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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Radiometric values on digital imagery are affected by several sources of uncertainty. A practical, comprehensive and flexible procedure to analyze the radiometric values and the uncertainty effects due to the camera sensor system is described in this paper. The procedure is performed on the grey level output signal using image raw units with digital numbers ranging from 0 to 2^{12}-1. The procedure is entirely based on statistical and experimental techniques. Design of Experiments (DoE) for Linear Models (LM) are derived to analyze the radiometric values and estimate the uncertainty. The presented linear model integrates all the individual sensor noise sources in one global component and characterizes the radiometric values and the uncertainty effects according to the influential factors such as the scene reflectance, wavelength range and time. The experiments are carried out under laboratory conditions to minimize the rest of uncertainty sources that might affect the radiometric values. It is confirmed the flexibility of the procedure to model and characterize the radiometric values, as well as to determine the behaviour of two phenomena when dealing with image sensors: the noise of a single image and the stability (trend and noise) of a sequence of images.

Photogrammetric applications require calibrated sensors, not only in geometry but in radiometry and color. Sensor evaluation in operational and laboratory conditions is essential to characterize the many factors affecting the radiometric and geometric properties and to find out the limitations of the systems. Geometric processing applications have reached a high maturity level, but the radiometric processing applications are still in its infancy [

The characterization of the radiometric values of a sensor is a preliminary stage of calibration of photogrammetric sensors [^{2}·sr·nm)). Relative calibration normalizes the output of the sensor so that a uniform response is obtained in the entire image area when the focal plane of the sensor is irradiated with a uniform radiance field.

The characterization concerns the knowledge of the factors and the quantification of the effects on radiometric values of a sensor [

The radiometric response is an observational process that encompasses different responses from different sources such as electromagnetic radiation, optical system, electronics and object scene. The uncertainty of radiometric values fundamentally limits the distinguishable content in an image and can significantly reduce the robustness of an image processing application. It is important to analyze and characterize the uncertainty effects of the radiometric values.

The spatial variability component is caused by the sensor system (spatial non-uniformities) [

In this paper we analyze the noise component due to the sensor system, the spatial non-uniformity effect of a single image, and the temporal non-uniformity and trend effects of a sequence of images (

A practical and comprehensive experimental and statistical procedure, the DoE for LM, is presented in the next sections to analyze and characterize the grey level values, in particular, the linearity, the noise and the temporal trend of the sensor. It is tested on a Foveon X3 CMOS sensor featured in a Sigma SD15 digital single lens reflex (SLR) camera.

The presented new approach to evaluate image sensors adds flexibility to define parameters and relationships in the mathematical model. Furthermore, its implementation is easy and practical, and can be carried out in many laboratories, which is essential for applied disciplines such as photogrammetry and computer vision.

The radiometric properties and their uncertainty effects on digital images are widely studied and well understood in the literature. There are many scientific articles evaluating these topics [

The spatial non-uniformities in CMOS and CCD sensor are different, being higher in CMOS. Also, the spatial non-uniformitiy is not distributed randomly across the image in CMOS sensors. In [

There exist standards for electronic noise characterization of image sensors [

ISO 15739:2003 Photography—Electronic still-picture imaging—Noise measurements.

European Machine Vision Association (EMVA) Standard 1288, Standard for Characterization of Image Sensors and Cameras.

Both standards agree on the definition and assignment of the noise components and are focused mainly on the noise and the spatial non-uniformity due to the sensor system (

Linear sensitivity (photo-response) of the sensor,

All noise components are stationary and white with respect to time and space. The parameters describing the noise component are invariant with respect to time and space.

Only the total quantum efficiency is wavelength dependent,

If these conditions are not fulfilled, the computed parameters are meaningless [

Thus, the temporal noise represents the different photo-response among pixels and also, equivalently, the different photo-response among different exposures, both under the assumption that the mean response is stable with respect to space and time, that is, the temporal noise is stationary in space and time. Consequently, the temporal noise does not consider the non-uniformity of the mean response.

