Propagation of sensor noise in oceanic hyperspectral remote sensing

David B. Gillis; Jeffrey H. Bowles; Marcos J. Montes; Wesley J. Moses

doi:10.1364/OE.26.00A818

1. Introduction

The use of hyperspectral remote sensing to determine the concentrations of biophysical parameters such as chlorophyll and dissolved organic materials in the world’s oceans is an important and challenging problem, and a variety of methods for doing so have been introduced in the literature [1–4]. The basic premise underlying most of these algorithms is to posit the existence of a forward model that can calculate a remote-sensing reflectance spectrum $R_{r s} (λ)$ from the concentrations of the various constituents; the algorithms are then designed to invert this model and derive the concentrations from a given spectrum.

It is usually assumed that the inputs into the inversion process – that is, the measured spectra – are essentially ‘correct’ in the sense that they represent the true $R_{r s} (λ)$ of the water under study. Unfortunately, this assumption is in general very difficult to verify. The process of converting a raw hyperspectral measurement - digital numbers from the sensor - to a $R_{r s} (λ)$ spectrum is non-trivial, and susceptible to a number of potential errors, including, among others, sensor calibration issues, imperfect knowledge of the atmosphere, and the intrinsic statistical variability that arises in measuring at-sensor radiance. In addition, the vast majority of remote sensing images have little to no in situ data that can be used to verify that the final calculated reflectance spectrum is correct.

Unfortunately, any errors (regardless of source) that arise in measuring the reflectance spectrum are propagated through to the retrieved parameters. This is mainly due to the fact that the spectrum is a complicated, highly nonlinear function of the individual optically active components [5,6]. As a result, relatively small changes in the concentrations can lead to relatively large percentage changes in the reflectance spectrum; reasoning backwards, then, it is reasonable to believe that uncertainties in the spectrum will lead to commensurate uncertainties in the estimated concentrations. This observation has been verified in a number of studies [7–16].

The usual method for dealing with these kinds of issues is to perform an error analysis [17] on the entire system. In particular, one attempts to estimate the uncertainty in the measurement, and then propagate this uncertainty through the inversion process and thereby put bounds on the variation expected to be seen in the retrieved parameters. For hyperspectral data, the three main sources of uncertainty are in atmospheric correction, sensor calibration (especially for satellite systems that undergo changes over time), and sensor noise. Of the three, sensor noise is the most tractable [18–21] and open to analysis. In [22], the authors of this work developed a sensor noise model and showed via numerical simulations that sensor noise alone could cause errors in the estimated constituent concentrations to be as high as 80%; in [23], they showed that these errors could be reduced by incorporating knowledge of the sensor noise into the retrieval process by using the Mahalanobis distance. In both cases, the authors relied on Monte Carlo simulation in order to estimate the statistical parameters (mean and variance) needed to describe the uncertainty in the data.

The aim of this paper is to demonstrate an approach that removes the need for Monte Carlo simulation, and, in particular, to derive closed-form formulas that can be used to calculate the expected variability of remote-sensing reflectance spectra for a given sensor and the conditions under which the data was collected. As part of this derivation, we also show that, in addition to the sensor noise characteristics, the amount of uncertainty in the data can be strongly affected by the conditions (atmospheric parameters and view geometry) under which the data was collected. We also note that, although this paper focuses on the impact of sensor noise on water column retrievals, the basic approach is relevant to any use of reflectance value(s) in a retrieval, such as vegetation indices [24].

2. Theory and modeling

2.1 Uncertainty in radiance

In this section we present formulas for describing how measurement uncertainty at the sensor level propagates to derived remote sensing reflectance signatures. To be more precise, we assume that a given body of water has an idealized ‘noise free’ (or constant) remote sensing reflectance that we wish to determine using hyperspectral measurements. What is actually measured at the sensor, of course, is the incoming upwelling radiance (which includes light upwelling from the water body as well as light reflected by the water surface and light scattered by the atmosphere) that is reported as ‘digital numbers’ (or counts). The measured signal must first be converted from counts to radiance units (using a calibration procedure), and then to the desired remote sensing reflectance (via atmospheric correction).

With that in mind, we let $R_{r s}^{*} (λ)$ represent our idealized remote sensing reflectance. This reflectance is propagated through the atmosphere to a (total) at-sensor radiance spectrum which we denote $L_{t}^{*} (λ)$ . Note that the value of $L_{t}^{*}$ will depend on not only the given $R_{r s}^{*}$ but also the atmospheric conditions, the illumination and viewing geometry, and the altitude of the sensor. Since our focus in this work is on the variation introduced by the sensor, we also treat the parameter $L_{t}^{*}$ as an idealized (constant) value that represents the ‘true’ radiance.

