Secchi depth in the Oslofjord – Skagerrak area : theory , experiments and relationships to other quantities

Abstract. The Secchi depth and its relationships to other properties of the sea water in the Oslofjord–Skagerrak area have been investigated. White and black disks of different sizes have been applied, and the Secchi depth has been observed with the naked eye, through colour filters and with a water telescope. Spectral luminances and illuminances have been calculated from recordings of radiance and irradiance, and attenuation coefficients have been determined. A theoretical expression for the Secchi depth based on luminances has been tested against field observations, and it is found that the field results for the product of Secchi depth and attenuation coefficients are on average only 4% less than the predicted value for the white disk. For the Secchi depths observed through colour filters or for the black disk, the average field results are more than 30% smaller than the theoretical estimates. The reduction in the disk diameter from 30 to 10 cm should theoretically reduce the Secchi depths by 13–22%, while the field observations show an average reduction of 10–20%. Similarly we find from theory that the removal of sun glitter should increase the Secchi depth by 12%, while the observed increase is 14% on average for the white disk. Our overall conclusion is that the theoretical expression works well for the white disk, but less so for the colour filter observations and the black disk. Statistical relationships between Secchi depths and attenuation coefficients have been determined, and it is found that the root-mean-square errors relative to the mean value are smaller for the beam attenuation coefficients (12–24%, white disk) than for the vertical attenuation coefficients (16–65%, white disk). The depth of the 1% level of surface quanta irradiance (PAR) can be estimated with a relative root-mean-square error of 23% from observations of the white Secchi depth. Similar estimates of chlorophyll a and total suspended material will have rms errors in the range 40–90%. Our conclusion becomes that the Secchi depth observation is a very useful tool for checking the value and order of magnitude of other related quantities in the Oslofjord–Skagerrak area.


Motivation for the present study
The threshold depth of observation for the Secchi disk is a direct measure of the vertical visibility in water, and it is one of several parameters used by environmental authorities to describe water quality.In some branches of aquatic science it is termed transparency.The depth is determined by the optical properties of the water and can therefore be related to these properties.Observations of the Secchi depth can never be satisfactory substitutes for direct recordings of the other optical properties, but they can serve as independent checks of these properties.
The Norwegian Institute for Water Research and the University of Oslo have collected numerous observations of the Secchi depth as well as recordings of other optical quantities.We have analysed parts of this database to check a theoretical expression for the Secchi depth, and we have checked some of the assumptions on which the Secchi depth theory is based, as well as some results that can be deduced from the theory.Different disks and a water telescope have been applied, and threshold depths have been determined by the naked eye as well as with coloured glass filters in front of the eye.The results are based on observations from the Oslofjord-Skagerrak area.The test of the theoretical expression, as well as the statistical relationships found between Secchi depth and other optical properties, has been the main motivation for this study.
In Sect. 2 we present what is essentially Tyler's (1968) theory of the Secchi depth, and Sect. 3 describes some of the regional environmental characteristics and the applied data sets and instruments.Section 4 estimates the different optical properties appearing in the theoretical expression for the Secchi depth, and the expression as well as an assumption about the applied attenuation coefficients are compared to observations.Empirical relationships between the Secchi depth and other optical properties are discussed in Sect. 5.The results are summed up in Sect.6.

History
The visibility of the Pacific Ocean was studied by the Russian naval officer Otto von Kotzebue as early as 1817 by means of a red piece of cloth being lowered into the sea, and on one occasion by a white plate, according to Krümmel (1886).This is probably the first known scientific investigation of the optical properties of the ocean.Other early transparency measurements and occasional observations are mentioned by Boguslawski (1884), a later work by Krümmel (1907), and by Wernand (2010).
In 1866, almost fifty years after von Kotzebue's measurements, Alessandro Cialdi, Commander of the Papal Navy, published a report containing a section by Frater Pietro Angelo Secchi, where the factors influencing the visibility in the sea of submerged disks of different sizes and colourings were discussed (Secchi, 1866).In the years to come the white version of this device became a standard instrument in marine investigations.In some scientific communities the disk is referred to as "the white disk", and in others as "the Secchi disk", although the suggestion of using a white disk came from Cialdi.
Today the marine standard method of measurement is to lower a white disk, with a diameter of approximately 30 cm, supported on a cord and with its plane horizontal, from the ship rail and into the sea to a depth where the disk cannot be seen any longer.The disk is then hauled upwards to a depth where it can be recognised once again.The mean value of the two threshold depths is termed "the Secchi depth".In limnology some communities prefer using a 20 cm disk, painted in black and white quarters, but this device will not be discussed in this paper.
However, during the first century of Secchi depth measurements a satisfactory theory describing the relationship between the threshold depth and the optical properties of the sea was missing.The factors influencing the depth were known (Krümmel, 1889), and Sauberer and Ruttner (1941) had set up the equations governing the contrast and the upward directed light, but the final step connecting the Secchi depth and the attenuation coefficients was not taken.According to Shifrin (1988) Gershun had solved the problem in 1940, but his results were published in Russian and therefore not well known outside the Soviet Union.In the western world the breakthrough came when Tyler (1968) applied a contrast formula, presented sixteen years earlier by Duntley (1952), to derive an expression for the Secchi depth.Holmes (1970) tested the constant of this expression by field measurements.A different contrast formula, including the halo of scattered light around the disk, was suggested by Levin (1980).Preisendorfer (1986) discussed the assumptions and limitations of the Secchi depth theory and procedure, using attenuation coefficients of photopic quantities.Krümmel (1889) referred to observations made by Austrian oceanographers in the Adriatic and Ionian seas in 1880 with disks of different metals and paintings.Lisitzin (1938) observed the Secchi disk through coloured glass filters in the Baltic Sea as early as in the 1920s, and Takenouti (1950) performed similar measurements in Japanese lakes.Højerslev (1977Højerslev ( , 1978) ) succeeded in relating such glass filter observations to other marine-optical properties.Levin (1980) mentioned recordings with glass filters in the Black Sea.An interesting study of the black disk was made in 1988 by Davies-Colley.Haltrin (1998) reported observations with disks that were painted blue, green and red.
For more than two decades remote sensing of water colour has been used to estimate the Secchi depth (http://www.globcolour.info/data_access_demo.html), which is one of the ESA GlobColour products.