PTM is based on the photo-response noise (temporal noise) with and without light to determine all the parameters characterizing completely the sensor radiometry. These parameters are: the overall system gain, the quantum efficiency, the saturation signal, the temporal dark noise, the absolute sensitivity threshold, the saturation capacity, the signal-to-noise ratio and the dynamic range.

The spatial non-uniformity is commonly called

If temporal non-uniformity is present in the behaviour of a sensor, that is, the mean response is not stationary with respect to time, then the temporal noise does not represent the different photo-response among pixels. Therefore, the computed parameters by the PTM are meaningless.

There also exist patterns or periodic spatial variations due to the optical system (shading and vignetting effects), manufacturing imperfections and electronic interferences, causing non-stationary and non-white signal in space. To detect and correct the periodic variations, the standards propose the computation of the spectrogram. Finally, the standards propose plotting the logarithmic and accumulated histograms to detect defect pixels (outliers).

First of all, the procedure developed in this paper is not intended to replace any standard. Basically, the approach deeps in the grey level output signal (instead of physical radiometric properties) that interests professionals and users of digital cameras. Furthermore, an exploratory point of view is presented to analyze the noise of the image sensor reporting total quantities of noise according to the experimental factors of practical imaging processes. The strength is the flexibility of the linear models formulation which allows the noise to be explained and reported according to factors such as reflectance, wavelength, time, space and others.

The only requirement that the sensor must satisfy for the application of the DoE for LM is to provide independent and identically distributed measurements. Therefore, each pixel measurement is an independent realization of the random variable (grey level).

We are interested in quantifying the noise of the grey values according to the reflectance, wavelength and time factors. Furthermore, the noise is analyzed considering just a single image and a sequence of images.

Following the standards described in Section 2.1, PTM can be used to characterize sensor radiometry, quantum efficiency, system gain, saturation signal,

It is worth noting that the combination of several images does not imply any arithmetical operation among images, it only implies to consider jointly the data of the images.

In this paper, the evaluation of the total quantities of noise present on both a single image and a sequence of images are presented, rather than the evaluation and decomposition of the total noise into its components: shot noise, dark noise, quantification noise, DSNU and PRNU; quantum efficiency, overall system gain, saturation signal, absolute sensitivity threshold, and other parameters characterizing the radiometry of a sensor can be computed following the standards.

The single image noise of our procedure (Section 8.2) includes the temporal noise and the fixed pattern noise described by the standards. Likewise, the image sequence noise (Section 8.3) includes the temporal noise and the fixed pattern noise described in the standards and, in addition, temporal non-uniformity random effects on the signal. It should be noted that the standard does not consider possible temporal non-uniformity on the signal, that is, the photo-response noise (temporal noise) is considered as stationary and white with respect to time, as mentioned in Section 2.1. If temporal non-uniformity random effects are present, then the fixed pattern noise and the temporal noise cannot be separated and, therefore the PTM cannot be applied [

The digital imaging process is a spatial sampling of the electromagnetic radiation from an object scene into the camera. For each sampling unit (_{i}, y_{j}_{i}, y_{i}

An electromagnetic radiation function

The electromagnetic radiation can be absorbed, transmitted, or reflected by an object. We consider the reflected energy case. If

The nature of

However,

The radiometric response of an image sensor element, at a certain spatial unit on the image (

In an image, the original continuous spatial distribution (

The radiometric response involves two processes: image sensing and quantization. The former refers to converting the electromagnetic energy incident at the sensor element to a proportional output electrical signal and the latter to transforming the output electrical signal generated by each sensor element into a digital code value. Therefore,

The developed procedure models and characterizes the radiometric response (grey level values) as well as the noise of a single image and the noise and trend of a temporal sequence of images due to the sensor system of a camera.