In very general terms, a hyperspectral sensor measures this light by counting incoming photons and converting them to electrons. To be more precise, incoming light is gathered by the sensor fore optics and focused onto a slit. Light that passes through the slit is dispersed, often by a grating, and then focused onto the focal plane. The focal plane converts impinging photons to electrons with a particular probability known as the quantum efficiency (which is wavelength dependent). The discrete nature of photons dictates that the number of photons incident on the focal plane (and hence the number of electrons generated) for a given unit of time will vary according to a Poisson distribution. In statistical terms, if we let $S_{e}$ be the number of electrons generated, then $S_{e}$ is a random variable whose variance $σ^{2} (S_{e})$ is exactly equal to the mean $μ (S_{e})$ . The standard variation of this variable $σ (S_{e}) = \sqrt{μ (S_{e})}$ is known as the ‘shot noise’ of the sensor, and in many hyperspectral sensors is the dominant source of measurement variation (the sensor is ‘shot noise limited’).

It can be shown [22] that the average number of electrons generated by a given radiance $L_{t}^{*}$ at wavelength $λ$ can be calculated as

μ (S_{e}) = g \cdot L_{t}^{*} = \frac{λ}{h c} \cdot \frac{π}{4} \cdot \frac{D^{2}}{f^{2}} \cdot p^{2} \cdot T \cdot η_{s y s} \cdot L_{t}^{*} .

Here

D, f, p, T

and

η_{s y s}

are the various sensor parameters (defined in Table 1), h is Planck’s constant, and c is the speed of light. The main thing to note here is that g is dependent only on the sensor parameters and wavelength; in particular, it is independent of the incoming source. In mathematical terms, the function g can be considered as simply a scaling factor that converts radiance into electrons.

Table 1. Sensor parameters.

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The total number of electrons $N_{e}$ will also include additional electrons generated by the sensor (e.g. dark noise, read noise, etc.) that we denote by $ν_{e}$ . If we let $γ = 1 / g$ , then we can convert the total signal in electrons back to radiance:

L = γ \cdot N_{e} = γ \cdot S_{e} + γ \cdot ν_{e}

Here

L

represents the measured radiance, which is also a random variable. The term

η_{R} \equiv γ \cdot ν_{e}

represents the non-shot noise of the sensor, in radiance units. Note that, in general, the mean value of

η_{R}

will not be zero. To get around this, a series of ‘dark’ (no incoming light) images can be taken and averaged together. By subtracting off this average dark level, we may assume without a loss of generality that the dark noise is zero-mean

μ (η_{R}) = 0

with some (measured) variance

σ^{2} (η_{R})

.

Equation (2) allows us to easily calculate the mean and variance of the measured radiance:

\begin{matrix} μ (L) = γ \cdot μ (S_{e}) + μ (η_{R}) = L_{t}^{*} \\ σ^{2} (L) = γ^{2} \cdot σ^{2} (S_{e}) + σ^{2} (η_{R}) \\ = γ \cdot γ \cdot μ (S_{e}) + σ^{2} (η_{R}) \\ = γ \cdot L_{t}^{*} + σ^{2} (η_{R}) . \end{matrix}

Note that the sensor signal-to-noise ratio (SNR) – defined as the mean of the signal divided by the noise (which equals the standard deviation) – is the same in both electrons and radiance (as expected):

S N R (electrons) = \frac{μ (S_{e})}{σ (S_{e})} = \frac{S_{e}}{\sqrt{S_{e}}} = \frac{γ \cdot S_{e}}{\sqrt{γ^{2} \cdot S_{e}}} = \frac{L_{t}^{*}}{\sqrt{γ \cdot L_{t}^{*}}} = \frac{μ (L)}{σ (L)} = S N R (radiance)

We have assumed here that the read noise is negligible. In particular, it is clear that the sensor SNR depends not only on the sensor parameters (the

γ

term) but also on the amount of incoming light; to put this another way, the same remote sensing reflectance can generate very different SNR values when the environmental conditions have changed. As an example, Fig. 1 shows estimated SNR curves for a given

R_{r s}

measured by a spaceborne sensor (at an altitude of 600 km) under different atmospheric conditions.

Fig. 1 Each curve represents the estimated spaceborne sensor signal-to-noise ratio (SNR) for a fixed remote sensing reflectance under various atmospheric conditions. In the legend,τ is the aerosol optical thickness at 550 nm and the text term indicates the aerosol model. In all cases the relative humidity was 50% and the viewing geometry ID (defined in Table 4) was 4.