Theory of the Secchi depth
An important factor that enters the theory of the Secchi depth is the properties of the human eye as a contrast sensor.Studies in air (Blackwell, 1946) have demonstrated that the human eye is able to distinguish a target from its background down to a lower limit or threshold value of the contrast between the target and its background.In our case the target is the Secchi disk, and the definition of the contrast C becomes C = (L D − L)/L, where L D is the luminance from the disk and L the luminance from the background.The eye integrates the total spectrum of radiances within the direction to the disk and the background, and weights these radiances by the eye's spectral sensitivity.During daylight conditions this is represented by the photopic sensitivity of the eye, and we have expressed it by the CIE 1924 curve for luminous efficiency (e.g.Walsh, 1958).This curve, which is also termed the photopic luminosity function, has its maximum at 555 nm, half-peak values at 510 and 610 nm, and 1 % of the maximum at 438 and 687 nm.The resulting integrals become the luminances L D and L. If the eye of the observer is at the depth z = 0, which in this paper will mean just beneath the surface, or if a water telescope is being used, that is a tube through the water surface with a glass window at the bottom, then the observed contrast C(0) at this depth between the nadir luminances L D (0) and L(0) from the white Secchi disk and the background, respectively, will be When the Secchi disk is lowered, L(0) remains constant while L D (0) is reduced until the disk reaches the depth where it disappears from sight, namely the Secchi depth.Then C(0) has been reduced to the threshold value C t .According to Blackwell (1946) C t is not a constant, but depends on the angle α subtended by the observed target, the luminance L(0) of the background, the probability of detection and the exposure time.C t decreases with decreasing detection probability and with increasing α, L(0) and exposure time.
Photometric luminance is not a practical quantity if we want to use the equation of radiative energy transfer, because the luminance represents a spectral integral, and therefore its attenuation coefficients are not constant in space, even in optically homogeneous waters.However, when we determine the Secchi depth, we notice that the colour of the disk is the same as that of the surrounding waters.This colour corresponds to the wavelength region where the water has its maximum beam transmittance, which is also where the upward scattered radiance obtains its spectral maximum.We can therefore assume as a first approximation that the observed luminances in Eq. (1) correspond to the radiances at the wavelength of the spectral transmittance maximum.In the present discussion we have let the symbol L represent these monochromatic radiances as well as the spectrally narrow-banded photopic luminances.The magnitude of the errors introduced by the monochromatic assumption will be discussed in Sect.4.8.
The equation of radiative energy transfer for the nadir radiance L D from the Secchi disk sounds where z is the vertical coordinate with zero at the surface and positive downwards, c is the beam attenuation coefficient at the wavelength of maximum transmittance, and L * D is the path function along the path from the disk to the observer.The same equation for the nadir radiance L from the background becomes Here L * is the path function along an upward directed path outside the disk.
Tyler (1968) applied Duntley's (1952) contrast formula and assumed that at each depth z the path function L *D (z) above the Secchi disk is practically uninfluenced by the disk, and therefore approximately equal to the path function L * (z) of the background.This assumption may be questioned, and especially at positions close to the disk.However, let us assume that it is still valid for most of the path between the disk and the surface.By making the approximation L *D (z) ≈ L * (z) and subtracting Eq. (3) from Eq. ( 2) we obtain Provided c is constant with depth the integration of this equation between the surface and the Secchi depth (5) We recognise the left side of this equation as the numerator of Eq. ( 1).On the right-hand side we have the radiance L D (Z D ) which is the reflected part of the downward irradiance E d (Z D ) incident at the disk.The reflectance of this radiance may be termed ρ DL , defined by At the same level outside the disk the upward radiance L(Z D ) will be related to the downward irradiance by the radiance reflectance R L : In Eqs. ( 6)-( 7) it is assumed that the downward irradiance at the Secchi depth is practically unaffected by the presence of the disk, and has the same value above as outside the disk.By using Eqs.( 6)-( 7) the radiances on the right side of Eq. ( 5) may be written as functions of E d (Z D ), and Eq. ( 5) becomes At this point we see that it would be very convenient if we could also express the denominator L(0) on the righthand side of Eq. ( 1) as a function of E d (Z D ).L(0) can be transformed in several ways, but the simplest expression is obtained by using E d (z) and R L (z) and writing L(z) as an exponential function of z: where K L is the average vertical attenuation coefficient of L(z) in the depth range 0-Z D .
If we insert for the numerator in Eq. (1) from Eq. ( 8) and for the denominator from Eq. ( 9), we obtain: which can be more conveniently written as The contrast C(0) is equal to the threshold value C t , provided the eye of the observer is below the surface.Tyler (1968) presented a version of Eq. ( 11), and Hou et al. (2007) obtained a similar expression by using modulation transfer theory.
If the observer's eye is above the surface, the radiance from the direction of the disk will consist of two terms.The first is L D (0) τ/n 2 , where τ is the radiance transmittance for a ray of normal incidence at the water-air interface, and n is the index of refraction for water.This is the transmitted part of the radiance L D (0).The other term is the radiance L r from sun and sky reflected at the surface towards the observer.The sum of the terms becomes L D (0) τ/n 2 + L r .Similarly the radiance from the background will now be L(0) τ /n 2 + L r .It should be noted that the first part of the latter sum is the quantity that is usually termed the water-leaving radiance L w The contrast of the Secchi disk observed in air through a flat sea surface becomes If capillary waves are present, they will have a blurring effect on the image of the disk, and as a result the apparent contrast will be reduced.Preisendorfer (1986) expressed the transmittance W of C(0) at the water-air interface as a function of the angular subtense of the Secchi disk as seen from just below the surface, and the variance of the slopes of the capillary waves.In our notation his expression can be written where the symbols k and U represent the constant 787 m s −1 and the wind speed, respectively, and D is the diameter of the disk.When W is included in Eq. ( 13), the equation becomes In this case it is C air that represents the threshold value C t .Equation ( 15) can also be written as The contrast C(0) has to be greater than C t , and in inverse proportion to the blurring effect W . C(0) will also be a linear function of the ratio L r /L w between the reflected and waterleaving radiances.The greater the surface reflection is, the greater C(0) has to be.It will be convenient for our purposes to express L r and L w as functions of the same input above the surface, namely the downward irradiance E air .L r can be written as where ρL,air = L r /E air is the Fresnel-reflected radiance towards the zenith, depending on the slope distribution of the surface and the angular distribution of radiance from sky and sun.L w can be transformed by using the quantity R introduced by Morel and Gentili (1996): By inserting for L r and L w from Eqs. ( 17) and ( 18) into Eq.( 16) we obtain Equation ( 11) can then be written, by substituting for C(0) from Eq. ( 19), as where A is defined as the product of the two fractions inside the greater parentheses.This is our basic equation for the threshold depth of the white Secchi disk observed from above the surface.The numerator of the first fraction is a function of the reflective properties ρ DL of the disk and the radiance reflectance R L (Z D ) of the background at the Secchi depth, the denominator is a function of the ratio between the surface-reflected and water-leaving radiances, W represents the influence of capillary waves on the transmittance of C(0) through the surface, while C t is the threshold contrast of the human eye.Observation of the disk through coloured glass filters will change the threshold depths, and the coefficients c, K L , ρ DL , R L , ρL,air and R will refer to other wavelengths than the spectral region of maximum transmittance.Otherwise Eq. (20) remains the same.
Equation (20) contains 10 variables, and if we had had precise values of these at a large number of stations, it would have been very interesting to test the equation.However, the usefulness of an expression needing the input of nine variables to predict the one remaining would be limited.It would have been far more convenient if the right-hand side of Eq. ( 20) had been a constant.The Secchi depth Z D would then become linearly proportional to 1/(c +K L ), or (c +K L ) would be proportional to 1/Z D .Unfortunately, R L , ρL,air and R are variables determined by the optical properties of the sea water and sea surface, C t is a function of the Secchi depth and the background luminance, while W varies with the wind speed and the Secchi depth.Thus the variables on the righthand side of Eq. ( 20) are neither independent nor constant, but the logarithmic function will reduce the variation of the expression inside the parentheses, which is why the equation may still provide a useful support for other marine-optical observations.
A more practical form of Eq. ( 20) for our purposes is where A 1 and A 2 are defined by the expressions inside the parentheses.In order to test Eq.( 21) we will calculate the mean value of the expression on the left-hand side of the equation, as well as the mean values of the different logarithmic functions on the right-hand side.Some special cases can be pointed out: -For a perfectly black disk, that is a disk where ρ DL = 0 (Eq.6), the quantity A 1 of Eq. ( 21) obtains the value −1, while C t will be a negative number.The two negative signs will cancel each other inside the parenthesis of ln(A).
-When Z D is observed by using a water telescope, the effects of waves and surface-reflected sky radiance are eliminated, so that W = 1 in addition to ρL,air ≈ 0, and ln(A 2 ) = ln(W ) = 0.
The different steps leading up to Eq. ( 21) make it possible to discuss the different factors that influence the Secchi depth.We will determine the attenuation coefficients c and K L from calculated luminances in Sect.4.1, estimate mean values of the quantities ln(A 1 ), ln(A 2 ), ln(W ) and ln(C t ) in Sects.4.2-4.3,and then see how the mean value of (c + K L ) phot Z D,white relates to the combined effects of these quantities according to Eq. ( 21) in Sect.4.4.In Sects.4.5-4.7 we will look at the effects of colour filters, disk size, sun glitter, ship shadow and waves.Section 4.8 will check how the assumption made early in Sect.2, about attenuation coefficients of luminances in photopic units corresponding to monochromatic coefficients at the wavelength of maximum transmittance, agrees with our results.

Environment, data sets and methods
In this section we describe briefly the environmental conditions of the investigated area, the data sets and the applied instruments and methods.

Environmental conditions
Our main area of investigation has been the Oslofjord, with additional data from the Skagerrak and two stations from the Kattegat (Fig. 1).The waters of the Oslofjord are in general eutrophic due to a supply of nutrients from the surrounding settlements (Ibrekk and Holtan, 1988).There is an estuarine circulation in the fjord, but rather weak in the inner part of the fjord.There is usually an upper and a lower layer separated by a transition layer, the pycnocline.The surface layer may sometimes be well-mixed, but often there will be a gradual change of properties from the upper 1-2 m down to the pycnocline (Aure et al., 1996;Staalstrøm et al., 2012).Salinities in the surface layer are typically in the range 20-29, and in deep waters up to 34 (Gade, 1963(Gade, , 1967;;Staalstrøm et al., 2012).Such values are of interest because a low salinity means a high content of fresh water, and this may indicate a high content of yellow substance, which is a significant optical component in these areas.The Secchi depth is found above the lower layer in the fjord.Maps of the Secchi depth, observed in 1988, have been presented by Aas et al. (1989), andAndresen (1993) has studied seasonal and annual changes of the Secchi depth in the period from 1936 to 1992.

