waters

Shallow water bathymetry modeling of multi/hyperspectral imagery using 4SM produces:

• One 8-bits coded image_Z250 of computed depths in decimeters up to 25 meters

• One 8-bits coded image_Zsensivity (left) on computed depths in decimeters.

• In the N-bands case (N>2): N spectral images_LBS of computed spectral bottom reflectance .

• In all cases: one 8-bits coded image_B of average computed bottom reflectance.

• Spectral K: the values of the operational spectral attenuation coefficient of light in water for the area studied, i. e. the spectral two-ways operational coefficients K at the specified wavelength of each of the available waveband

• One 16-bits coded image_ZG of computed depths in centimeters.

where N is the number of wavebands in the spectral image

If the raw image is 16 bits,

then a 16_to_8 bits scaled spectral image is first written in channels 1_to_N.

U8_Image_mSE in channel_N+1

This Special Effects mask that has to be prepared manually.

U8_Image_WZ in channel_N+2

• This image_WZ makes a note of which waveband with longest wavelength exhibited bottom detection in excess of its threshold Lm value.For example, if bottom detection starts with band 3 and the BOA radiance in this band is higher than Lm ₃, then a value of 3 for that pixel is written in Image_WZ.

U8_Image_WR in channel_N+3

This image_WR makes a note of the radius of the circular smoothing window. For example, a value of 3 means a 7*7 circular smoothing window.

U8_Image_Z250 in channel_N+4

• Shallow depths are coded in decimeters from 1 to 250

• Depths in excess of 250 are coded 253.

• Deep waters are coded 254.

• No_Data areas are coded 255.

U8_Image_Z in channel_N+5

• This image is for screen display purpose only, in the case when depths exceed 25 meters.

• Depth in shallow water area is written in decimeters down to a certain MAXimum. Beyond this maximum, depth is written as MAX+(depth-MAX/10).

• Image_Z20 : MAX is 200 decimeters. Beyond 20 m, depth is coded as 200+(depth-20), like 137 stands for 13.7 m and 204 stands for 24 m. Maximum is 210 for 30 meters.Use thepctZ20 color pallet provided.

• Image_Z10 : in very clear waters, the depth range may exceed 30 m. In such case, MAX is 100 decimeters. Beyond 10 m, depth is coded as 100+(depth-10), like 37 stands for 3.7 m and 137 stands for 37 m. Maximum is 210 for 110 meters.Use thepctZ10 color pallet provided.

• Land area is delineated with high precision using the NIR band (XS3 of SPOT XS or TM4 of Landsat TM). The green-vegetated land area is interpreted into Normalized Difference Vegetation Index (NDVI) and coded into shades of green; Perpendicular Vegetation Index (PVI) also available upon request. The red-vegetated land area is coded into shades of brown. The non-vegetated land area is coded in shades of gray.

• Non-shallow water area is masked and coded separately: clouds and shadows, breaking waves, etc.

U8_Image_B in channel_N+6

• It is coded for easy visual inspection on screen. An average bottom brightness, i. e. a shade of gray, is computed for every pixel which exhibits bottom detection by at least two bands.

• For non-shallow water area, the coding scheme described above for imageZ applies.

• Shallow bottom brightness is scaled from 0 for black bottoms to 200 for the brightest type of shallow bottom that exists in the area studied. Pixels for which computed bottom reflectance exceeds 200 are saturated at 201.

• If the depth has been computed using only the last band exhibiting bottom detection and through the arbitrary choice LBref[1] or LBref[2] of a prevailing uniform bottom reflectance (the 1-band case), then corresponding pixels in image_B are coded at the value of LBref.

• PctB : the image_B file may be screen-displayed using the pctB pseudocolor table provided.

U8_Image_BSC in channel_N+7

• This channel is mostly a visual display and control of how good -or bad- the spectral bottom signature might be.

• This output channel is mostly a display of the ratio LB _green/LB _blue.

• If a Red waveband exhibits bottom detection and the LB _redexceeds (LB _blue+ LB _green)/2, then the ratio LBS _red/(LB _blue+LB _green)/2 may be written instead.

• Underestimated depths tend to yield a low LB _green/ LB _blueratio == show as a dense-blue artifact in image_BSC

• Overestimated depths tend to yield a high LB _green/ LB _blueratio == show as a dense-green artifact in image_BSC.

U8_Spectral_Image_X in channel_N+12+1 to channel_N+12+N

Spectral linearized radiances are multiplied by 40 in order to use the whole 8-bits range of 0_to_255

U8_Spectral_Image_Deglinted in channel_N+12+N+1 to channel_N+12+N+N

Spectral radiances are scaled, smoothed, deglinted, normalized as BOA radiances, then saved for a reference to just prior to modeling.

U8_Spectral_Image_LBS in channel_N+12+N+N+1 to channel_N+12+N+N+N

• Image_LBS is a "water column corrected " spectral view of the scene. It intends to map the spectral reflectance of the whole area, whether dry land or shallow water, "like if there were no water", so as to allow for thematic multispectral classification of the shallow bottom using the same suite of classifiers as typically used for land cover typing.

• These radiances are BOA radiances, i. e. after removal of the atmospheric path radiance

• Scaling of image_LBS

  • Bottom reflectance in each of the images that are stored in image_LBS is scaled from 0 for a black bottom to 250 for the brightest type of shallow water bottom that exists in the area studied.

  • Pixels for which reflectance exceeds 251 are saturated at 251.

  • These images may be screen-displayed individually as shades of gray,

  • Or any three of them may be screen-displayed as a RGB color composite.

• Information content of image_LBS

  • Depth dependence : because the attenuation of the sun light through water is strongly wavelength-dependent, the information content of image_LBS decreases gradually as the water depth increases.

  • In very shallow waters , all bands exhibit bottom detection and each of them in image_LBS carries its own contribution to the spectral signature of the shallow bottom.

  • In deeper waters , some bands at longer wavelength in the red part of the spectrum are extinct, and may only be allocated a bottom reflectance which is computed to be the average of spectral reflectances for other bands.

  • Average brightness : this process may finally end with only two bands exhibiting bottom detection with suitable color separation.

16S_Image_ZG channel_N+12+N+N+N+1

A 16-bits-Signed channel: depths are coded in centimeters. Non-marine areas are coded at -1.

If the image is 16 bits, then come the N 16U raw data channels, like for a CASI or Ikonos image.