Regular spacing of drainage outlets from linear mountain belts

Abstract Straight sections of many actively uplifting mountain belts have simple patterns of drainage, transverse to their main structural trend. Streams rising near or beyond the topographic ridgepole of these sections are spaced at seemingly regular intervals. To test whether this regularity exists, morphometric aspects of drainage networks were measured in 11 mountain belts. The spacing of drainage basins can be expressed using a spacing ratio, which in effect is the ratio of the length and the width of the catchments under consideration. Average spacing ratios for most linear mountain belts are within a narrow range of values between 1.91 and 2.23. A linear relationship exists between the spacing of catchment outlets and the distance between the main divide and the front of the mountain belt in which they have developed. The Nepalese Himalaya form an exception to this regular pattern. In this mountain belt drainage is blocked and diverted by structures that have developed in relation to the Main Boundary Thrust. Structural complications cause drainage patterns to become less regular, introducing important non trans verse components. The linear relationship between spacing of catchment outlets and half-width of the mountain belt may be expressed in an equation of the same general form as Hack's law of stream length and drainage basin area. It seems likely that the mechanism underlying Hack's law also explains the consistent regularity of drainage spacing in active mountain belts. However, no generally accepted explanation for Hack's law has been offered. The narrow range of spacing ratios found for drainage networks in active orogens may represent an optimal catchment geometry that embodies a ‘most probable state’ in the uplift-erosion system of a linear mountain belt. The linear relationship between the half-width of a mountain belt and spacing of catchment outlets has profound implications for the modelling of erosion of orogenic topography, and for the formation and filling of foreland basins.     

River spacing and drainage network growth in widening mountain ranges

Drainage networks in linear mountain ranges always display a particular geometrical organisation whereby the spacing between the major drainage basins is on average equal to half the mountain width (distance from the mountain front to the main drainage divide), independent of climate and tectonics. This relationship is valid for mountains having different widths and is thus usually thought to be maintained by drainage reorganisation during mountain belt widening. However, such large‐scale systematic drainage reorganisation has never been evidenced. In this paper, we suggest an alternative explanation, namely that the observed drainage basin relationships are an inherent property of dendritic river networks and that these relationships are established on the undissected, lowland margins outside mountain ranges and are progressively incorporated and quenched into uplifted topography during range widening. Thus, we suggest that the large‐scale geometry of drainage networks in mountain ranges is mainly antecedent to erosion. We propose a model in which the large‐scale drainage geometry is controlled mainly by the geometrical properties of the undissected surfaces (in particular, the ratio of the regional slope to the local slope related to roughness) over which rivers are flowing before uplift, and is therefore independent of climate and tectonics.     

Discrete-element modelling of extensional fault-propagation folding above rigid basement fault blocks

We employed a discrete‐element technique to investigate the effects of cover strength and fault dip on the style of fault‐propagation folding above a blind normal fault. Deformation in the cover is initially characterised by an upward‐widening monocline that is often replaced, with continued slip on the basement fault, by a single, through‐going fault. Localisation on a single fault produces hangingwall synclines and footwall anticlines as a result of breaching of the earlier monocline and which do not represent ‘drag’ against the fault. As basement fault dip decreases the width of the monocline at the surface increases. Experiments varying the strength of the overburden material illustrate the control that cover strength has on both fault propagation and folding in the cover. Reduction of the strength of the cover results in: (1) the width of the monocline above the fault tip increasing, and (2) more marked footwall thinning and hangingwall thickening of beds. In contrast, an increase in cover strength results in a narrower monocline and rapid propagation of the basement fault into the cover. In multi‐layer (variable strength) experiments simultaneous faulting of competent layers and flow of weaker layers produces complex structural relationships. Faults in the cover die out up and down section and do not link to the basement fault at depth. Similarly, complex macroscopically ductile characteristics such as footwall thinning and hangingwall thickening can be juxtaposed against simple brittle fault cut‐offs. These relationships must be borne in mind when interpreting the field and seismic expression of such structures. We discuss the modelling results in terms of their implications for structural interpretation and the surficial expression of fault‐related folding in extensional settings.     