E. Aas et al.: Secchi depth in the Oslofjord-Skagerrak area
The Skagerrak serves as a transition zone between the North Sea and the Baltic (Aarup et al., 1996a, b;Højerslev et al., 1996), and it also supplies the more saline waters to the Oslofjord.A very thorough analysis of the Secchi depths in the North Sea-Baltic Sea region has been made by Aarup (2002).Surface salinities in the northeastern part of the Skagerrak are in the range 25-32, and below the surface layer the waters become more Atlantic, with salinities up to 35 (Højerslev et al., 1996;Aarup et al. 1996a).
The environmental parameters salinity, wind speed, cloudiness and Secchi depth are presented for the different parts of our investigated area in Table 1.The salinity ranges are based on observations from different years.The Secchi data for the Inner and Outer Fjord are based on observations from 1982-1992, tabulated by Andresen (1993), while the data for the Skagerrak are observations from 1900-1999, based on a figure presented by Aarup (2002).The Norwegian Meteorological Institute has calculated mean values and standard deviations of wind speed and cloudiness at two locations: Fornebu, representing the Inner Fjord, and the Faerder Lighthouse at 59.03 • N, representing the border between the Outer Fjord and the Skagerrak.The input data are diurnal means from the entire 30 yr period .It should be pointed out, however, that cloudiness in these regions does not follow a Gaussian distribution.
Average values for the inherent optical properties of the Skagerrak-Outer Oslofjord area were found by Sørensen et al. (2007).At 442 nm the absorption coefficient of yellow substance was a y (442) = 0.62 m −1 , the bleached particle absorption a bp (442) = 0.065 m −1 , and the particle scattering b p (442) = 0.645 m −1 .The approximate spectral variations of these coefficients within the range 400-550 nm were a y (λ) = a y (442) e −(0.0105 nm −1 )(λ−442 nm) , a bp (λ) = a bp (442) e −(0.0089 nm −1 )(λ−442 nm) and b p (λ) = b p (442) 442 nm λ 0.376 where λ is the wavelength, implying that the contributions from yellow substance and particles to the attenuation coefficient c are of the same order of magnitude in the blue part of the spectrum, while particles will tend to dominate in the red part.
During 2002-2003 spectra of upward radiance just beneath the surface in the Oslofjord and Skagerrak were recorded for calibration and validation purposes (Sørensen et al., 2003(Sørensen et al., , 2004(Sørensen et al., , 2007)), related to the ESA committee MAVT.The peak values of the upward radiance spectra usually occurred in a wavelength range from 480 to 570 nm, with the mean wavelength around 525 nm.The bluish maxima were only observed at stations that were strongly influenced by Atlantic waters (salinity close to 35).The half-peak bandwidths of these spectra were typically 150 nm. Figure 2 shows the mean value of the spectra at the MERIS channels, normalised at 555 nm, as well as the standard deviation.An example of the rather rare Atlantic spectrum is included for comparison.When the radiance spectra are multiplied by the CIE photopic efficiency function described in Sect.2, the shapes of the resulting spectral upward luminances are significantly narrowed.The half-peak bandwidths will now be typically around 80 nm, and the peak values will be situated between 550 and 570 nm.

First data set and methods
The first data set consists of data from 26 stations, collected in 1992 between May and December, in the Inner Oslofjord.The set contains the threshold depths of different types of Secchi disks observed by the open eye as well as with coloured glass filters.Two sizes of white and black disks were used, the standard size with a diameter of 30 cm, and a smaller one with a diameter of 10 cm.It is difficult to obtain a perfectly black disk, defined as a disk with no reflectance at all.Disks painted black or made from black materials may still have a radiance reflectance ρ DL that is greater than the corresponding reflectance R L of the background water.The closest approximation to the non-reflecting disk that we have used has been a bowl-shaped lamp shade of brass, painted black for the scientific purpose, with diameter 30 cm.This device was originally acquired by the University of Copenhagen, but later on kindly donated to the University of Oslo by N. K. Højerslev.The instrument is supposed to work in principle very much like the light trap termed Rayleigh's Horn.Measurements were also taken with a 3.0 m-long water telescope (donated by the University of Copenhagen) that reached from the ship rail into the sea.The properties of the photopic sensitivity of the eye alone and in combination with blue, green and red glass filters (Mikaelsen and Aas, 1990) are shown in Table 2.The filters were produced by Schott and termed BG12, VG9 and RG1.The latter filter corresponds to OG590 in the latest Schott catalogue.The combined effect of Table 1.Environmental conditions in the Oslofjord-Skagerrak area.S is salinity range between the surface and the Secchi depth, U is mean ± standard deviation of wind speed in m s −1 , C is mean ± sd of cloudiness in octas, and Z D,white is mean ± sd of Secchi depth in m.

Area
Inner Fjord Outer Fjord Skagerrak north of 59.67 • N 59.00 • -59.67 • N 57.00 • -59.00 • N S (all year) a,b 15-28 16-30 20-32 U (summer) c 2.8 ± 0.5 6.7 ± 1.1 U (winter) c 2.0 ± 0.7 7.9 ± 1.6 C (summer) c 5.0 ± 0.8 4.8 ± 0.7 C (winter) c 5.5 ± 0.9 5.6 ± 0.9 Z D,white (summer eye sensitivity, filter transmittance and a typical spectrum of upward luminances L(0) is also shown, and we see that the half-peak bandwidth of the naked eye is reduced from 100 to 78 nm, otherwise the bandwidths and peak wavelengths remain the same.The Secchi depth was observed on both the sunlit and shaded sides of the ship.
The radiance reflectance ρ DL of the white disk was determined in 1992 from laboratory recordings of the spectral downward irradiance E d and the corresponding reflected upward radiance L D from the white disk, as expressed by Eq. ( 6).Both were measured in water just above the submerged disk.In order to avoid the shadow of the radiance meter on the disk, the instrument could not be held directly above it, but at an angle of 30 • -40 • away from the axis of the disk.Still, we think that the obtained values of ρ DL were close to the correct ones, being 0.25-0.29-0.30at the wavelengths 450-520-550 nm, respectively.These values will be discussed in Sect.4.2.
The accuracy of a Secchi depth observation depends mainly on the wind conditions.Winds will produce waves and may make the ship drift, with a sloping line from the observer down to the disk as a result.It will then become more difficult for the observer to estimate the correct depth below the waves.Other effects of waves are discussed in Sects.4.2 and 4.3, and values of observed wind speed are presented for the different investigated areas in Table 1.Based on experience we estimate the possible error of Z D,white to be in the range 0.2-0.5 m, depending on wind speed.It may be mentioned that the observations in this and the other data sets were made by experienced oceanographers.
In addition to the Secchi observations quanta irradiance was recorded in the sea with the Underwater Quantum Sensor LI-192SB from LI-Cor, inc., Lincoln, Nebraska, and with the LI-190SB Quantum Sensor on deck as a reference.

Second data set and methods
A second data set consists of 143 stations from the Inner and Outer Oslofjord, collected between 1973 and 2008 by the University of Oslo during courses and project excursions, with recordings of quanta irradiance and Secchi depths.The period of observation included all months from February to December.The Secchi depth was determined with the 30 cm white disk on the sunlit side of the ship, and quanta irradiance was recorded with the Li-Cor instrument mentioned above.

Third data set and methods
The third data set consists of observations at 79 stations.Secchi depths, spectral irradiances, radiances, and absorption and attenuation coefficients were measured in the Oslofjord-Skagerrak area during the years 2002-2003, in the summer period from early May to early September.
The Secchi depth was observed with the 30 cm white disk, by the naked eye and with colour filters, often on both the sunlit and shaded sides of the ship.Vertical sub-surface profiles of the downward and upward irradiances and the upward radiance were measured with the PRR-600 from Biospherical Instruments, San Diego, California, with the deck instrument PRR-610 used as a reference.The spectral channels were 412, 443, 490, 510, 555 and 665 nm, corresponding to the channels of the satellite sensors SeaWiFS and MERIS.Vertical profiles of the upward radiance L and the downward and upward irradiances E d and E u were recorded.According to the radiance model of Aas and Højerslev (1999), the quantity Q = L/E u should obtain values in the interval from π to 2π only, and a few stations where Q lay outside this range have been omitted.Immersion coefficients determined by the manufacturer and self-shading effects (Gordon and Ding, 1992;Zibordi and Ferrari, 1995;Aas and Korsbø, 1997)   up to the surface.The uncertainty of the resulting surface values was estimated to be ±10 %.A few series of spectral upward radiance just beneath the surface, recorded with the hyperspectral Ramses-ARC radiance sensor from TriOS, have been included to examine the wavelength of the spectral peak.
Vertical profiles of the spectral absorption and scattering coefficients were recorded with an ac-9 from WET Labs, Philomath, Oregon.The applied instrument records at 412,440,488,510,532,555,650,676, and 715 nm.The data were cleaned for obvious noise, and unrealistic spikes were avoided by using a median filter, resulting in an estimated uncertainty of ±10 %.The recordings with the Biospherical instrument and the ac-9 were analysed and sent to the ESA for calibration and validation purposes (Sørensen et al., 2003(Sørensen et al., , 2004(Sørensen et al., , 2007)).
At 19 stations from the 2002-2003 data set chlorophyll a (Chl) and total suspended material (TSM) were sampled on glass fibre filters.The concentrations of Chl were determined by the high-performance liquid chromatographic (HPLC) method, and the TSM by a gravimetric method (Sørensen et al., 2007, and references therein).
Figure 1 shows the locations of the stations for the three data sets.The Inner Fjord contains data from all three sets (168 stations), the Outer Fjord data from the second and third sets (48 stations), while the Skagerrak only contains data from the third set (32 stations).
Tables 3, 5, 6, 7, and 9 are based on data solely from the third set, Table 4 on data from the first and third sets, while Table 8 has applied data from all three sets.