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2.2 Uncertainty in remote sensing reflectance

Once the signal measured by the sensor has been converted to radiance, the next step is to convert the radiance to remote sensing reflectance via atmospheric correction. In mathematical terms, we can think of this step as a function $F$ that converts radiance to reflectance, that is $R_{r s} = F (L)$ . Our aim in this section is to describe how the uncertainty in the measured $L$ , as described in the previous sections, is propagated to uncertainty in the derived $R_{r s}$ . To be more precise, we think of atmospheric correction as transforming the random variable $L$ (the measured radiance) to a new random variable $R_{r s}$ ; our goal is to derive the mean and standard deviation of the transformed variable, given that we know the mean and standard variation of the radiance (Eq. (3)).

The answer to this question will in general depend on which type of correction algorithm is applied. Relatively simple algorithms such as ELM [25] and QUAC [26] are linear – that is, $F (L) = a \cdot L + b$ for some constants $a, b$ . In these cases, it is easy to show [27] that

\begin{array}{l} μ (R_{r s}) = a \cdot μ (L) + b \\ σ^{2} (R_{r s}) = a^{2} \cdot σ^{2} (L) . \end{array}

Unfortunately, if the function $F (L)$ is non-linear, which is the case in radiative transfer-based correction algorithms, then there is in general no exact way of calculating either the mean or standard variation of the transformed variable. The usual way of handling this is to expand the function $F (L)$ in a first-order Taylor approximation about the mean $μ (L) = L_{t}^{*}$ (sometimes known as the delta method in statistics [28])

R_{r s} = F (L) \approx F (L_{t}^{*}) + F^{'} (L_{t}^{*}) \cdot (L - L_{t}^{*}) = F^{'} (L_{t}^{*}) \cdot L + F (L_{t}^{*}) - F^{'} (L_{t}^{*}) \cdot L_{t}^{*} .

Here

F^{'} (L_{t}^{*})

is the derivative of

F (L)

evaluated at the point

L_{t}^{*}

. Note that Eq. (5) is linear in

L

and therefore we can easily calculate the mean of the measured remote sensing reflectance as:

\begin{matrix} μ (R_{r s}) \approx F^{'} (L_{t}^{*}) \cdot μ (L) + F (L_{t}^{*}) - F^{'} (L_{t}^{*}) \cdot L_{t}^{*} \\ = F^{'} (L_{t}^{*}) \cdot L_{t}^{*} + F (L_{t}^{*}) - F^{'} (L_{t}^{*}) \cdot L_{t}^{*} \\ = R_{r s}^{*} . \end{matrix}

Here the last equality follows from the fact that

L_{t}^{*}

is the ‘forward propagation’ of our idealized (noise-free) reflectance

R_{r s}^{*}

. Note that Eq. (6) is just the intuitively expected result that the average of our derived reflectances is equal to the ‘true’ value

R_{r s}^{*}

(at least to first order).

Similarly, we can calculate the variance as:

\begin{matrix} σ^{2} (R_{r s}) \approx [^{F^{'} (L_{t}^{*})] 2} \cdot σ^{2} (L) \\ = [^{F^{'} (L_{t}^{*})] 2} \cdot (γ \cdot L_{t}^{*} + σ^{2} (η_{R})) . \end{matrix}

Note that Eqs. (6) and (7) provide us with a complete mathematical answer to the question of how the sensor noise affects the derived $R_{r s} (λ)$ spectra, and provides a formula for estimating the variance which can then be used in statistical measures such as the Mahalanobis distance. We also emphasize that Eq. (7) provides a ‘best-case’ scenario in the sense that only sensor noise is included; in particular, we assume perfect knowledge of the atmosphere. In the real world, uncertainty in the choice of atmospheric parameters will lead to greater uncertainty in the remote sensing reflectance.

It is worth noting here the physical meaning of the terms in Eq. (7). The $γ$ and $σ^{2} (η_{R})$ terms are characteristics of the sensor system (see Table 1), and are independent of the data being measured. Note that, in a shot noise limited system, the second term can be ignored, and the variance in reflectance is seen to scale linearly with $γ$ . Intuitively, $γ$ can be thought of as a measure of the amount of radiance needed to generate a single electron; an increase in the overall system efficiency will lead to a decrease in $γ$ and thus a decrease in the uncertainty in the reflectance.

Conversely, the two remaining terms, $L_{t}^{*}$ and $F^{'} (L_{t}^{*})$ , are independent of the sensor, and are controlled by the environmental parameters (including the reflectance of the water body, the altitude of the sensor, and various atmospheric terms) at the time of the measurement.