Universality and variability in basin outlet spacing: implications for the two-dimensional form of drainage basins

It has been observed that the distance between the outlets of transverse basins in orogens is typically half of the distance between the main divide and the range front irrespective of mountain range size or erosional controls. Although it has been suggested that this relationship is the inherent expression of Hack's law, and/or possibly a function of range widening, there are cases of notable deviations from the typical half‐width average spacing. Moreover, it has not been demonstrated that this general relationship is also true for basins in morphologically similar nonorogenic settings, or for those that do not extend to the main drainage divide. These issues are explored by investigating the relationship between basin outlet spacing and the 2‐dimensional geometric properties of drainage basins (basin length, main valley length and basin area) in order to assess whether the basin outlet spacing‐range width ratio is a universal characteristic of fluvial systems. We examined basins spanning two orders of magnitude in area along the southern flank of the Himalayas and the coastal zone of southeast Africa. We found that the spacing between basin outlets (L_os) for major transverse basins that drain the main divide (range‐scale basins) is approximately half of the basin length (L_b) for all basins, irrespective of size, in southeast Africa. In the Himalayas, while this ratio was observed for eastern Himalayan basins (a region where the maximum elevations coincided with the main drainage divide), it was only observed in basins shorter than ～30 km in the western and central Himalayas. Our analysis indicates that basin outlet spacing is consistent with Hack's law, apparently because the increase in basin width (represented by outlet spacing) with basin area occurs at a rate similar to the increase in main stream length (L_v) with basin area. It is suggested that most river systems tend towards an approximately diamond‐shaped packing arrangement, and this applies both to the nonorogenic setting of southeast Africa as well as most orogenic settings. However, in the western Himalayas shortening associated with localised rock uplift appears to have occurred at length scales smaller than most the basins examined. As a result rivers in basins longer than ～30 km have been unable to erode in a direction normal to the range front at a sufficiently high rate to sustain this form and have been forced into an alternative, and possibly unstable, packing arrangement.     

The role of fault length,overlap and spacing in controlling extensional relay ramp fluvial system geometry

Relay ramps are integral components of normal fault systems that control sediment transport pathways in evolving rifts. We attribute differences in the geometry of fluvial systems that drain relay ramps to the scale of the ramp bounding fault segments, the spacing between segments and the amount of overlap between segments. Previous conceptual models for relay ramp geomorphological evolution have assumed that ramp fluvial catchments develop on the ramp surfaces and flow parallel to fault strike into the adjacent basin. Numerous examples exist in nature, however, that show that this is not ubiquitous. The fundamental question of what drives differences in fluvial geometry in these settings has, to date, not been fully addressed. We selected 27 relay ramps across the Basin and Range, western North America, and mapped, via GPS and remote sensing, the faults and ramp fluvial systems associated with each site. The sites represent a range of fault scales, which we define by the total outboard fault length, and a range of spacing and overlap values in order to better understand the structural controls on differences among ramp fluvial systems. Results show that the majority of a relay ramp surface drains parallel to fault strike when the outboard fault is less than about 15 km long. High overlap/spacing ratios are associated with relays along shorter (<15 km long) outboard faults, whereas lower overlap/spacing ratios are associated with relays along longer faults. Relays with lower overlap/spacing values may be more common along longer outboard faults because they survive for longer periods of time in the landscape. Our geomorphological observations can be used to predict synrift depocenter locations along segmented faults, but these observations only apply if the faults are short (<15 km long) and in early rifting stages. At longer fault lengths, ramp fluvial system geometry has no discernable relationship with any specific structural parameter.     

Sediment flux from an uplifting fault block

The stratigraphy of rift basins is a direct result of sediment liberation and transport through catchment–fan systems whose dynamics are controlled by both external and internal factors. We investigate the response of catchment–fan systems established across an active normal fault to variations in both tectonic and climatic boundary conditions. Numerical experiments show that the ratio of fan area to catchment area provides a sensitive indicator of tectonic activity. A step decrease in fault slip rate results in a delayed response by the catchment–fan systems; the response time is ∼50 kyr for a variety of parameter values. Decreased slip rate also gives rise to an abrupt but transient pulse in sediment discharge from the fans due to a drop in the hangingwall subsidence rate. In contrast, variations in climatic activity, using precipitation rate as a proxy, produce extremely rapid responses throughout the catchment–fan system. Thus, high-frequency climatic changes will overprint lower frequency tectonic variations in the stratigraphic record of fan deposits. Finally, we map out possible combinations of fault geometry, fault slip rate and precipitation rate that allow fan progradation and high rates of sediment discharge from the system.     

Seismic reflection coefficients from mantle fault zones

Summary. Several bright reflections from structures within the mantle can be seen on BIRPS' deep seismic reflection profiles. We have calculated apparent reflection coefficients for the brightest of these events and obtain values around 0.1. It is not possible to produce such large reflections by either compositional layering or seismic anisotropy if olivine and pyroxene are the only significant minerals in the mantle. These large reflections can be produced by a mafic layer or a partially hydrated layer within normal peridotite. The brightest reflections seem to be best explained as major faults or shear zones within the mantle.     