Test of Eq. (21) in photopic units
In Sects.4.1-4.3we will obtain mean values for the different terms in Eq. ( 21), and in Sect.4.4 we will test the equation with these values.Other results of our Secchi depth experiments, related to Eq. ( 21), will be discussed in Sects.4.5-4.8.

Values of c phot and K L,phot
In Sect. 2 it was pointed out that the luminance is not a practical quantity in our marine-optical research, because the corresponding coefficient of beam attenuation will depend on the spectral shape of the luminance at the point where the attenuation starts and the distance along the beam.The coefficient c of Eq. ( 4) describes the beam attenuation of two different luminances: (1) the upward luminance from the disk and (2) the corresponding upward luminance from the surrounding waters.The radiance reflectance ρ DL of the white Secchi disk is defined by Eq. ( 6).If the albedo of the disk had been 1 and the disk had acted like a perfect Lambert diffuser, the reflected radiance would have been constant for all directions with a radiance reflectance equal to 1/π = 0.32.A real non-perfect Secchi disk will not have this value, but a number of the same order of magnitude.Tyler (1968) applied an estimate of ρ DL equal to 0.82/π = 0.26.Haltrin (1998) presented albedos for white and coloured disks, and provided the disks had acted like perfect diffusers, the values of ρ DL for the white disk would have been 0.23-0.26-0.26 at 450-520-550 nm.Our earlier determinations of ρ DL , described in Sect.3.2, are very close to the values presented above, being 0.25-0.29-0.30at the wavelengths 450-520-550 nm, respectively.The latter values have been applied here.
From the recorded spectral downward irradiances E d (Z D ) at the Secchi depth in the Oslofjord and Skagerrak the reflected upward radiances L D (Z D ) from the disk were then estimated by Eq. ( 6).The beam attenuations of the spectral radiances from this depth and up to the surface were calculated by applying recorded values of spectral c, and the resulting spectrum of radiances just beneath the surface was determined.The radiance spectra at the Secchi depth and at the surface were then integrated spectrally by using the photopic efficiency function, and from the resulting two luminances the efficient attenuation coefficient c phot,disk could be obtained.The attenuation coefficient c phot of the upward background luminance L(z) between the Secchi depth and the surface was calculated in a similar way.All values of the ratio c phot,disk /c phot have been found to lie within the range 1.00 ± 0.01.Thus the two attenuation coefficients are practically equal.
The upward luminances at the surface and the Secchi depth, L(0) and L(Z D ), were found by integrating the spectral upward luminances at the two depths, and then K L,phot could be determined by using Eq. ( 9).Because the observed values of Z D for the black disk were less than for the white disk (Table 3), the spectral luminance L(Z D ) at this smaller depth differed from the former luminance, and accordingly c phot and K L,phot for the black disk also became slightly different (Table 3).The attenuation coefficients c phot and K L,phot for the luminances observed through colour filters were calculated in the same way as for the open-eye coefficients.

4.2
Values of ρ DL , R L , R, ρL,air , ln(A 1 ) and ln(A 2 ) It was explained in Sect.4.1 how the downward illuminance and the reflected luminance from the disk were determined, and which spectral values of ρ DL we have applied.From the latter values ρ DL was estimated equal to 0.30 for the white disk observed through the red filter, and by definition equal to 0 for the black disk.The values of ρ DL for the white disk observed with the open eye and through the blue and green filters were determined by using the spectral recordings of downward illuminance, and the results are 0.29, 0.27 and 0.29, respectively (Table 3).
The mean value of R L (0), defined as the ratio between upward luminance and downward illuminance just beneath the surface (Eq.7), has been calculated from 32 stations as 0.56 % for the open eye.This value is valid for both the white and black disks (Table 3).However, the mean value of R L (Z D ) at the Secchi depth becomes different for the white and black disks, being 0.83 % and 0.61 %, respectively, and the reason for this is that Z D differs for these disks.The estimated values of R L for the colour filters are shown in Table 3.
It was found by Aas et al. (2009) that an average value for the quantity R in the Oslofjord-Skagerrak area was 0.506 ± 0.045 for solar altitudes between 15 • and 60 • , which is close to similar results found by Morel and Gentili (1996) and Mobley (1999).R was assumed to be independent of wavelength, and the found value has been applied for both the white and black disks (Table 3).
The radiance reflected towards the zenith at the surface of the sea is described by the radiance reflectance ρL,air .Its value for a clear sky depends on the angular distribution of sky radiance, the solar altitude, the ratio between diffuse sky irradiance and direct solar irradiance, and the statistical distribution of surface slopes.Polynomials for calculating ρL,air were obtained during a previous work (Aas, 2010).The problem was simplified by looking at average values for all possible angles between wind direction and solar azimuth, using the Cox and Munk model with a one-dimensional Gaussian distribution for the surface slopes (Cox and Munk, 1954a, b).The input was spectral sky and solar radiance data from the Oslo region (Høkedal and Aas, 1998;Aas and Høkedal, 1999).However, there is the problem that while the wave height and wave spectrum are functions of wind speed, duration and fetch, only the wind speed appears in the Cox-Munk model.A comparison between wave heights and wind speeds at those of our stations where both quantities were observed, shows that in many cases either the wind duration or the fetch must have had a limiting effect on the wave height.Still, the mean values ± the standard errors of wind speed and wave height, 5.5 ± 0.4 m s −1 and 0.7±0.1 m, respectively, are consistent with the conditions for a fully developed sea shown in the diagram by Gröen and Dorrestein (1976, also shown by WMO, 1998).Accordingly we have tentatively chosen the wind speed 5.5 m s −1 to represent the average conditions.The mean values and standard deviations of ρL,air shown in Table 3 are based on 10 different cases of atmospheric radiance distribution.The values of ρL,air for the open eye (the white and black disks) and for the blue, green and red filters were obtained from atmospheric radiance distributions at 550, 470, 540 and 620 nm, respectively (Table 3).
The estimated mean value and standard deviation of ln(A 1 ) were obtained by using the value of ρ DL shown in Table 3, and varying values of R L (Z D ) for each station.Similarly the values of ln(A 2 ) were found from the values of ρL,air and R suggested by Table 3 and varying values of R L (0).The results for ln(A 1 ) and ln(A 2 ) are presented in Table 3.
The size distributions of the optical quantities estimated here are often highly asymmetric, implying that the standard deviation does not always provide a satisfactory description of the range of variation, like when the standard deviation is greater than the mean value.