The procedure is entirely based on statistical and experimental techniques, in particular, Design of Experiments (DoE) for Linear Models (LM) [

The spatial non-uniformity due to the sensor system (

The uncertainty sources of scene changes in illumination or reflectance, geometric shooting conditions, integration bidirectional reflectance of the objects, time integration and additional imaging operators (

The sequential factor

The theoretical and experimental approach developed to analyze the radiometric values, the noise of a single image and the noise and trend of a temporal sequence of images consists of four steps. The procedure is depicted in

The first step is to characterize the radiometric values. The radiometric response

The second step is to characterize the sensor noise of a single image. A new linear model relating the adjusted residuals (resulting from the first step) to the influential factors is formulated (Section 6.2) and analyzed (Section 8.2), resulting in the quantification of the sensor noise of a single image according to the influential factors.

The third step models the adjusted residuals according to the influential factors except the sequential factor

The fourth step is to characterize the temporal trend effect due to the sensor nonstationary of the sensor in a sequence of images. The signal to temporal trend ratio (

A linear model (LM) relating the radiometric values (grey level values) to a linear combination of all factors that affect the radiometric response is formulated.

In this study, the light source

The spectral response factor

The reflectance factor

The reflectance factor

The sequential factor _{ijn}^{th}-observation of the radiometric value ^{th}-level of the factor ^{th}-level of the factor _{i}^{th}-level of the reflectance factor _{j}^{th}-level of the spectral response factor _{ijn}_{ijn}_{ijn}^{2}_{ij}_{i}·t_{j}·t_{ij}·t_{ijn}_{ijn}

The whole uncertainty sources due to the optical camera system, the changing external conditions, the changing geometric shooting conditions, the bidirectional reflectance of the objects, the time integration and the additional imaging operators (

The accurate characterization of the sensor noise of a single image is carried out with a new linear model relating the adjusted residuals resulting from the previous radiometric response model (Section 6.1) to the influential factors

Furthermore, the adjusted radiometric value _{in}^{th}-observation of the residual standard deviation ^{th}-level of the factor _{i}^{th}-level of the wavelength factor _{in}_{in}_{in}_{in}_{i}·f_{i}·t_{i}·f·t_{in}_{in}

The linear model relates the residual adjusted standard deviation

The adjusted radiometric value factor _{in}^{th}-observation of the residual standard deviation ^{th}-level of the factor _{i}^{th}-level of the wavelength factor _{in}_{in}_{i}·f_{in}_{in}

The temporal sensor trend effect can be quantified by means of the adjusted model coefficients resulting from the radiometric response model (Section 6.1). The

The time interval between images can be a relevant factor on the _{ijn}^{th}-observation of the ^{th}-level of the factor ^{th}-level of the factor _{i}^{th}-level of the spectral response factor _{j}^{th}-level of the time interval factor _{jn}_{ijn}_{i}·f_{j}·f_{ij}_{ij}·t_{ijn}_{ijn}

The performed DoE is based on a crossover design, therefore

As mentioned in Section 4.2, each pixel is a realization of the radiometric variable of a sensor element. If the sensor is trichromatic, there will be three radiometric values for the red, green and blue bands.

Furthermore, by imaging a colorcheker we can obtain the radiometric response for different reflectance values, since it is made of different regions with different reflectance characteristic,

The reflectance factor is a physical characteristic of the scene objects and is defined on the continuous open interval [0,1] (

The colorcheker was sequentially shot, resulting in

An example of the arranged data structure after observation is shown in ^{12}-1. The

The independency and randomness hypothesis of the observations are guaranteed because, as mentioned above, each pixel corresponds with a realization of the radiometric variable by a sensor element and all of them are considered equal and independent [

Four corresponding linear models are formulated for the radiometric values, the sensor noise of a single image, the sensor noise of a sequence of images and the sensor temporal trend of a sequence of images. The four linear models will be adjusted by least squares and analyzed by the resulting ANOVA table, the adjusted model coefficients and the interaction plots.