It is a little difficult so see what the derivative term actually means without actually calculating the value. As a concrete example, we show in the Appendix the results derived when using the Tafkaa atmospheric correction algorithm [29]. After a bit of algebra, we end up with the following result:

F^{'} (L_{t}^{*}) = \frac{{(R_{r s}^{*} \cdot \bar{s} \cdot π - 1)}^{2}}{μ_{0} E_{0} \cdot T_{g} \cdot t_{u} \cdot t_{d}}

here,

\bar{s}, T_{g}, t_{u}, t_{d}

are all atmospheric transmission terms, and

μ_{0} E_{0}

is the incident flux at the top of the atmosphere (see Table 2 for definitions). It is worth emphasizing here that Eq. (8) implies that the variation seen in the derived remote sensing reflectance can be significantly impacted by the atmospheric conditions in effect when the data was collected. We will examine this effect in the next section.

Table 2. Tafkaa parameters.

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3. Experimental results and discussion

In this section, we apply the formulas from the previous section to a variety of simulated data modeled with a wide range of atmospheric and viewing conditions. We use this data to verify the mathematics of the previous sections, and also to provide evidence of how the error in derived remote sensing reflectance depends on both the noise characteristics of the sensor, as well as the environmental conditionals in which the data was collected.

3.1 Experimental data

To execute the study, we began by generating 5 remote sensing reflectance spectra (Fig. 2, Table 3) using the ‘Case 2’ four-component model of the Hydrolight radiative transfer program [30]. We then chose 432 atmospheric / viewing geometry combinations (Table 4) to model, for a total of 2160 test cases. Two different sensor models (with differing SNR levels) were used: the first is a space-borne sensor modeled for a low-earth orbit at an altitude of 600 km, using the characteristics of NRL’s HICO sensor [31]; the second is an airborne sensor modeled at two altitudes (1.5 and 5 km) that uses the characteristics of the PIXIS sensor [32].

Fig. 2 The five remote sensing reflectance spectra used in the study. Each spectrum was modeled using the ‘Case 2’ four-component model in the Hydrolight software package. Parameters for each spectrum are given in Table 3.

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Table 3. Component concentrations for the remote-sensing spectra.

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Table 4. Atmospheric and sensor view parameters.

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For each test case, the surface $R_{r s}^{*} (λ)$ spectrum was propagated through the given atmosphere via Tafkaa to an at-sensor radiance spectrum $L_{t}^{*} (λ)$ . We then created 10,000 ‘noisy’ radiance spectra by repeatedly sampling from a Gaussian distribution with parameters given by Eq. (3). In order to apply the equation, we used Eq. (1) together with the parameters in Table 1 to calculate the ‘gain’ term $γ (λ)$ . The dark noise term $σ^{2} (η_{R})$ was derived from existing (real-world) HICO dark data (both sensors used the same dark noise values). Finally, the noisy radiance spectra were atmospherically corrected via Tafkaa, using the same atmospheric parameters of the forward model. We note that, over all cases and all wavelengths, the sensor noise averaged about 1% of the total signal, with a maximum of about 6%. An example of the relative noise for a particular radiance case is shown in Fig. 3 below.

Fig. 3 An example of an at-sensor radiance spectra (spaceborne sensor) and associated sensor noise. To help with visualization, the error bars in this example represent ten standard deviations of the noise.

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3.2 Validation of the variance formula

We begin this section by verifying that the formula for estimating the variance in remote sensing reflectance (Eq. (7)) presented in the previous section is correct and remains valid over all test cases.

As in the previous section, we treat atmospheric correction as a function $F$ that maps radiance values to reflectance values. Our main challenge is to verify that the linear approximation (Eq. (5) and hence Eq. (7)) remains valid over the areas of interest. To be more precise, in purely mathematical terms (that is, without any physical constraints), the graph of $R_{r s} = F (L)$ for each wavelength defines a hyperbola whose vertical and horizontal asymptotes are defined by the atmospheric parameters (Fig. 4). In reality, of course, radiance values are constrained and cannot run off to infinity. We can bound the radiance by switching to reflectance units; in particular, if we let $μ_{0} E_{0}$ be the incident flux at the top of the atmosphere, then $ρ_{o b s}^{*} = π L / (μ_{0} E_{0})$ is the at-sensor apparent reflectance that we assume lies between 0 and 1. It follows that the radiance is bounded above by $μ_{0} E_{0} / π$ . As long as the function $F$ is (at least approximately) linear in this regime (the red box in Fig. 4), then Eq. (7) should be a valid approximation. In Fig. 5, we plot the function $F$ for different wavelengths in the bounded regime; it is clear from the graph that the linearity assumption is valid in this case.

