InSAR-DInSAR 技術細節（二） 干涉測量的條件（木有免費的午餐，以及晚餐））

1 幾個條件

干涉測量的條件，或者說干涉測量的要求，主要從以下幾個方面進行考慮：

* 物理過程對應的形變的大小；
* 像素大小、幅寬、以及系統性能能；
* 時間間隔（時間主要會導致去相關，似乎正是這個限制導出了形變大小、像素大小、坡度角等限制）；
* 坡度角（斜坡效應）；
* 基線限制（跟軌道數據的限制和平臺的限制也是有聯繫的）
* limitations in distinguishing between the different geophysical or induced sources of coherent phase variation (由於干涉相位中包含了有很多因素，某一種應用的成功取決於先驗條件的精確性，比如說用於輔助形變測量的DEM的精度)；
* Other limiting conditions can occur in the interferometric processing sequence（eg. phase unwarping）；
跟相位解纏相近的：The second type of ambiguity arises because interferograms record relative changes in phase, not absolute changes. In other words, we cannot identify the fringe corresponding to zero change in phase, i.e., the contour of null deformation. Mathematically, we are free to add a constant value (or “offset”) to all the pixels in an interferogram. This ambiguity persists even if the interferogram has been unwrapped. Usually, we can determine this constant by assumption (e.g., null deformation at one point) or independent measurement (e.g., a GPS survey).
* Others depending on the application of the products.

2 利用圖形表示這些conditions

3 conditions 之間的聯繫

這些限制大多都是相通的，畢竟其本質上是對相干性的限制。一個很好的切入角度是：
干涉測量的關鍵在於相減的兩個相位的隨機性能夠被抵消。
For this, the elementary targets must remain stable. This condition is met if the two images are taken at the same time. A more geometric condition requires that elementary targets within a pixel contribute the same way to both images. Therefore the pixel must not stretch or shrink by more than a fraction of the wavelength from one image to the other. Otherwise, targets at both ends of a given pixel will add differently in each image, leading to internal phase contributions that do not cancel by subtraction. Mathematically, let L represent the length perpendicular to the trajectory of a pixel on the ground (20 m for ERS-1), let λ represent the wavelength (56 mm for ERS-1), and let θ1 and θ2 represent the angles of incidence (from local vertical) in the first and second image, respectively. The difference in round trip distance of targets at both ends of a pixel is 2Lsinθ . Hence the fundamental condition for interferometry（這個基本的公式）

2L(sinθ1−sinθ2)<λ

restricts the separation between the satellite’s orbital trajectories during the two image acquisitions to typically less than 1 km (for ERS-1).
Fortunately, satellite orbits are designed to repeat identically after a period of time called the orbital cycle and generally meet this condition.
The local slope of the terrain influences this condition. Close to the interferometric limit (1), even a moderate slope with the wrong orientation will blur fringes. According to (1), steep incidence, coarse resolution, and short wavelength all make the condition harder to satisfy.
這個基本的公式由相干性導出，包含的限制有：基線、坡度角、像素大小、波長以及形變。相干性是根本要求，坡度角、軌道數據精度、形變大小等只是他的一個側面。
Similarly, the direction of observation must also be identical for the two images; otherwise, elementary targets will sum differently in the along-track direction of the pixel. The interferogram degrades linearly with the angle between the two directions of observation. The degradation becomes total when this angle exceeds the width of the antenna beam (typically 0.3 for ERS-1). In signal-processing terms, this happens as soon as the illuminated areas (Figure 1) cease to overlap in the along-track direction, creating an excessively large difference between the “mean Doppler” of the two images. （這個與方位向頻譜偏移是否是一回兒事？）
這個式子還可以看出形變測量的限制：
The necessary condition for interferometry (relation (1)) implies that the maximum detectable deformation gradient is one fringe per pixel, or the dimensionless ratio of the pixel size to the wavelength. This value depends on the satellite; it is 3×10−3 for ERS and 13×10−3 for JERS. For instance, the coseismic deformation in the Landers earthquake locally exceeded this threshold, creating incoherence.
Similarly, block rotation can change the radar observation direction sufficiently to violate the necessary condition for interferometry. Such a change of direction of observation produces a set of parallel fringes oriented perpendicular to the satellite track. As for the gradient limit, where we cannot exceed one fringe of range change per range pixel, we cannot accept more than one fringe per azimuth pixel. Areas close to this limit appear in the vicinity of the Landers fault. The limit is found when a round trip range change of one wavelength is created across the azimuth pixel size. （這個與方位向頻譜偏移是否是一回兒事？）

4 說明

The user may also spoil the interferometric effect by applying slightly different processing procedures to each image. Such a slight mistake can damage the interferogram more than a huge error consistently applied to both images in the pair.

Abrupt changes in topography, such as volcanic eruptions, can exceed this limit（the gradient of the surface deformation）. Discontinuities such as surface rupture also exceed this limit because they produce an infinite gradient. In this case, however, we can still measure the relative motion on opposite sides of a fault by counting fringes along a path which runs around the end of the fault [Massonnet et al., 1993b, Figure 3b].

這些限制條件並不絕對
Even if the range gradient caused by deformation on the ground exceeds the limit of one fringe per pixel, a judicious choice of image pairs may yield a usable interferogram. For example, a large orbital separation can create a fringe gradient that just compensates that of the ground deformation, bringing the geometric configuration back within the interferometric limits. Similarly, a difference in the mean Doppler between two images could compensate for rotation on the ground. In this way it should be possible to calculate useful fringes in parts of a high gradient deformation field using an otherwise unusable image pair. However, the topographic sensitivity associated with a larger orbital separation would prevent the use of this technique in hilly terrain because it would require removing topographic effects with a proportionally accurate DEM.

5 參考文獻

Hooper, A., Bekaert, D., Spaans, K., & Arıkan, M. (2012). Recent advances in SAR interferometry time series analysis for measuring crustal deformation. Tectonophysics, 514, 1-13.
Zhong, L., & Dzurisin, D. (2014). Insar imaging of aleutian volcanoes. Springer Praxis Books, 2014(8), 1778–1786.
Ketelaar, V. (2009). Satellite radar interferometry : subsidence monitoring techniques.
RADAR INTERFEROMETRY Data Interpretation and Error Analysis.pdf