The model was adjusted using 60 (

The resulting ANOVA table is shown in

The validation plot of residual

The statistical significance tests in the ANOVA table are based on two basic hypotheses: the normality of the conditional distribution of the residuals and the homogeneity of the residual variance,

The slight deviation from the homoscedasticity hypothesis does not invalidate the significant tests (

The heterogeneity of the residual variance causes the slight non-normality of the distribution for the totality of the residuals,

The heterogeneity of the residual variance makes the standard error statistic and the mean absolute error statistic in

The accurate characterization of the sensor noise of a single image is carried out with a new linear model to explain the adjusted standard deviations

The resulting ANOVA table is shown in

Eliminating the factor

The interaction plot from the adjusted model,

The signal-noise ratio (

The linear model relates the residual adjusted standard deviation

The resulting ANOVA table is shown in

The interaction plots from the adjusted model,

The signal to noise ratio (

The

The resulting ANOVA table is shown in

The ultimate model relates the

The knowledge and the assessment of the behaviour of the grey level values registered by an image sensor as well as the uncertainty effects are essential keys to improve image processing applications, to finding out the limitations of the imaging systems and to calibrate imaging systems.

In this paper we performed the evaluation of an electro-optical sensor under laboratory conditions to characterize the grey level values on image sequences. The grey level value is modelled and characterized according to the influential factors in practical imaging processes. The grey level noise due to sensor system present in a single image and the grey level noise and temporal trend present in a sequence of images are analyzed. The noise is distributed randomly and the temporal trend is a systematic effect. The procedure is tested on a Foveon X3 image sensor featured in a Sigma SD15 digital SLR camera using raw units. The performance of the system can be carried out with standard laboratory equipment.

A practical and comprehensive characterization of the radiometric values (grey level values) is achieved under ideal laboratory conditions, that is, uniform lighting, controlled reflectance values and ideal shooting conditions. The characterization concerns the knowledge of the factors affecting the radiometric values under the specified laboratory conditions, namely the reflectance factor, the wavelength factor and the sequential factor. A linear model has been formulated to characterize the radiometric values achieving a goodness-of-fit of 99.97%, a standard error of 7.65 raw units and a mean absolute error of 5.45 raw units. The high goodness-of-fit percentage indicates that the linear model characterizes well the digital numbers under uniform external conditions, explaining the totality of the variability present in the radiometric response. The variability unexplained by the linear model, 0.03%, corresponds to the residual variability. Since all the factors affecting the performed experimental process have been modelled, this residual variability can be assigned to the noise of the sensor system. This residual variability is composed of both the noise of a single image and the temporal non-uniformity random effect due to the sequential factor. The standard error statistic is a measurement of the residual variability of the model and represents grey level noise due to the sensor system. But this statistic is a rough measure because the adjusted residuals are not homocedastic and it is also conditioned to the experimented sample of the reflectance factor

The third objective is to obtain an accurate quantification of the grey level noise of the sensor system present in a sequence of images according to the influential factors. The noise in a sequence of images is composed of the noise of a single image plus the temporal non-uniformity random effects. Therefore, the data has been joined by the sequential factor

Finally, the sequential factor of a sequence of images has a trend effect on the radiometric values. This effect is similar to a logarithmic trend until the stabilization of the signal. The proportion of signal temporally degraded until its stabilization (

The temporal trend effect can be corrected on images since it is a systematic effect. However, the random effects, like the noise, cannot be corrected; they can only be taken into account in image processing applications.

The noise quantities computed in this paper are complementary to the standards ISO and EMVA. Whether the sensor does not exhibit temporal non-uniformity random effect, the PTM can be applied to compute the totality of the parameters characterizing the radiometry of a sensor. Furthermore, if the temporal non-uniformity random effect is present in an imaging process, the noise components and quantities analyzed in this paper are completely useful. The noise quantities are computed and reported according to the influential factor and continuously across all theirs ranges.

The presented approach is very flexible. It can be used to define different relations between the grey level values and the influential factors, and to model non-stationary behaviour regarding time and space, for example temporal trend, spatial trend (vignetting, shading and non-uniform illumination effects), in contrast to the standards. Also, the procedure is useful to model and analyze the effects of other factors such as the time integration, geometrical shooting conditions and so on. The unique condition that has to be satisfied in the application of our procedure is that each pixel provides independent and identically distributed measurements of the grey level value.

Future research will deal with the extension of the presented approach to model spatial variability effects such as vignetting and non-uniform illumination, as well as wavelength relationships.