Fig. 4 (a). The Tafkaa correction algorithm for a particular test case at $λ = 525 n m .$ The red box indicates physically realizable at-sensor radiance values.

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Fig. 5 (b). Graph of Tafkaa atmospheric correction for various wavelengths in the red box area of the previous figure.

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To validate this assumption, we calculated for each test case the sample variance of the 10,000 noisy reflectance spectra and compared this with the estimated variance as calculated in Eq. (7). In each test case, the estimated and empirical variances were essentially identical; a typical case is shown in Fig. 6 below. Although it is difficult to easily summarize all 2160 cases, we show in Fig. 7 the average relative error over all cases, where the relative error is defined

relative error = 100 \times \frac{| emperical - estimated |}{emperical} .

We note that in 99% of the cases the relative error was less than 5%. The maximum over all test cases and all wavelengths was 8%.

Fig. 6 Empirical (sample) variance (blue) from 10,000 simulations compared with the variance estimated by Eq. (7) (red).

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Fig. 7 The average relative error in predicted variance.

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3.3 Effect of environmental conditions on errors in remote sensing reflectance

The formulas presented in Section 2 are helpful in that they allow an analyst to predict the amount of uncertainty that can be expected in a given measurement. There is also an interesting physical interpretation; intuitively, one would expect the amount of noise in a given measurement to be dependent on the characteristics of a given sensor (e.g. the $γ$ term in Eq. (7)). What may not be immediately obvious is that the collection parameters (atmospheric conditions, solar and view angles, etc.) can also affect strongly the accuracy of a given measurement, due to the derivate term $F^{'} {(L_{t}^{*})}^{2}$ in Eq. (7). To put this another way, the same sensor, flown on different days and times, can exhibit very different noise values, even when measuring the same scene. This is particularly true for space-borne systems that must look through the entire atmospheric column.

Examples of this behavior are shown in Figs. 8, 9, 10, and 11 below. In each figure, the solid line represents the true reflectance and the error bars represent plus and minus one standard deviation of the noisy samples. Figure 8 represents a ‘typical’ case for the spaceborne sensor flown at 600 km; by typical we mean that most of the test cases (~90%) exhibited variation of the same general magnitude. In Fig. 9, we show a ‘worst case’ version for the same sensor and same input reflectance; the only change being the atmospheric and viewing parameters. Similar results for the airborne sensor are shown in Figs. 10 and 11. We note for completeness that, in both examples, the atmospheric parameters for the ‘worst case’ are $τ = 1.5$ , relative humidity = 98%, the aerosol model is urban, and the view ID (Table 4) is 2.

Fig. 8 Variance in derived $R_{r s}$ spectra (spaceborne sensor at 600 km) (typical case). The error bars represent one standard deviation of the data.

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Fig. 9 Variance in derived $R_{r s}$ spectra (spaceborne sensor at 600 km) (worst case). The error bars represent one standard deviation of the data.

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Fig. 10 Variance in derived $R_{r s}$ spectra (airborne sensor at 5 km) (typical case). The error bars represent one standard deviation of the data.

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Fig. 11 Variance in derived $R_{r s}$ spectra (airborne sensor at 5 km) (worst case). The error bars represent one standard deviation of the data.

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4. Summary

The impact of sensor noise described in this paper is unavoidable when measuring real-world data, since sensor noise represents the minimum noise level that is inevitably present in the data. Measurement error will be propagated through the atmospheric correction process, even in the (unlikely) case of perfect knowledge of the atmosphere; several studies have shown that the induced uncertainty in remote sensing reflectance can have a strong effect on the retrieval of the associated environmental parameters.

In this work, we have derived explicit formulas for estimating the uncertainty in $R_{r s} (λ)$ spectra that are due to sensor noise. The formulas show that the magnitude of the error is a function of both the noise characteristics of the sensor and the atmospheric and environmental conditions under which the data is collected. These formulas were tested and validated over a wide range of environmental conditions, and we showed by example how changing the collection conditions can lead to a large change in the error.

Appendix derivation of the formula using Tafkaa

To keep the notation as simple as possible, we begin by noting that a general rational function can be put into a standard form:

f (x) = \frac{A x + B}{C x + D} = \frac{\frac{B C - A D}{C^{2}}}{x + \frac{D}{C}} + \frac{A}{C} = \frac{a}{x - h} + k

It is then easy to check that

\begin{matrix} f^{'} (x) = - \frac{a}{{(x - h)}^{2}} \\ g (x) = f^{- 1} (x) = \frac{a}{x - k} + h \\ f^{'} (g (x)) = - \frac{{(x - k)}^{2}}{a} \end{matrix}

Note that, in the context of atmospheric correction,

g (x)

is the ‘forward model’ that takes

R_{r s}

to

L

.

