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For satellite remote sensing, radiances received at the sensor are not only affected by the atmosphere but also by the topographic properties of the terrain surface. As a result, atmospheric correction alone does not yield output images that truly reflect terrain surface properties, namely surface reflectance (bidirectional reflectance factor, BRF) of objects on the earth surface. Following the concept of the radiometric control area (RCA)-based path radiance estimation method, we herein propose a statistical approach for surface reflectance estimation utilizing DEM data and surface reflectance of selected radiometric control areas. An algorithm for identification of shaded samples and a shape factor model were also developed in this study. The proposed RCA-based surface reflectance estimation method is capable of achieving good reflectance estimates in a region where elevation varies from 0 to approximately 600 m above the mean sea level. However, further study is recommended in order to extend the application of the proposed method to areas with substantial terrain variation.

Remote sensing images have been widely used for applications of earth surface monitoring such as landslide sites identification, land use/land cover (LULC) classification and change detection, crop yield estimation, reservoir and coastal water quality monitoring,

Theory and models of radiometric propagation from the sun to the sensor which are essential for remote sensing image processing have been well developed. For example, Slater [

Since radiances leaving the object surface are affected by the atmosphere and surrounding topographic features prior to being received at the sensor, these effects (in the form of path radiance and shape factors) must be taken into account in estimation of surface reflectance. Methods of in-scene estimation of path radiances have been proposed, with the dark object subtraction (DOS) method being most widely applied [

In contrast to the above in-scene estimation methods, Cheng

An area of approximately 750 km^{2} in northern Taiwan was chosen for this study (

A set of Formosat-II multispectral images (including blue band: 450–520 nm, green band: 520–600 nm, red band: 630–690 nm, and near infrared band: 760–900 nm, with 8 m spatial resolution) of the study area acquired at 01:57 GMT (9:57 a.m. local time) on December 11, 2008 was collected (

For a target of Lambertian surface, the at-sensor solar radiance of spectral wavelength _{sλ}

(

_{pλ}

_{oλ}

_{dλ}

_{dλ}

_{1}_{λ}

_{2}_{λ}

_{i}

Radiances reaching the sensor are recorded and linearly converted to digital numbers (DNs) by the following equation using the band-specific gain and offset parameters of the sensor:

The dependence of _{pλ}_{dλ}_{1}_{λ}_{2}_{λ}^{−1}·m^{−2}·sr^{−1}) for the blue, green, red and near infrared bands, respectively. For local remote sensing applications which do not cover extensively wide study areas, the exoatmospheric solar irradiance, downwelled irradiance, path radiance, and atmospheric transmittances can all be assumed to be spatially homogeneous. In other words, _{pλ}_{1}_{λ}_{2}_{λ}_{i}

Cheng

Although in general the at-sensor radiance can be expressed by _{i}_{i}

Under either situation, digital numbers of these ground samples (hereinafter referred to as the _{i}_{o}_{i}_{s}_{i}

The shape factor

For convenience of subsequent explanation, we first define the following notations which will be used later. Consider a group of _{i}_{0} = _{i}_{1} = _{2} represents and elevation differences (_{i}^{2}–1). Elevations of the target ground sample and its surrounding samples in the sample group are represented by an elevation matrix

Spatial variations of the shape factor _{1} and elevation difference _{2} within the study area can be considered as two random fields with their mean vectors and covariance matrices defined as
_{1}. For target samples with larger elevation differences between the target sample and its neighboring samples, _{2} = (_{1}, _{2}, ⋯, _{p2–1}), we can expect more significant obstruction effect and lower values of the shape factor. From the fundamental theorem of estimation theory [_{2} associated to the target ground sample, is the conditional expectation of _{2} and _{2} in _{2}, respectively.

The mean vector and covariance matrix (_{2} and Σ_{22}) of _{2} in _{2} using DEM data of a total of 388,000 ground samples within the study area. With the very large number of ground samples, estimates of _{2} and Σ_{22} can be expected to be nearly unbiased and with very small variance, even though _{2} is likely to exhibit spatial dependence. As for _{i}^{2}–1) in _{22} and thus are readily available once Σ_{22} has been obtained.