Value of C t and W
It was mentioned in Sect. 2 that according to Blackwell (1946) the contrast threshold C t will depend on the angle α subtended by the observed target, the luminance L(0) of the background, the probability of detection, and the exposure time.In Part I of Blackwell's investigation circular targets, brighter than the background, had an exposure time of six seconds, and the contrast thresholds were those that corresponded to a 50 % detection probability.In Part II the targets were darker than the background, and the conclusion was that in most cases negative stimuli are equivalent to positive stimuli of the same area and contrast.The results in Part III were obtained when an indefinitely long exposure time was used.A comparison between Blackwell's experiments and Secchi disk observations is not straightforward, because Blackwell's target images were constant during the time of exposure, while at sea the image of the Secchi disk is usually varying.We have chosen an indefinitely long exposure time and the highest detection probability, 100 %.
The apparent angle α from the observer's eye across the Secchi disk is a function of the diameter D of the disk, the height H of the observer's eye above the surface of the sea, the Secchi depth Z D and the refractive index n.Because α is a small angle, tan(α/2) can be approximated by α/2, and 7.3 ± 0.9 7.9 ± 0.7 7.8 ± 0.8 8.4 ± 0.7 4.7 ± 0.1 Z D (c + K L ) phot 7.0 ± 1.3 4.8 ± 1.2 5.1 ± 0.9 5.5 ± 0.7 1.9 ± 0.3 tan(j ), where j is the corresponding angle of refraction in water, can be approximated by α/(2n), using Snell's Law.The angle α in radians becomes If we use Eq. ( 22) with D = 0.3 m, H = 3 m, and n = 1.33, and if Z D lies in the interval from 0.5 to 14 m, then the angle α across the Secchi disk will lie between 1.3 • and 5.1 • .This is within the range in Blackwell's study where the angles of α extended from 0.01 • to 6 • .Our recordings showed that the background luminances L(0) in the Oslofjord-Skagerrak area were in the range 0.4-1600 cd m −2 for the observations with open eye and colour filters.Blackwell's luminances extended from 3 × 10 −6 to 3400 cd m −2 .Thus our environmental conditions were within the ranges described by Blackwell's experiments.We have selected those of Blackwell's data that are closest to our observed background luminances and angles of subtense, and interpolated these data to fit our ranges.The estimated mean values of C t and ln(C t ) for the 30 cm disk, observed with the open eye and through colour filters, are presented in Table 3.
Rather than calculating C t for each single case and then finding the mean value and standard deviation from the resulting data set, we have found it necessary to restrict the calculations of C t to the mean, maximum and minimum conditions.The mean conditions were assumed to produce the mean value of C t , and a crude estimate of the standard deviation was obtained by the expression (C t,max -C t,min )/4.For a normal distribution, 95 % of the observations will fall within a range of ±2 standard deviations from the mean value.
In Table 3 the mean value of C t is 0.56 % for the white disk, and the mean value of ln(C t ) becomes −5.2.Krümmel (1889) referred to Helmholtz for the threshold contrast 1/133 = 0.75 % which produces ln(C t ) = −4.9.The relative difference between the last number and our estimate is only 6 %.Tyler (1968) applied C t ≈ 0.66 %, leading to ln(C t ) = −5.0,which is even closer to our estimate.Gordon and Wouters (1978) applied C t values from 0.15 to 0.6 % or ln(C t ) from −5.1 to −6.5 in their model studies.Højerslev (1986) deduced from his Baltic recordings a threshold contrast slightly greater than our estimate, C t = (0.70 ± 0.03) %, resulting in ln(C t ) = −(4.96± 0.04).Considering that C t is not supposed to be a constant, these values are surprisingly similar.
It is noteworthy that Blackwell's experiments involved both the colour and night visions of the human eye.It is well known that night vision requires some time to be activated and adapted, and that a sudden bright glint may temporarily change the vision from night to colour mode.The range where both colour and night visions are active, termed mesopic vision, is 0.001-10 cd m −2 according to the CIE.Thus some of our colour filter observations are within this range, but to what extent the varying light conditions have influenced the resulting Secchi depths, we cannot say.
Since our Secchi depths usually are from coastal areas where the surrounding land masses reduce the influence of the wind on the ordinary sea waves, the use of Eq. ( 14) may perhaps also be overestimating the effect of the wind on the capillary waves.Still, we have tentatively estimated W by using the observed wind speeds and Secchi depths.The average values of W and ln(W ), based on the 32 stations constituting the data set where we have complete observations of c, K L and R L , are shown in Table 3.

Value of ln(A) and comparison to observations of Z D,white (c + K L ) phot
There are different ways to test Eq.( 21) and to estimate a value for ln(A), and the result will depend on the chosen method.By using the estimated mean values of ln(A 1 ), ln(A 2 ), ln(W ) and ln(C t ) from Table 3, ln(A) in Eq. ( 21) obtains the value 7.3 for the 30 cm white disk.The mean value of Z D,white (c + K L ) phot , found from observations, is 7.0 (Table 3), which is 4 % less than the value predicted by ln(A).
Early in the analysis it was discovered that whenever Z D,white was in the range 1-2 m, the values of Z D,white (c + K L ) phot tended to increase to 10-14, and consequently the few stations where Z D,white < 2 m have been omitted.The reason for the discrepancy between theory and observations for small Secchi depths may be that one or more of the assumptions on which Eq. ( 21) is based becomes invalid.For instance, the assumption L * D (z) ≈ L * (z) made for Eq. ( 4) may be less good when Z D is small.Figure 3 presents (c + K L ) phot as a function of Z D,white at the 32 stations where we have sufficient observations down to the Secchi depth, with the curve added.The curve represents the mean value (c + K L ) phot Z D,white = 7.0, according to Table 3.It has been extended to values of Z D,white < 2 m to illustrate that it will significantly underestimate (c + K L ) phot in this range.The symbol R 2 = 0.87 in Fig. 3 is the coefficient of determination.The general definition of this quantity for a data set (x, y) and a chosen trendline , where ε is the root mean square of the residuals y-f (x), and s y is the standard deviation of y.It should be pointed out that while the quantity R 2 depends on the chosen trendline, the coefficient of correlation r is solely a function of the data set (x, y), since r is defined by r = S x,y /(s x s y ), where S x,y is the covariance of x and y, and s x is the standard deviation of x.
Another way to test the mean value ln(A) = 7.3 in Eq. ( 21) is to correlate (c + K L ) phot with 1/Z D,white .The best-fit line through the origin obtains the slope 7.5, and if (c + K L ) phot is estimated by 7.5/Z D,white , the rms of the error [(c + K L ) phot − 7.5/Z D,white ] becomes 0.23 m −1 .The mean value of (c + K L ) phot is 1.09 m −1 (Table 3), and the rms error relative to this value represents 21 %.It is interesting that if we estimate (c + K L ) phot by using Eq. ( 23), the rms error only changes to 0.24 m −1 or 22 %.The coefficient of determination in this case becomes R 2 = 0.88.If the observer's eye is below the surface, ρL,air = 0 and W = 1, and the value of ln(A) in Eq. ( 21) becomes 8.7.Tyler (1968) estimated the number 8.69 for this case, while Holmes (1970) found for sub-surface observations in turbid coastal waters that the average value of Z D,white (c + K L ) phot was 9.4.Højerslev (1977) obtained the same value for the sub-surface case.Thus our theoretical estimate of ln(A) for the sub-surface case agrees with Tyler, but deviates 8 % from the estimates by Holmes and Højerslev. Højerslev (1977) found that for observations above the surface, the product Z D,white (c + K L ) phot should be in the range 7.9-9.4.This is 8-29 % above our estimate in Table 3.

Effect of colour filters and the black disk
When the white Secchi disk is observed through the coloured glass filters, the optical coefficients and ln(A) change values, as shown by Table 3.The table demonstrates that in this case the observed mean values of Z D (c + K L ) phot are more than 30 % smaller than the estimated mean values of ln(A).
The best-fit correlation lines y = A + Bx and y = B 0 x for the observations of Z D,blue , Z D,green and Z D,red as functions of Z D,white have been determined, and the found constants A, B and B 0 as well as the mean values of the ratios (y/x) = Z D,filter /Z D,white are presented in Table 4.We see that the colour filters reduce the Secchi depths to 50-70 % of the depths for the open eye.The result that Z D,green on average is reduced to almost 70 % of Z D,white , although the wavelengths of peak photopic sensitivity for Z D,green and Z D,white are very close (Table 2), may be due to the half-peak bandwidth of the green filter, which is only half of that for the open eye.
Table 4 also shows the correlation lines y = A + Bx with the constant term A. When the values of B 0 , B and (y/x) mean are close to each other, it means that the observations lie close to a straight line through the origin.
Mikaelsen and Aas (1990) (bowl) observed by the open eye.Mean values of y/x and y and their standard deviations are included.The size of the disk is 30 cm if not otherwise indicated.r is the correlation coefficient, ε is the root-mean-square error of the deviations (y-A-Bx), and ε 0 is the rms of (y-B 0 x).N is the number of data pairs (y, x).1985) yielded mean values of the ratios equal to 0.90, 0.91 and 0.29 for the blue, green and red filters.The difference from our results in greenish coastal waters is clearly a result of the difference in water colour.Table 3 predicts that the average value of Z D,black (c + K L ) phot should be ln(A) = 4.7 for the 30 cm black disk, but the observed value is less than half of this: 1.9.This could be because an ordinary disk of black plastic was used in these observations, and not the bowl-shaped device.The surface of the plastic disk seemed to be entirely black on deck, but looked dark grey in the sea, and brighter than its background.In 1992 a series of measurements at 19 stations was taken with the black-painted bowl together with the black and white disks.The depth Z B,black of the black bowl was on average 20-30 % greater than the depth Z D,black of the black disk (Table 4).Thus the bowl produces a smaller reflection and a greater contrast than the disk.Since the estimated attenuation (c + K L ) phot is approximately the same for the black and white disks as shown by Table 3 (the minor difference being due to Z D,black being smaller than Z D,white ), the values of ln(A) in Table 3 imply that the ratio Z B,black /Z D,white should be approximately 4.7/7.3= 0.64.However, the observed ratio in Table 4 is only 0.23-0.26.This could mean that our black bowl is not perfectly black, but it could also be that the use of colour filters and the black disk or bowl introduces effects that are not included in Eq. ( 21).