The authors would like to thank the support provided by the Spanish Ministry of Science and Innovation to the project HAR2010-18620. The authors also acknowledge the chance to take pictures with the lighting equipment in the Soils Laboratory at the Universitat Politècnica de València.

(

Goodness-of-fit graphic of the lineal model. Observed

Residuals

(

Descriptive statistics and plots for the totality of the residuals. (

Descriptive statistics (

Interaction plot. Standard deviation

Signal to noise ratio for the three level of the wavelength factor: R (red colour), G (green colour) and B (blue colour).

Interaction plot. Standard deviation

Signal to noise ratio for the three level of the wavelength factor: R (red color), G (green color) and B (blue color).

Temporal evolution of the radiometric values in the sequence of images. Evolution for the spectral response factor

Scheme of the approach developed in this paper to analyze the radiometric values (grey level values), the sensor noise of a single image, the sensor noise of a sequence of images and the sensor temporal trend of a sequence of images.

Outline of the most important uncertainty sources and effects affecting the radiometric values in digital cameras. The uncertainty sources analyzed in this paper are highlighted.

Noise | Random | Single images | ||

Systematic | ||||

Random | ||||

Systematic | ||||

Systematic | ||||

Random | ||||

Random | ||||

Systematic | ||||

Spatial Variability | Spatial non-uniformity | |||

Trend | ||||

Trend | ||||

Random | ||||

Systematic | ||||

Trend | ||||

Trend | ||||

Temporal Variability | Random |
Comparing images at different times | ||

Trend | ||||

Trend | ||||

Trend |

Data structure example. (Note that the

^{12} |
^{12} |
||||||||
---|---|---|---|---|---|---|---|---|---|

660 | 1 | 101 | R | 1 | 895 | 2 | 101 | B | 1 |

661 | 2 | 101 | R | 1 | ‥ | ‥ | 101 | B | 1 |

‥ | ‥ | 101 | R | 1 | ‥ | N | 101 | B | 1 |

‥ | N | 101 | R | 1 | 800 | 1 | 102 | B | 1 |

231 | 1 | 102 | R | 1 | 797 | 2 | 102 | B | 1 |

230 | 2 | 102 | R | 1 | ‥ | ‥ | 102 | B | 1 |

‥ | ‥ | 102 | R | 1 | ‥ | N | 102 | B | 1 |

‥ | N | 102 | R | 1 | ‥ | ‥ | ‥ | B | 1 |

‥ | ‥ | ‥ | R | 1 | ‥ | ‥ | R | B | 1 |

‥ | ‥ | R | R | 1 | 900 | 1 | 101 | R | 2 |

265 | 1 | 101 | G | 1 | 895 | 2 | 101 | R | 2 |

263 | 2 | 101 | G | 1 | ‥ | ‥ | 101 | R | 2 |

‥ | ‥ | 101 | G | 1 | ‥ | N | 101 | R | 2 |

‥ | N | 101 | G | 1 | 800 | 1 | 102 | R | 2 |

502 | 1 | 102 | G | 1 | 797 | 2 | 102 | R | 2 |

506 | 2 | 102 | G | 1 | ‥ | ‥ | 102 | R | 2 |

‥ | ‥ | 102 | G | 1 | ‥ | N | 102 | R | 2 |

‥ | N | 102 | G | 1 | ‥ | ‥ | ‥ | R | 2 |

‥ | ‥ | ‥ | G | 1 | ‥ | ‥ | R | R | 2 |

‥ | ‥ | R | G | 1 | ‥ | ‥ | ‥ | ‥ | ‥ |

900 | 1 | 101 | B | 1 | ‥ | ‥ | ‥ | ‥ | ‥ |

ANOVA table after adjusting the radiometric response function model. All the effects are statistically significant (P-Value). The standard error results 7.65 raw units. The mean absolute error results 5.57 raw units. The goodness-of-fit is 99.97%.