Starting from Eq. (10) in the Tafkaa users guide [33], converting $ρ_{o b s}^{*}$ back to radiance, and rearranging, we find

R_{r s} = f (L_{t}) = \frac{L_{t} - \frac{μ_{0} E_{0}}{π} \cdot T_{g} \cdot ρ_{a t m}^{*}}{\bar{s} \cdot π \cdot L_{t} + μ_{0} E_{0} T_{g} \cdot (t_{u} t_{d} - \bar{s} \cdot ρ_{a t m}^{*})}

where the atmospheric parameters are defined as in Table 2. Putting this equation in standard form, we have

\begin{matrix} a = {(\frac{1}{\bar{s} \cdot π})}^{2} \cdot (- \frac{μ_{0} E_{0}}{π} T_{g} ρ_{a t m}^{*} \cdot \bar{s} \cdot π - μ_{0} E_{0} T_{g} \cdot (t_{u} t_{d} - \bar{s} \cdot ρ_{a t m}^{*})) \\ = - \frac{1}{π} \cdot \frac{1}{{\bar{s}}^{2}} \cdot \frac{μ_{0} E_{0}}{π} T_{g} t_{u} t_{d} \\ h = - \frac{1}{\bar{s}} \cdot \frac{μ_{0} E_{0}}{π} \cdot T_{g} \cdot (t_{u} t_{d} - \bar{s} \cdot ρ_{a t m}^{*}) \\ k = \frac{1}{\bar{s} \cdot π} . \end{matrix}

It follows that

\begin{matrix} f^{'} (g (x)) = - \frac{{(x - \frac{1}{\bar{s} \cdot π})}^{2}}{- \frac{1}{π} \cdot \frac{1}{{\bar{s}}^{2}} \cdot \frac{μ_{0} E_{0}}{π} T_{g} t_{u} t_{d}} \\ = \frac{{(x \cdot \bar{s} \cdot π - 1)}^{2}}{μ_{0} E_{0} T_{g} t_{u} t_{d}} \end{matrix}

and, therefore,

f^{'} (L_{t}^{*}) = f^{'} (g (R_{r s}^{*})) = \frac{{(R_{r s}^{*} \cdot \bar{s} \cdot π - 1)}^{2}}{μ_{0} E_{0} T_{g} t_{u} t_{d}}

References and links

1. Z. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters. 2. Deriving bottom depths and water properties by optimization,” Appl. Opt. 38(18), 3831–3843 (1999). [CrossRef] [PubMed]

2. W. J. Klonowski, P. R. Fearns, and M. J. Lynch, “Retrieving key benthic cover types and bathymetry from hyperspectral imagery,” J. Appl. Remote Sens. 1(1), 011505 (2007). [CrossRef]

3. V. E. Brando, J. M. Anstee, M. Wettle, A. G. Dekker, S. R. Phinn, and C. Roelfsema, “A physics based retrieval and quality assessment of bathymetry from suboptimal hyperspectral data,” Remote Sens. Environ. 113(4), 755–770 (2009). [CrossRef]

4. C. D. Mobley, L. K. Sundman, C. O. Davis, J. H. Bowles, T. V. Downes, R. A. Leathers, M. J. Montes, W. P. Bissett, D. D. R. Kohler, R. P. Reid, E. M. Louchard, and A. Gleason, “Interpretation of hyperspectral remote-sensing imagery by spectrum matching and look-up tables,” Appl. Opt. 44(17), 3576–3592 (2005). [CrossRef] [PubMed]

5. International Ocean-Colour Coordinating Group, “Remote sensing of ocean colour in coastal, and other optically-complex, waters,” in Reports of the International Ocean-Colour Coordinating Group, No.3, S. Sathyendranath, ed. (IOCCG, 2000).

6. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

7. R. J. W. Brewin, S. Ciavatta, S. Sathyendranath, T. Jackson, G. Tilstone, K. Curran, R. L. Airs, D. Cummings, V. Brotas, E. Organelli, G. Dall’Olmo, and D. E. Raitsos, “Uncertainty in ocean-color estimates of chlorophyll for phytoplankton groups,” Front. Mater. Sci. 4, 104 (2017).