Considering Lambertian targets and isotropic diffuse irradiance, surface reflectance of the target ground sample _{dλ}_{i}^{2}–1). Thus, if only shaded ground samples are considered, we have

In the above equations ^{shade}_{λ}_{dλ}

Shaded ground samples account for approximately one sixth of the total number of ground samples in the study area. With this very large sample size and also based on the asymptotic distributional properties of the sample correlation coefficients [^{2}–1) can be estimated with near zero bias and very small variance. Thus, for every ground sample the sample-specific
_{λ}_{λ}

Substituting

Suppose that a total of _{i}

In the above matrix equation, digital numbers and surface reflectances of RCA samples are known and cos_{i}_{pλ}_{1}_{λ}_{2}_{λ}_{1} and _{2}_{λ}

In previous sections we have demonstrated that values of _{pλ}_{1}_{λ}_{2}_{λ}_{1} and _{2}_{λ}_{λ}

The topographic effect of cos_{i}

As was explained in Section 3.3, the correlation coefficients between digital numbers of shaded ground samples (^{shade}_{i}^{2}–1) can be estimated with near zero bias and very small variance. These correlation coefficients
^{2}–1) correspond to a group of ^{2}–1). We shall refer to the correlation matrix P_{λ}

The correlation maps show a pattern nearly symmetric to the direction of incoming solar radiation for all spectral bands. Digital numbers (or corresponding radiances) of the shaded ground samples are contributed by the downwelled radiance from portion of the sky dome above the target sample. Downwelled radiances are the results of atmospheric scattering, most importantly the Rayleigh and Mie scatterings, whose effects are symmetric to the direction of solar irradiance. The symmetric pattern of the correlation map correctly reflects the characteristics of atmospheric scattering. The correlation map also shows that positive and negative correlation coefficients are associated with the samples falling in front of and in the back of the target sample (with respect to the target sample and the direction of solar irradiance), respectively. Such a pattern is also consistent with the topographic effects on radiance received at the target ground as explained below.

As shown in _{i}^{shade}_{i}^{2}–1) are negatively correlated. In contrast, solar irradiances reaching the obstacle samples which situate in front of the target sample may be reflected to the target sample. The higher the obstacle samples (_{i}^{shade}_{i}^{2}–1) are positively correlated.

Prior to calculation of surface reflectances using _{pλ}_{1}_{λ}_{2}_{λ}_{1} and _{2}_{λ}_{pλ}_{1}_{λ}_{2}_{λ}_{1} and _{2}_{λ}

Among these constants, values of _{pλ}_{pλ}

Since the study area encompasses an area of 750 km^{2} with different land cover conditions, it is difficult to conduct extensive

True color satellite image of the subregion B which is an area with high mountains and substantial terrain variation (elevation varies from approximately 200 to 2000 m) shows significant topographic effect with visually apparent shaded areas. Although the topographic effect has also been largely eliminated in the color reflectance image of subregion B, the reflectance image still shows different levels of reflectance for the shaded (in purple color) and non-shaded (in green color) areas. Such results may arise from the unaccounted shade effect of very high mountains in subregion B. In our study an elevation matrix (

This study is an extension of the RCA-based path radiance estimation algorithm for surface reflectance estimation using digital numbers and surface reflectance of selected radiometric control areas. A few concluding remarks are drawn as follows:

A shaded sample identification algorithm using DEM data is proposed in this study.

The correlation maps demonstrate a pattern that not only is consistent with the atmospheric scattering effect but also characterizes the effect of neighboring samples on radiance received at the target sample. Such result is an indication that the proposed shape factor model (

The proposed RCA-based surface reflectance estimation method is capable of achieving good reflectance estimates in a region where elevation varies from 0 to approximately 600 m above the mean sea level. Further study on variable size of the elevation matrix with respect to the degree of terrain variation is recommended in order to extend application of the proposed method to areas with substantial terrain variation.

The corresponding author is grateful to the National Science Council (NSC) of Taiwan, ROC for its financial support of his sabbatical leave during which this manuscript was completed. The corresponding author would like to dedicate this paper to the Late Professor Toshiharu Kojiri of the Disaster Prevention Research Institute, Kyoto University, who hosted the sabbatical visit of the corresponding author and had many fruitful and interesting discussions with him. We are also grateful to two anonymous reviewers for their constructive comments which led to a much better presentation of the final form of this paper.

(

Sun/shade occurrences: (

An illustrative sketch for calculation of cos_{i}

Images showing topographic effects. _{i}_{i}_{i}

Correlation map P_{λ} calculated from satellite images of different spectral bands.

Illustration of the topographic effects of the forward and backward samples on radiance (or digital number) received at the target sample.

Comparison of the true color satellite images (

Band-dependent constants (_{pλ}_{1}_{λ}_{2}_{λ}_{1} and _{2}_{λ}

_{pλ} |
59 | 19 | 8 |

_{1}_{λ} |
346.011 | 358.119 | 279.024 |

_{2}_{λ}_{1} |
269.372 | 333.391 | 544.729 |

_{2}_{λ} |
47.662 | 36.112 | 28.326 |

Comparison of path radiances (_{pλ}

59 | 19 | 8 | |

69 | 36 | 26 | |

29 | 19 | 13 |

Note: Path radiances of the DOS and AERONET methods were estimated by Cheng