Number of dependent variables: 1 | |||||

Number of categorical factors: 2 ( | |||||

Number of quantitative factors: 2 (^{2}) | |||||

| |||||

| |||||

5.96068E9 | 11 | 5.4188E8 | 9,261,608.77 | 0.0000 | |

| |||||

3.74283E7 | 2 | 1.87142E7 | 319,855.19 | 0.0000 | |

| |||||

5,351.49 | 1 | 5,351.49 | 91.47 | 0.0000 | |

| |||||

16,931.4 | 1 | 16,931.4 | 289.39 | 0.0000 | |

| |||||

1.96456E8 | 22 | 8.92981E6 | 152,624.99 | 0.0000 | |

| |||||

21,604.7 | 11 | 1,964.07 | 33.57 | 0.0000 | |

| |||||

902,480 | 2 | 451,240 | 7,712.42 | 0.0000 | |

| |||||

633.214 | 22 | 28,782.5 | 491.94 | 0.0000 | |

| |||||

Residual | 8.42091E6 | 143,927 | 58.5082 | ||

| |||||

Total (corrected) | 2.52327E10 | 143,999 | |||

| |||||

R-Squared = 99.97% | |||||

R-Squared (adjusted for d.f.) = 99.97% | |||||

Standard Error of Est. = 7.65 | |||||

Mean absolute error = 5.57 | |||||

Durbin-Watson statistic = 1.58 (P = 0.0000) |

Resulting ANOVA table of the sensor noise model.

Number of dependent variables: 1 | |||||

Number of categorical factors: 1 ( | |||||

Number of quantitative factors: 2 ( | |||||

| |||||

| |||||

2.25918 | 2 | 1.12959 | 3.42 | 0.0327 | |

| |||||

1,588.72 | 1 | 1,588.72 | 4,816.04 | 0.0000 | |

| |||||

0.298784 | 1 | 0.298784 | 0.91 | 0.3412 | |

| |||||

5.85899 | 2 | 2.9295 | 8.88 | 0.0001 | |

| |||||

1.15191 | 1 | 1.15191 | 3.49 | 0.0617 | |

| |||||

0.465308 | 2 | 0.232654 | 0.71 | 0.4941 | |

| |||||

0.924773 | 2 | 0.462387 | 1.40 | 0.2464 | |

| |||||

Residual | 946.099 | 2,868 | 0.329881 | ||

| |||||

Total (corrected) | 7,949.27 | 2,879 | |||

| |||||

R-Squared = 88.10% | |||||

R-Squared (adjusted for d.f.) = 88.05% | |||||

Standard Error of Est. = 0.57 | |||||

Mean absolute error = 0.45 | |||||

Durbin-Watson statistic = 2.04 (P = 0.1477) |

Resulting ANOVA table of the sensor noise model without the effect of the sequential factor

Number of dependent variables: 1 | |||||

Number of categorical factors: 1 ( | |||||

Number of quantitative factors: 1 ( | |||||

| |||||

| |||||

17.5761 | 2 | 8.78803 | 26.63 | 0.0000 | |

| |||||

6,771.23 | 1 | 6,771.23 | 20,516.05 | 0.0000 | |

| |||||

12.7137 | 2 | 6.35687 | 19.26 | 0.0000 | |

| |||||

Residual | 948.551 | 2,874 | 0.330046 | ||

| |||||

Total (corrected) | 7,949.27 | 2,879 | |||

| |||||

R-Squared = 88.07% | |||||

R-Squared (adjusted for d.f.) = 88.05%t | |||||

Standard Error of Est. = 0.58 | |||||

Mean absolute error = 0.45 | |||||

Durbin-Watson statistic = 2.04 (P = 0.1557) |

Resulting ANOVA table of the image sequence sensor noise model for a Foveon X3 sensor (dynamic range 2^{12} raw units).

