Abstract: The topography and geomorphology of active orogens result from the interaction of tectonics and climate. In most orogens, a fluvial channel is most sensitive to the coupling between tectonics, lithology, and climate. Meanwhile, the related signals have been recorded by both the drainage geometry and channel longitudinal profile. Thus, how to extract tectonic information from fluvial channels has been a focused issue in geologic and geomorphologic studies.
The well known stream-power river incision model bridges the gap between tectonic uplift, river incision and channel profile change, making it possible to retrieve rock uplift pattern from river profiles. In this model, the river incision rate depends on the rock erodibility, contributing drainage area and river gradient. The steady-state form of the river incision model predicts a power-law scaling between the drainage area and channel gradient. Via a linear regression to the log-transformed slope-area data, the slope and intercept are channel concavity and steepness indices, respectively. The concavity relates to lithology, climatic setting and incision process while the channel steepness can be used to map the spatial pattern of rock uplift. For its simple calculation process, the slope-area analysis has been widely used in the study of tectonic geomorphology during past decades.
However, to calculate river slope, the coarse channel elevation data must be smoothed, re-sampled, and differentiated without any reasonable smooth window or rigid mathematical fundamentals. One may lose important information and derive stream-power parameters with high uncertainties. In this paper, we introduce the integral approach, a procedure that has been widely used in the latest four years and demonstrated to be a better method for river profile analysis than the traditional slope-area analysis. Via the integration to the steady-state form of the stream-power river incision equation, the river longitudinal profile can be converted into a straight line of which the independent variable is the integral quantity χ with the unit of distance and the dependent variable is the relative channel elevation. We can calculate the linear correlation coefficient between elevation and χ based on a series of concavity values and find the best linear fit to be the reasonable channel concavity index. The slope of the linear fit to the χ value and elevation is simply related to the ratio of the uplift rate to the erodibility.
Without calculating channel slope, the integral approach makes up for the drawback of the slope-area analysis. Meanwhile, via the integral approach, a steady-state river profile can be expressed as a continuous function, which can provide theoretical principle for some geomorphic parameters (e.g., slope-length index, hypsometric integral). In addition, we can determine the drainage network migration direction using this method. Therefore, the integral approach can be used as a better method for tectonogeomorphic research.

Key words: stream-power river incision model, integral approach, Chi-Plot, steepness index

摘要： 以稳态的河水动力侵蚀方程为基础，分析河道高程剖面，可以得到河道陡峭系数，从而反映区域基岩隆升速率的空间分布特征。利用传统的坡度-面积分析法计算陡峭系数时，需要对原始高程数据进行平滑、重采样和微分等一系列操作来计算坡度，这会造成信息丢失和重复引入误差。而近5a来逐渐得到推广应用的积分方法，通过直接求解河水动力侵蚀方程，将稳态的河道高程剖面变换成1条直线，直线斜率即河道陡峭系数，避免了计算坡度带来的缺点。同时，该方法用积分函数表示基岩河道高程剖面，可以将陡峭系数和其他一些地貌参数（坡度长度指数、面积高程曲线积分）联系起来，为用这些参数表示区域构造活动信息提供理论依据。此外，该方法还可用于判别分水岭迁移方向。因此，综合这些优势，积分法在分析流域地貌特征和进行构造地貌的研究中具有广阔的应用前景。

关键词: 河水动力侵蚀方程, 积分法, Chi-plot, 陡峭系数