Automatic determination of quantitative parameters describing the pattern of urban expansion is extremely important for urban planning, urban management and civic resource configuration. Though the widely adopted LEI (landscape expansion index) has exhibited the potential to capture the evolution of urban landscape patterns using multi-temporal remote sensing data, erroneous determination still exists, especially for patches with special shapes due to the limited consideration of spatial relationships among patches. In this paper, we improve the identification of urban landscape expansion patterns with an enhancement of the topological relationship. We propose MCI (Mean patch Compactness Index) and AWCI (Area-Weighted Compactness Index) in terms of the moment of inertia shape index. The effectiveness of the improved approach in identifying types of expansion patches is theoretically demonstrated with a series of designed experiments. Further, we apply the proposed approaches to the analysis of urban expansion features and dynamics of urban compactness over Urumqi at various 5-year stages using available SUBAD-China data from 1990–2015. The results achieved by the theoretical experiments and case application show our approach effectively suppressed the effects induced by shapes of patches and buffer or envelope box parameters for the accurate identification of patch type. Moreover, the modified MCI and AWCI exhibited an improved potential in capturing the landscape scale compactness of urban dynamics. The investigated 25-year urban expansion of Urumqi is dominated by edge-expansion patches and supplemented by outlying growth, with opposite trends of increasing and decreasing, with a gradual decrease in landscape fragmentation. Our examination using the proposed MCI and AWCI indicates Urumqi was growing more compact in latter 15-year period compared with the first 10 years studied, with the primary urban patches tending to be compacted earlier than the entire urban setting. The historical transformation trajectories based on remote sensing data show a significant construction land gain—from 1.06% in 1990 to 6.96% in 2015—due to 289.16 km² of recently established construction, accompanied by fast expansion northward, less dynamic expansion southward, and earlier extension in the westward direction than eastward. This work provides a possible means to improve the identification of patch expansion type and further understand the compactness of urban dynamics.

As the inherent attribute of land cover, anisotropy leads to the heterogeneity of directional reflection; meanwhile, it creates the opportunity for retrieving characteristics of land surface based on multi-angle observations. BRDF (Bidirectional Reflectance Distribution Function) is the theoretical expression of anisotropy and describes the reflectance in terms of incident-view geometry. Prior BRDF knowledge is used to achieve the multi-angle retrieval for earth observation systems with a narrow FOV (Field of View). Shape indicators are a feasible way to capture the characteristics of BRDF or to build an a priori database of BRDF. However, existing shape indicators based on the ratio of reflectance or the weight of scattering effects are too rough to describe the BRDF’s shape. Thus, it is necessary to propose new shape vectors to satisfy the demand. We selected six typical land covers from MODIS-MCD12 on the homogeneous underlayers as the study sites in North America. The daily BRDF is retrieved by MODIS-BRDF parameters and the RossThick-LiSparseR model. When the SZA (Solar Zenith Angle) is set at 45°, seven directions (−70°, −45°, −20°, 0°, 20°, 45°, and 70°) including edge spot, zenith spot, hot spot and approximate dark spot of the BRDF principal plane were selected to construct two vectors by the change rate of reflectance and angle formulation: Partial Anisotropic Vector (PAV) and Angular Effect Vector (AEV). Then, we assessed the effectiveness of PAV and AEV compared with ANIX (Anisotropic Index), ANIF (Anisotropic Factor) and AFX (Anisotropic Flat Index) by two typical BRDF shapes. The representativeness of PAV and AEV for the original BRDF was also assessed by cosine similarity and error transfer function. Lastly, the application of hot spot components in AEV for land cover classification, the monitoring of land cover in mining areas and the adjustment effect by NDVI (Normalized Difference Vegetation Index) were investigated. The results show that (1) the shape vectors have good representativeness compared with original BRDF. The representativeness of PAV assessed by cosine similarity is 0.980, 0.979 and 0.969, and the representativeness of AEV assessed by error transfer function is 0.987, 0.991 and 0.994 in the three MODIS broadbands of Near Infrared (NIR, 0.7–5.0 µm), Short Wave (SW, 0.3–5.0 µm) and Visible (VIS, 0.3–0.7 µm). (2) Some components of shape vectors have high correlation with AFX. The correlation coefficient between hot spot components in AEV and AFX is 0.936, 0.945 and 0.863, respectively, in NIR, SW and VIS bands. (3) The shape vectors show potentiality for land cover classification and the monitoring of land cover in mining areas. The correlation coefficients of hot spot components in AEV for MODIS-pixels with the same types (0.557, 0.561, 0.527) are significantly higher than MODIS-pixels with various types (0.069, 0.055, 0.051) in NIR, SW and VIS bands. The coefficients of variation for hot spot components are significantly higher after land reclamation (0.0071, 0.0099) than before land reclamation (0.0020, 0.0028). (4) The correlation between NDVI and the BRDF shapes is poor in three MODIS broad bands. The correlation coefficients between NDVI and the BRDF shapes in three temporal scales of annual, seasonal and monthly phases are only 0.134, 0.063 and 0.038 (NIR), 0.199, 0.185 and 0.165 (SW), and 0.323, 0.320 and 0.337 (VIS), on average.