8. Z. Lee, R. Arnone, C. Hu, P. J. Werdell, and B. Lubac, “Uncertainties of optical parameters and their propagations in an analytical ocean color inversion algorithm,” Appl. Opt. 49(3), 369–381 (2010). [CrossRef] [PubMed]

9. T. S. Moore, J. W. Campbell, and M. D. Dowell, “A class-based approach to characterizing and mapping the uncertainty of the MODIS ocean chlorophyll product,” Remote Sens. Environ. 113(11), 2424–2430 (2009). [CrossRef]

10. C. Hu, K. L. Carder, and F. E. Muller-Karger, “How precise are SeaWiFS ocean color estimates? Implications of digitization-noise errors,” Remote Sens. Environ. 76(2), 239–249 (2001). [CrossRef]

11. D. Antoine, F. d’Ortenzio, S. B. Hooker, G. Bécu, B. Gentili, D. Tailliez, and A. J. Scott, “Assessment of uncertainty in the ocean reflectance determined by three satellite ocean color sensors (MERIS, SeaWiFS and MODIS‐A) at an offshore site in the Mediterranean Sea (BOUSSOLE project),” J. Geophys. Res. Oceans 113(C7), 4472 (2008).

12. J. Wei, Z. Lee, and S. Shang, “A system to measure the data quality of spectral remote-sensing reflectance of aquatic environments,” J. Geophys. Res. Oceans 121, 8189–8207 (2016).

13. F. E. Hoge and P. E. Lyon, “Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: an analysis of model and radiance measurement errors,” J. Geophys. Res. Oceans 101(C7), 16631–16648 (1996). [CrossRef]

14. J. Beekhuizen, G. B. M. Heuvelink, I. Reusen, and J. Biesemans, “Uncertainty Propagation Analysis of the airborne hyperspectral data processing chain,” In Proceedings of First Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (IEEE, 2009) pp. 1–4. [CrossRef]

15. R. W. Gould Jr, S. C. McCarthy, E. Coelho, I. Shulman, and J. G. Richman, “Combining satellite ocean color and hydrodynamic model uncertainties in bio-optical forecasts,” J. Appl. Remote Sens. 8(1), 083652 (2014). [CrossRef]

16. M. S. Salama and A. Stein, “Error decomposition and estimation of inherent optical properties,” Appl. Opt. 48(26), 4947–4962 (2009). [CrossRef] [PubMed]

17. J. Taylor, Introduction to error analysis, the study of uncertainties in physical measurements (University Science Books, 1997).

18. I. Levin, E. Levina, G. Gilbert, and S. Stewart, “Role of sensor noise in hyperspectral remote sensing of natural waters: application to retrieval of phytoplankton pigment,” Remote Sens. Environ. 95(2), 264–271 (2005). [CrossRef]

19. I. M. Levin and E. Levina, “Effect of atmospheric interference and sensor noise in retrieval of optically active materials in the ocean by hyperspectral remote sensing,” Appl. Opt. 46(28), 6896–6906 (2007). [CrossRef] [PubMed]

20. D. Liu, Y. Zhang, B. Zhang, K. Song, Z. Wang, H. Duan, and F. Li, “Effect of sensor noise in spectral measurements on chlorophyll-a retrieval in Nanhu Lake of Changchun, China,” J. Electromagnet Wave. 20(4), 547–557 (2006). [CrossRef]

21. D. Jorge, C. C. F. Barbosa, L. A. S. De Carvalho, and E. M. L. De Moraes Novo, “Snr (signal-to-noise ratio) impact on water constituent retrieval from simulated images of optically complex amazon lakes,” Remote Sens. 9(7), 644 (2017). [CrossRef]

22. W. J. Moses, J. H. Bowles, R. L. Lucke, and M. R. Corson, “Impact of signal-to-noise ratio in a hyperspectral sensor on the accuracy of biophysical parameter estimation in case II waters,” Opt. Express 20(4), 4309–4330 (2012). [CrossRef] [PubMed]

23. D. B. Gillis, J. H. Bowles, and W. J. Moses, “Improving the retrieval of water inherent optical properties in noisy hyperspectral data through statistical modeling,” Opt. Express 21(18), 21306–21316 (2013). [CrossRef] [PubMed]

24. J. Verrelst, G. Camps-Valls, J. Muñoz-Marí, J. P. Rivera, F. Veroustraete, J. G. Clevers, and J. Moreno, “Optical remote sensing and the retrieval of terrestrial vegetation bio-geophysical properties–A review,” ISPRS J. Photogramm. Remote Sens. 108, 273–290 (2015). [CrossRef]

25. F. A. Kruse, K. S. Kierein-Young, and J. W. Boardman, “Mineral mapping at Cuprite, Nevada with a 63-channel imaging spectrometer,” Photogramm. Eng. Remote Sensing 56(1), 83–92 (1990).