Effect of size
The observations with the 10 cm disks were made in 1992, when instruments for spectral recordings of c and K L were not available.Consequently the two sides of Eq. ( 21) cannot be tested directly for the 10 cm disk, but other experiments have been made.The angle α of the total field of view across the 10 cm Secchi disk can be calculated from Eq. ( 22) with D = 0.1 m, H = 3 m and n = 1.33.Because the observed values of Z D for this smaller disk lie in the interval from 0.5 to 12 m, the angle α across the Secchi disk becomes 0.5 • -1.7 • .The threshold contrast for the smaller disk becomes greater than for the 30 cm disk, while the value of W will be reduced according to Eq. ( 14), because D 2 is reduced more than Z 2 D .The combined effect is that the values of ln(A) in Table 3 for the 30 cm white and black disks will be reduced by 22 and 13 %, respectively.For the blue, green and red colour filters the decreased size produces similar reductions of 18, 21, and 17 %.This can be tested directly by comparing the values of Z D for disk diameters of 10 and 30 cm.The results in Table 4 show that the relative influence of size is greater for the open eye than for the colour filters, and that the decrease in diameter from 30 to 10 cm reduces the Secchi depth by an average of 10-20 %, in agreement with our estimates.Figure 4 presents all open-eye and colour filter observations for 10 and 30 cm disks put together, and the best fit line through the origin obtains a slope of 0.83, indicating an average Secchi depth reduction of 17 %.

Effects of sun glitter, water telescope and ship shadow
The effect of a water telescope is to eliminate sun glitter and skylight reflection at the surface, and to reduce the blurring effect of capillary waves.If we insert a surface reflectance ρL,air = 0 and a wave factor W = 1 into Eq.( 21), we find that ln(A) should increase by 12 % for the 30 cm white disk, based on the estimated quantities in Table 3.For the colour filters and black disk the effect is in the range ±6 %.Mikaelsen and Aas (1990) tested the effect at four stations in 1987 with a 30 cm disk.According to their observations Z D,white increased by 11 % on average using the water telescope, and by including the observations with the colour filters the increase became 15 %.New experiments were done in 1992, and a 10 cm disk was used in order to see it properly within the field of view of the telescope.The results, listed in the last five rows of Table 4, show that the telescope increased Z D,white,10 by an average of 14 %, and the effect on Z D,filter,10 was of the same magnitude.If all open-eye and filter observations are put together (Table 4), we see that the average increase in the Secchi depth by using the telescope is the same, 14 %.The best-fit line through the origin obtains the slope 1.19 (Fig. 5), indicating an average increase of 19 %.Holmes' use of a water telescope (1970) only increased the observed depth by 2-4 %, which is significantly less than our result.Højerslev (1986) found for Baltic waters that the ratio between the depth Z D,tel observed with a water telescope and the ordinary Z D could be approximated by a function of the wave height H in units of metres: Z D,rel /Z D = 1 + 0.4 H .This expression underestimates the effect at our 11 water telescope stations where H was < 0.1 m, but we have not tested it in more rough seas.Sandén and Håkansson (1996) investigated the effect of wind on Z D , and it seems like the presence of wind tended to reduce the depth by ∼ 10 %.
If we observe the Secchi disk on the sunlit side of the ship, sun glitter from the sea will reduce its threshold depth.On the shadow side there may be less glitter, but the ship shadow will also reduce the luminances from both the background and the disk, and this may require a greater contrast, and thus lead to a smaller depth.The balance between gains and losses related to the absence or presence of direct sunlight is demonstrated by Fig. 6, which presents the Secchi depths observed on both sides of the ship with the white disk.The best-fit line through the origin obtains the slope 0.97, which is close to 1.If we do the same experiment for all depths observed with colour filters and the open eye (Table 4), we see that the slope becomes 0.93, that is Z D is now on average reduced by 7 % on the shadow side.

The monochromatic assumption
In Sect. 2 it was assumed as a first approximation that the beam attenuations of the upward luminances resembled the attenuations of the radiances at the wavelength of maximum spectral transmittance, and Sect.4.1 described how the efficient attenuation coefficients c phot,disk and c phot of upward luminance from the disk and background could be estimated.In this section we will investigate the relationships between the photopic and monochromatic coefficients as well as the errors introduced by the monochromatic assumption.The coefficients c phot,disk and c phot have been correlated with the monochromatic beam attenuation coefficient c 555 at 555 nm, and the results are presented in Table 5.Similarly, the vertical attenuation coefficient K L,phot has been correlated with K L,555 .The correlations are described by best-fit lines on the forms y = A+Bx and y = B 0 x.Table 5 also displays values of (y/x) mean and y mean .We see that the values of B, B 0 and (y/x) mean are close to each other, indicating that the points (x, y) lie close to the lines.This is confirmed by the very small errors ε and ε 0 introduced by using the correlation lines (≈ 0.01 m −1 ).ε and ε 0 are defined as the root mean squares of the deviations (y-A-Bx) and (y-B 0 x), respectively.The coefficient c phot , for instance, is almost identical to c 555 , with an average ratio of 1.02.However, Table 5 also demonstrates that the deviations between the photic and monochromatic coefficients are greatest for K L,phot versus K L,555 , which displays a slope of 1.12.The sum (c+K L ) phot versus (c + K L ) 555 obtains the slope 1.03.
It follows from the descriptions of c phot and K L,phot that these coefficients are apparent optical properties, depending on the ambient light field.It could not be stated a priori that they would be so strongly correlated with the monochromatic coefficients at 555 nm, with a coefficient of correlation equal to 1.00.We have not investigated how the found relationships will work in more clear and bluish sea waters, but we assume that the correlations may be weaker there.Holmes (1970) equipped his irradiance and beam attenuation meters with filters where the spectral shape of the transmittance resembled the spectral sensitivity of the eye, but the attenuation meter then recorded a lamp spectrum that probably was different from the natural spectrum within the sea.
The correlation results for attenuation coefficients derived from luminances observed through the colour filters (Table 5) versus monochromatic coefficients show that the correlation coefficients are all very close to 1.0.The slope B 0 for the blue and green filters obtains values between 0.95 and 1.01, while the red filter deviates more from 1.0 with the slopes 1.06 and 1.21.The slopes B 0 for the sum (c + K L ) are in the range 0.97-1.10.None of these deviations from the slope 1.0 explain the discrepancies between theory and observations displayed by Table 3 for the colour filters.Our overall conclusion becomes that the monochromatic assumption works satisfactorily, and that the mentioned discrepancies must have other causes.In Blackwell's experiments (1946) the targets were projections of "white" light from electrical lamps onto a white screen.The different parts of the visible spectrum were not investigated separately, and it could perhaps be that C t also has a spectral dependence.
It remains to test the ability of Eq. ( 21) to describe the relationships between Z D and monochromatic values of c, K L and (c + K L ).This will be done in the next section.

Estimates of monochromatic coefficients
In Sect. 2 we assumed that the Secchi depth observed with the open eye might be determined by the attenuation coefficients at the wavelength of maximum water transmittance, that is around 555 nm.Section 4.4 demonstrated that the numerical values of the constants in the relationships between Z D and (c + K L ) would depend on the chosen method.If our intention is to estimate (c + K L ) 555 from observed Z D,white , the obvious choice, based on the form of Eq. ( 21), is to perform a linear correlation analysis between (c + K L ) 555 and 1/Z D,white .The slope of the line through the origin obtains the value 7.4, which is practically the same as the slope 7.5 found in Sect.4.4 for (c + K L ) phot .However, usually in marine-optical research the singular values of c and K L are more interesting than their sum (c + K L ), and therefore c 555 and K L,555 have been separately correlated with 1/Z D,white , as shown by Table 6.While the correlation coefficient is 0.95 for c 555 , which is almost as perfect as it can be expected to be, it is reduced to 0.72 for K L,555 .Similar results have been obtained for the channels at 412, 443, 490, 510, 620 and 665 nm (Table 6).The root-mean-square errors of K L and c, estimated from the correlation lines, are in the range 0.1-0.6 m −1 , but because K L is smaller than c the relative errors become greater for K L than for c.The result that the correlation coefficients are greater for c than for K L is as expected since c contributes more than K L to the sum (c + K L ) which determines Z D,white .
Often the vertical attenuation coefficient of downward irradiance, K d , will be a more useful quantity than K L .The correlation results for K d at the MERIS channels as a function of 1/Z D,white are presented in Table 6.Some of the values are remarkably similar to those obtained for K L , and the reason for this is that the mean value of the ratio K d /K L usually is close to 1.In order to complete the investigation, the coefficient K u of upward irradiance has been included in  5. Linear relationships of the forms y = A + Bx and y = B 0 x obtained by correlation analysis of attenuation coefficients of spectrally integrated luminance for the open eye and with blue, green and red glass filters and coefficients of monochromatic radiance at 470, 540, 555 and 620 nm, and mean values of y/x and y.The attenuation coefficients are averages over the depth range 0-Z D,white .r is the correlation coefficient.The error ε is the root mean square of the deviations (y-A-Bx), and ε 0 is the rms of (y − B 0 x).The analysis is based on 32 stations.6. Relationships of the forms y = A + Bx and y = B 0 x, where A, B and B 0 are coefficients from the correlation analysis between y and x = 1/Z D,white .The error ε is the root mean square of the deviations (y-A-Bx), and ε 0 is the rms of (y-B 0 x).r is the correlation coefficient and N is the number of stations.7. Relationships of the forms y = A + Bx and y = B 0 x obtained by correlation analysis of attenuation coefficients of monochromatic radiance and irradiance and Secchi depths observed with the open eye and with blue, green and red glass filters.The attenuation coefficients are averages over the depth range 0-Z D,white , and r is the correlation coefficient.The error ε is the root mean square of the deviations (y-A-Bx), and ε 0 is the rms of (y-B 0 x).The analysis is based on 25 stations.1/Z D,red 0.77 0.17 1.0 1.5 1.7 ± 0.5 0.47 ± 0.22 0.14 0.16 K u,620 1/Z D,white 0.82 0.12 2.2 2.8 3.1 ± 1.0 0.47 ± 0.22 0.13 0.14 Table 6.The results indicate that the errors of the estimated vertical coefficients are greatest for K u and smallest for K L , and less in the green and red part of the spectrum than in the blue part.We suspect that the rms errors at 412 and 443 nm of 90-100 % are due to varying amounts of yellow substance which will influence the coefficients more than the Secchi depth.In addition weak signals compared to noise and detection level at these wavelengths create errors.Gordon and Wouters (1978) found in their model study that in relatively turbid water, defined by the authors as water where the backscattering probability is constant, the product cZ D,white would be approximately constant, while (c + K d )Z D,white would vary more.This result is in agreement with the results in Table 6.