Number of dependent variables: 1 | |||||

Number of categorical factors: 1 ( | |||||

Number of quantitative factors: 2 ( | |||||

| |||||

| |||||

57.7505 | 2 | 28.8752 | 114.64 | 0.0000 | |

| |||||

15,205.2 | 1 | 15,205.2 | 60,367.49 | 0.0000 | |

| |||||

26.2726 | 2 | 13.1363 | 52.15 | 0.0000 | |

| |||||

Residual | 451.111 | 1,791 | 0.251877 | ||

| |||||

Total (corrected) | 15,920.1 | 1,796 | |||

| |||||

R-Squared = 97.17% | |||||

R-Squared (adjusted for d.f.) = 97.16% | |||||

Standard Error of Est. = 0.50 | |||||

Mean absolute error = 0.39 | |||||

Durbin-Watson statistic = 1.77 (P = 0.0000) |

Resulting ANOVA table of the temporal trend model.

Number of dependent variables: 1 | |||||

Number of categorical factors: 2 ( | |||||

Number of quantitative factors: 1 ( | |||||

| |||||

| |||||

0.00860736 | 2 | 0.00430368 | 1,724.75 | 0.0000 | |

| |||||

0.00177015 | 2 | 0.000885074 | 354.70 | 0.0000 | |

| |||||

0.0000105471 | 1 | 0.0000105471 | 4.23 | 0.0471 | |

| |||||

0.00130896 | 4 | 0.000327241 | 131.15 | 0.0000 | |

| |||||

0.00000794286 | 2 | 0.00000397143 | 1.59 | 0.2176 | |

| |||||

0.0000257446 | 2 | 0.0000128723 | 5.16 | 0.0107 | |

| |||||

0.00000503574 | 4 | 0.00000125894 | 0.50 | 0.7326 | |

| |||||

Residual | 0.000089829 | 36 | 0.00000249525 | ||

| |||||

Total (corrected) | 0.0404397 | 53 | |||

| |||||

R-Squared = 99.78% | |||||

R-Squared (adjusted for d.f.) = 99.67% | |||||

Standard Error of Est. = 0.002 | |||||

Mean absolute error = 0.0010 | |||||

Durbin-Watson statistic = 2.44 (P = 0.3465) |

Resulting ANOVA table of the temporal trend model without the interaction effects

Number of dependent variables: 1 | |||||

Number of categorical factors: 2 ( | |||||

Number of quantitative factors: 1 ( | |||||

| |||||

| |||||

0.0284155 | 2 | 0.0142077 | 5,719.15 | 0.0000 | |

| |||||

0.00177372 | 2 | 0.00088686 | 356.99 | 0.0000 | |

| |||||

0.00000822607 | 1 | 0.00000822607 | 3.31 | 0.0759 | |

| |||||

0.0044165 | 4 | 0.00110412 | 444.45 | 0.0000 | |

| |||||

0.0000237051 | 2 | 0.0000118525 | 4.77 | 0.0136 | |

| |||||

Residual | 0.000104338 | 42 | 0.00000248424 | ||

| |||||

Total (corrected) | 0.0404397 | 53 | |||

| |||||

R-Squared = 99.74% | |||||

R-Squared (adjusted for d.f.) = 99.67% | |||||

Standard Error of Est. = 0.002 | |||||

Mean absolute error = 0.0010 | |||||

Durbin-Watson statistic = 2.25 (P = 0.3152) |

Resulting ANOVA table of the temporal trend model, only with the effects of the spectral response factor

Number of dependent variables: 1 | |||||

Number of categorical factors: 2 ( | |||||

Number of quantitative factors: 0 | |||||

| |||||

| |||||

0.0288969 | 2 | 0.0144484 | 4,496.89 | 0.0000 | |

| |||||

0.00694348 | 2 | 0.00347174 | 1,080.53 | 0.0000 | |

| |||||

0.00445472 | 4 | 0.00111368 | 346.62 | 0.0000 | |

| |||||

Residual | 0.000144585 | 45 | 0.00000321299 | ||

| |||||

Total (corrected) | 0.0404397 | 53 | |||

| |||||

R-Squared = 99.64% | |||||

R-Squared (adjusted for d.f.) = 99.58% | |||||

Standard Error of Est. = 0.002 | |||||

Mean absolute error = 0.001 | |||||

Durbin-Watson statistic = 1.79 (P = 0.0251) |