26. L. S. Bernstein, X. Jin, B. Gregor, and S. M. Adler-Golden, “Quick atmospheric correction code: algorithm description and recent upgrades,” Opt. Eng. 51(11), 111719 (2012). [CrossRef]

27. K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis (Academic, 2003).

28. J. M. Ver Hoef, “Who invented the delta method?” Am. Stat. 66(2), 124–127 (2012). [CrossRef]

29. B. C. Gao, M. J. Montes, Z. Ahmad, and C. O. Davis, “Atmospheric correction algorithm for hyperspectral remote sensing of ocean color from space,” Appl. Opt. 39(6), 887–896 (2000). [CrossRef] [PubMed]

30. http://www.sequoiasci.com/product/hydrolight/ (accessed 17 Apr 2018)

31. R. L. Lucke, M. Corson, N. R. McGlothlin, S. D. Butcher, D. L. Wood, D. R. Korwan, R. R. Li, W. A. Snyder, C. O. Davis, and D. T. Chen, “Hyperspectral Imager for the Coastal Ocean: instrument description and first images,” Appl. Opt. 50(11), 1501–1516 (2011). [CrossRef] [PubMed]

32. https://www.princetonistruments.com/products/PIXIS-CCD (accessed 17 Apr 2018)

33. M. J. Montes, B.-C. Gao, and C. O. Davis, “NRL Atmospheric Correction Algorithms for Oceans: Tafkaa User’s Guide,” NRL Report NRL/MR/7230–04–8760 (2004). http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA422068 (accessed on 17 April 2018).

Parameter	spaceborne	airborne	Description
$D$	0.019	0.018	Diameter of aperture (m)
$f$	0.067	0.018	Focal length (m)
$p$	16	24	Pixel width (microns)
$T$	0.02	0.33	Integration time (s)
$η_{s y s} (λ)$	Wavelength dependent		Overall system efficiency
Altitude	600	1.5, 5.0	Sensor altitude (km)

Symbol	Description
$L_{t}$	Total at-sensor radiance
$ρ_{a t m}^{*}$	Reflectance from atmospheric scattering, direct sunlight, and surface reflection
$T_{g}$	Total atmospheric gaseous transmittance
$t_{u}, t_{d}$	Upward and downward atmospheric transmittance
$μ_{0} E_{0}$	Incident flux at the top of the atmosphere
$\bar{s}$	Average reflectivity at the base of the atmosphere

Parameter	Values
Aerosol Optical Thickness (τ)	0.1, 0.3, 0.7, 1.5
Aerosol Model	Maritime, Tropospheric, Urban
Relative Humidity	50%, 80%, 98%
Sensor Altitude (km)	1.5, 5, 600
Viewing Geometry ID:	Sun Zenith Angle (deg)	View Zenith Angle (deg)	Relative View Azimuth Angle (deg)
1	39.05	0	0
2	56.43	0	0
3	0.9	40	135
4	0.9	55	135

Parameter	spaceborne	airborne	Description
$D$	0.019	0.018	Diameter of aperture (m)
$f$	0.067	0.018	Focal length (m)
$p$	16	24	Pixel width (microns)
$T$	0.02	0.33	Integration time (s)
$η_{s y s} (λ)$	Wavelength dependent		Overall system efficiency
Altitude	600	1.5, 5.0	Sensor altitude (km)

Symbol	Description
$L_{t}$	Total at-sensor radiance
$ρ_{a t m}^{*}$	Reflectance from atmospheric scattering, direct sunlight, and surface reflection
$T_{g}$	Total atmospheric gaseous transmittance
$t_{u}, t_{d}$	Upward and downward atmospheric transmittance
$μ_{0} E_{0}$	Incident flux at the top of the atmosphere
$\bar{s}$	Average reflectivity at the base of the atmosphere

Propagation of sensor noise in oceanic hyperspectral remote sensing

Abstract

1. Introduction

2. Theory and modeling

2.1 Uncertainty in radiance

2.2 Uncertainty in remote sensing reflectance

3. Experimental results and discussion

3.1 Experimental data

3.2 Validation of the variance formula

3.3 Effect of environmental conditions on errors in remote sensing reflectance

4. Summary

Appendix derivation of the formula using Tafkaa

References and links

Cited By

Figures (11)

Tables (4)

Equations (16)

Optics Express

Spectral ID	Chl (mg m⁻³)	a_CDOM(440) (m⁻¹)	Sediments (g m⁻³)
1	1	0.1	1
2	2	1	10
3	10	0.5	5
4	15	0.5	15
5	50	2	25

Spectral ID	Chl (mg m⁻³)	a_CDOM(440) (m⁻¹)	Sediments (g m⁻³)
1	1	0.1	1
2	2	1	10
3	10	0.5	5
4	15	0.5	15
5	50	2	25