Ocean
In Table 6 the errors of the estimates produced by the bestfit lines y = A + Bx and y = B 0 x, where y is an attenuation coefficient and x = 1/Z D,white , are shown.Usually the line y = B 0 x through the origin works satisfactorily, but at some wavelengths the addition of a constant term A reduces the error of the estimates significantly.Figure 7 shows observations of K d at 555 and 665 nm versus x = 1/Z D,white , together with the best-fit lines y = A + Bx.The value of A is 0.02 m −1 at 555 nm, and 0.41 m −1 at 665 nm (Table 6).It is interesting that Jerlov's K d values for the clearest ocean water type I are of the same order of magnitude: 0.07 m −1 at 555 nm and 0.40 m −1 at 665 nm (e.g.Jerlov, 1976).Figure 7 indicates that in the green part of the spectrum the constant term can be omitted without any practical consequences, while in the red part a better fit is obtained if A is included.The inclusion of the constant term A reduces the error at 665 nm from 0.21 m −1 to 0.07 m −1 according to Table 6.For the vertical attenuation coefficients K L and K u of upward directed light the improvements by adding a constant term A to the correlation lines are less.Mikaelsen and Aas (1990) analysed five stations in the Oslofjord with recordings of c, and 11 stations with K d .Z D was observed by the naked eye and with colour filters.Most of their results overlap with the present findings.Sørensen et al. (1993) obtained from 189 stations in the Oslofjord-Skagerrak area the mean value and standard deviation c green Z D,white = 6.2 ± 1.3, which fall within the corresponding range of Table 6.
The relationships between the Secchi depths observed through the glass filters and the coefficients c, K L , K d and K u at 470, 540 and 620 nm are presented in Table 7.The results, based on 25 stations, show that the errors of the estimated coefficients are in the range 0.07-0.47m −1 , or 10-80 % of the corresponding mean values.It is noteworthy that compared to the observations with the open eye, the colour filter observations represent no improvement in the accuracy of the estimated vertical attenuation coefficients.Especially for the coefficients of beam attenuation it is clear that Z D,white produces better estimates than Z D,filter .We think this result is valid within our area of investigation, but not necessarily in other sea regions.
It may be noted that Table 7 contains analyses of attenuation coefficients at 620 nm that are parallel to those in Table 6, and that the results are different.The reason for this may be that Table 7 is based on stations with colour filter observations, and accordingly the number of applicable stations will be reduced to less than half of those in Table 6.Evidently the number of stations influences the statistical results.Højerslev (1977) suggested the approximate mean values 3, 6 and 9 for the products K d Z D , cZ D and (c + K d )Z D , respectively, where K d , c and Z D are observed through colour filters, but these findings are not confirmed by Table 7.
Because remote sensing of ocean colour is used to estimate coefficients like K d , the technique can also be used to estimate the Secchi depth.Within our area of investigation the relationship between 1/Z D,white and the water-leaving radiance L w recorded in the red part of the spectrum (630-690 nm) by the TM3 sensor at the Landsat satellite was found to be 1/Z D,white = [0.203m −1 ] + [0.072 W −1 m sr µm]L w , while the coefficient of correlation was 0.92 (Sørensen and Aas, 1994).For the TM1, TM2 and TM4 sensors in the bluegreen (450-520 nm), green (520-600 nm) and near-infrared (760-900 nm) parts of the spectrum the coefficients of correlation became 0.69, 0.87 and 0.65, respectively.Kratzer et al. (2003) and Zhang et al. (2003) have discussed estimates of 1/Z D,white based on satellite observation of the Baltic Sea, and Morel et al. (2007) have examined the Secchi depth estimates from various ocean colour sensors for the open ocean case.References to numerous investigations in lake waters can be found on the internet.

Quanta irradiance -PAR
The spectrally integrated quanta irradiance (400-750 nm), also termed PAR (photosynthetically available radiation), is one of several factors determining the primary production in the sea.If the transmittance of this irradiance between the surface and the Secchi depth is denoted T D , then the average vertical attenuation coefficient K q of the irradiance over the depth range 0 − Z D,white is related to Z D,white by The observed mean value ± the standard deviation of T D , obtained from our third data set of 143 stations from the Oslofjord inside the Faerder Light House at the Skagerrak border, is (9 ± 4) %.Based on the total data set of 205 stations, including stations from the nearby Skagerrak and Kattegat, the range becomes slightly greater: (9 ± 6) %.
Equation ( 24) can also be written as and by using T D ≈ 0.09 ± 0.06, the range of the product K q Z D,white becomes 1.9-3.5.This is consistent with the mean value and standard deviation of the product obtained from observed pairs of K q and Z D,white : 2.5 ± 0.5 (Table 8).
By linear correlation K q may be estimated from 1/Z D,white with an rms error that is less than 20 % of the mean value of K q .
The depth of the euphotic zone, defined as the surface layer where there is a net positive production from photosynthesis, is often estimated as the depth Z q (1 %) where the quanta irradiance is reduced to 1 % of its surface value.If K q is approximately constant with depth, Z q (1 %) can be determined from the equation 0.01 = e −K q Z q (1 %) (26) which can be transformed to By inserting for K q from Eq. ( 25), this expression becomes indicating a linear proportionality between Z q (1 %) and Z D,white , provided ln(T D ) is approximately constant.Similar expressions can be deduced for Z q (3 %), Z q (10 %) and Z q (30 %).The correlation results for Z q (p %) as a function of Z D,white are presented in Table 8, together with other related statistics.
Table 8.Linear relationships of the forms y = A + Bx and y = B 0 x, obtained by correlation analysis of the vertical attenuation coefficient K q of downward quanta irradiance, the depths Z q (p %) where the quanta irradiance is reduced to p percent of the surface value, and the Secchi disk depth Z D,white .The error ε is the root mean square of the deviations (y-A-Bx), ε 0 is the rms of (y-B 0 x), and r is the correlation coefficient.The analysis is based on 205 stations.Fig. 8.The depth of the euphotic zone, estimated by the 1 % level Z q (1 %) of the surface quanta irradiance, as a function of the Secchi depth Z D,white .The best-fit line through the origin is Z q (1 %) = 1.8 Z D,white .
An old rule of thumb for the Oslofjord says that Z q (10 %) corresponds roughly to Z D,white (also suggested by Paulson and Simpson, 1977), and Z q (1 %) to twice this depth.We have already found that the average percentage of the quanta irradiance at Z D,white was 9 %, and when we calculate the average ratios Z q (10 %)/Z D,white and Z q (1 %)/Z D,white , they become 0.9 and 2.2, respectively (Table 8). Figure 8 illustrates the significant scattering of points around the correlation line through the origin for Z q (1 %) as a function of Z D,white .If we apply the relationship y = A + Bx instead of the line through the origin, a slightly better fit is obtained, and the rms error for the estimates of Z q (1 %) as a function of Z D,white is reduced from 3.1 to 2.7 m (Table 8).Compared to the mean value 11.5 m of Z q (1 %), these numbers represent relative errors of 27 and 23 %, respectively.
Table 8 shows that estimates of Z q from observed Z D,white are likely to have relative mean errors in the range 20-30 %.This is because Z D,white is primarily a function of c + K L (Eq. 20), while Z q is a function of K q only (Eq.27).The vertical attenuation coefficients K L and K q are mainly functions of the absorption coefficient a, while the beam attenuation coefficient c consists of a + b, where b is the scattering coefficient.Thus Z q is less influenced by particle scattering than Z D , a property which reduces the correlation between the two quantities.It is our experience that an increased particle content in the sea may have a strong reducing effect on Z D , while the influence on Z q is much smaller.Still, the relationships in Table 8 will provide very useful checks of our irradiance measurements.
Not surprisingly the inter-correlations between Z q (1 %), Z q (3 %), Z q (10 %) and Z q (30 %) are stronger than between Z q and Z D,white (Fig. 9).The statistical relationships are shown in Table 8.Mikaelsen and Aas (1990) found K q Z D,white = 2.7 ± 0.6, Z q (10 %) = 0.66 Z D,white and Z q (1 %) = 1.7 Z D,white , while Sørensen et al. (1993) obtained K q Z D,white = 2.3 ± 0.4.All of these results are close to the values in Table 8.Kratzer et al. (2003) found for Baltic waters that K q ≈ 1.7/Z D,white , implying that for the same K q the Secchi depth Z D,white tends to be greater in the Oslofjord than in the Baltic.

Chlorophyll a and total suspended material
The chlorophyll content is perhaps the most used concept when the amount of algae in the sea shall be described.Unfortunately the concentration of chlorophyll a (Chl) is not an optical property, although it influences the absorption and scattering coefficients of sea water.As a result our estimates obtained by correlation analysis of Chl as a function of 1/Z D , based on 41 stations north of 57.0 • N, have average errors of Fig. 9.The depth of the p % levels Z q (p %) of the surface quanta irradiance, as a function of Z q (1 %) for p equal to 3, 10 and 30.The best-fit lines through the origin are Z q (3 %) = 0.69Z q (1 %), Z q (10 %) = 0.41Z q (1 %), and Z q (30 %) = 0.20Z q (1 %).
50-90 % relative to the mean value of chlorophyll content (Table 9, Fig. 10).The estimates of total suspended material TSM (51 stations) for the same area have average errors of 40-70 % (Table 9, Fig. 10).Still, such relationships provide useful checks because they quantify the order of magnitude of the concentrations, within the area of investigation.Better correlations may sometimes be found for smaller regions.
In general the relationships discussed in Sect. 5 are likely to show regional variations.
In the earlier investigation by Sørensen et al. (1993) the products Z D,white Chl (249 stations) and Z D,white TSM (275 stations) obtained the mean values and standard deviations (19 ± 16) mg m −2 and (7.8 ± 3.8) g m −2 , respectively, in agreement with the corresponding ranges of Table 9.An estimate of the product Z D,white Chl in the Baltic Sea, based on observations by Fleming-Lehtinen and Laamanen (2012), is ∼ 25 mg m −2 .
The turbidity Turb, expressed in nephelometric formazine units (NFU), has not been included in the present study, because the accuracy of the corresponding turbidity data was not satisfactory.However, in the mentioned investigation by Sørensen et al. (1993) the mean value and standard deviation of the product Z D,white Turb (308 stations) was (3.5 ± 1.4) m NFU −1 .

Summary and conclusions
We have analysed the Secchi depth Z D and its relationship to other properties of the sea water in the Oslofjord-Skagerrak area.White and black disks of different sizes have been applied, and the Secchi depth has been observed with the naked eye, through colour filters and with a water telescope.A few stations where Z D,white was less than 2 m or the Q factor was ]/Z D,white .
outside the range π − 2π have been omitted.The upward luminance will typically have its spectral maximum in the green part of the spectrum, close to 555 nm.An extended version of Tyler's equation for Z D , as expressed by Eq. ( 21), has been tested.The right-hand side of the equation, ln(A), is a function of the reflectance ρL,air at the air-sea interface, the reflectances R L (0) and R L (Z D ) of the sea water just below the surface and at the Secchi depth, the reflectance ρ DL of the disk, the contrast threshold C t of the human eye, the contrast transmittance W through the surface and the factor R. With the estimated mean values for these quantities (Table 3), ln(A) obtains the mean value ± the standard deviation 7.3 ± 0.9 for the 30 cm white disk observed with the naked eye.The left-hand side of the equation is the product of the observed Secchi depth Z D,white and the attenuation coefficients (c + K L ) phot , and its mean value becomes 7.0 ± 1.3 (Table 3, Fig. 3).Thus on average the observed value is only 4 % less than the one predicted for the white disk.The slope of the best-fit line through the origin for (c +K L ) phot as a function of 1/Z D,white becomes 7.5.The deviation between the observed (c + K L ) phot and its estimate 7.5/Z D,white obtains an rms value of 0.23 m −1 , which represents 21 % of the mean value of (c + K L ) phot (Table 3).
The agreement between ln(A) of Eq. ( 21) and the observed Z D (c + K L ) phot that we find for the white disk is reduced when the Secchi disk is observed through coloured glass filters.In this case the observed mean values of Z D (c+K L ) phot are more than 30 % smaller than the estimated mean values of ln(A) (Table 3).We speculate if there may be a spectral dependence of C t .
The observed depths Z D,black of the black disk and the related products Z D,black (c + K L ) phot were also smaller than predicted.According to Table 3 the average value of 620 nm (Table 2) have also been calculated (Table 7).The rms errors of the estimated coefficients are in the range 0.07-0.47m −1 , or 10-80 % of the corresponding mean values.The accuracy of the estimates based on the white disk and naked eye is equal to or better than the estimates from colour filter observations.
The mean vertical attenuation coefficient of quanta irradiance or PAR between the surface and the Secchi depth, K q , may be estimated from 1/Z D,white with an rms error less than 22 % of the mean value of K q (Table 8).The mean values of the ratios Z q (10 %)/Z D,white and Z q (1 %)/Z D,white are 0.9 and 2.2, respectively, or very close to 1 and 2. Table 8 shows that estimates of Z q from observed Z D,white will have relative errors in the range 20-30 %.The depth of the euphotic zone may be defined as Z q (1 %) (Fig. 8), and according to Table 8 this depth can be estimated by an rms error of 24 %.Consequently these relationships will provide very useful checks of our irradiance measurements.Relationships for Z q (3 %), Z q (10 %) and Z q (30 %) as functions of Z q (1 %) are presented in Fig. 9 and Table 8.
The Secchi depth may also be used to estimate the concentrations of chlorophyll a (Chl) and total suspended material (TSM) (Fig. 10).The estimates of Chl and TSM as functions of 1/Z D have average errors of 50-90 % and 40-70 %, respectively (Table 9, Fig. 10).Still, these estimates will provide the order of magnitude of the concentrations.
Our overall conclusion becomes that Eq. ( 21) quantifies well the relationships between the white disk and the optical attenuation coefficients, but less so for the colour filters and the black disk.The Secchi depths provide very useful checks of the monochromatic attenuation coefficients and of chlorophyll a and total suspended material.We assume that there may be regional differences for the found relationships, and they should be tested out for different oceanic regions and water types.

Fig. 2 .
Fig. 2.Mean value of the upward radiance spectra just beneath the surface, normalised at 555 nm and based on 48 stations, and the standard deviation of the normalised spectra.For comparison a station of typical Atlantic water is presented.

Fig. 3 .
Fig. 3. (c + K L ) phot as a function of Z D,white .The line is the function (c + K L ) phot = 7.0/Z D,white .See text in Sect.4.4 for discussion.R 2 is the coefficient of determination for this line.

Fig. 4 .
Fig. 4. All naked eye and colour filter observations of Z D with a 10 cm disk as a function of the observations with a 30 cm disk.The best-fit line through the origin is the function Z D,10 cm = 0.83 Z D,30 cm .

Fig. 5 .
Fig. 5.All naked eye and colour filter observations of Z D,10 cm,tel with a telescope as a function of Z D,10 cm without a telescope.The best-fit line through the origin is the function Z D,10 cm,tel = 1.19ZD,10 cm .

Fig. 6 .
Fig. 6.All naked eye and colour filter observations of Z D , on the shaded versus the sunlit side of the ship.The best-fit line through the origin is the function Z D,shade = 0.93Z D,sun .

Table 2 .
(Mikaelsen and Aas, 1990)lue and half-peak bandwidth λ of the eye's photopic sensitivity, alone and in combination with the upward spectral luminance L u ; for the open eye and with filters for the white disk(Mikaelsen and Aas, 1990).

Table 3 .
Mean value and standard deviation of the different quantities and terms of Eqs.(19)-(20) in the Oslofjord and Skagerrak, for the 30 cm white and black disks.Colour filters are for the white disk.Recordings of radiance and irradiance at 32 stations have been applied.
observed the Secchi depth in the Inner Oslofjord during 1986-1987.Based on their data from 11 stations the mean values of the ratio Z D,filter /Z D,white become 0.61, 0.81 and 0.62 for the blue, green and red filter, www.ocean-sci.net/10/177/2014/Ocean Sci., 10, 177-199, 2014

Table 4 .
Linear relationships of the forms y = A + Bx and y = B 0 x obtained by correlation analysis of Z D (white disk) observed by the open eye and with blue, green and red glass filters, and Z D,black (disk) and Z B,black