Statistical analysis of drainage density from digital terrain data

Drainage density (D_d), defined as the total length of channels per unit area, is a fundamental property of natural terrain that reflects local climate, relief, geology, and other factors. Accurate measurement of D_d is important for numerous geomorphic and hydrologic applications, yet it is a surprisingly difficult quantity to measure, particularly over large areas. Here, we develop a consistent and efficient method for generating maps of D_d using digital terrain data. The method relies on (i) measuring hillslope flow path distance at every unchanneled site within a basin, and (ii) analyzing this field as a random space function. As a consequence, we measure not only its mean (which is half the inverse of the traditional definition of drainage density) but also its variance, higher moments, and spatial correlation structure. This yields a theoretically sound tool for estimating spatial variability of drainage density. Averaging length-to-channel over an appropriate spatial scale also makes it possible to derive continuous maps of D_d and its spatial variations. We show that the autocorrelation length scale provides a natural and objective choice for spatial averaging. This mapping technique is applied to a region of highly variable D_d in the northern Apennines, Italy. We show that the method is capable of revealing large-scale patterns of variation in D_d that are correlated with lithology and relief. The method provides a new and more general way to quantitatively define and measure D_d, to test geomorphic models, and to incorporate D_d variations into regional-scale hydrologic models.     

High-frequency seismic radiation from a buried circular fault

Summary. We propose a simplified method for the calculation of near field accelerograms. It is based upon the hypothesis that, in the course of dynamic faulting, the dominating part of the seismic radiation is emitted by the rupture front. As the rupture moves smoothly it radiates continuously, generating the low-frequency part of the field. High-frequency waves are produced by jumps in the rupture velocity and abrupt changes in the stress intensity factor. The wave-front discontinuities created in this fashion are evaluated by asymptotic methods and may be propagated away from the source by ray theoretical methods. We apply our technique to the evaluation of asymptotic near field accelerograms for a circular fault buried in a half-space. The agreement with numerical accelerograms calculated by full-wave theory is very satisfactory. Two problems are given particular emphasis: (1) the phase shifts introduced by focusing and (2) a simpler method, based on dislocation theory, is proposed for the calculation of the radiation coefficients from a discontinuously moving rupture front.     

Drainage spacing regularity on a fault-block: A case study from the eastern Ruahine Range

Abstract:  Recent research has indicated river basin outlets draining linear sections of large, uplifting mountain belts often show a regularity of spacing, transverse to the main structural trend. A morphometric analysis of part of the Ruahine Range, on the North Island was undertaken to test whether drainage regularity may exist in smaller, younger mountain ranges. The ratio, R , of the half-width of the mountain belt, W , and the outlet spacing, S , was used to characterize drainage networks on the eastern side of the range. The spacing ratio for the range of 1.31 is lower than R results from studies of larger mountain belts ( R  = 1.91–2.23). We suggest the cause of this lower ratio is related to eastward migration of the Ruahine drainage divide.     

Shallow-marine sequences as the building blocks of stratigraphy: insights from numerical modelling

Two types of depositional sequences can be defined within the sequence stratigraphic framework: the parasequence and the high‐frequency sequence. Both sequences consist of stacked regressive and transgressive deposits. However, a parasequence forms under conditions of overall sea‐level rise, whereas a high‐frequency sequence forms as the sea level oscillates which results in typical forced regressive deposits during sea‐level fall. Both depositional sequences may develop over comparable temporal (10–100 kyr) and spatial (1–20 km wide and 1–40 m thick) scales. Numerical modelling is used to compare the architecture, preservation potential, internal volumes, bounding surfaces, condensed and expanded sections and facies assemblages of parasequences and high‐frequency sequences. Deposits originating from transgression are less pronounced than their regressive counterparts and consist of either preserved backbarrier deposits or shelf deposits. Shoreface deposits are not preserved during transgression. The second half of the paper evaluates in detail the preservation potential of backbarrier deposits and proposes a mechanism that explains the occurrence of both continuous and discontinuous barrier retreat in terms of varying rates of sea‐level rise and sediment supply. The key to this mechanism is the maximum washover capacity, which plays a part in both barrier shoreline retreat and backbarrier‐lagoonal shoreline retreat. If these two shorelines are not balanced, then the retreat of the coastal system as a whole is discontinuous and in time barrier overstep may take place.     

Palaeoseismicity of the Oquirrh fault, Utah from shallow seismic tomography

The magnitude and frequency of normal-fault palaeoearthquakes are usually determined by trenching studies that ascertain the size and number of colluvial wedges along the fault. Such information can be invaluable in predicting the seismic hazard and potential for a future earthquake in that region. Digging trenches across normal faults, however, is environmentally intrusive, expensive and limited in the penetration depth. To overcome these problems we propose the use of 3-D seismic tomography as a means to identify the shapes and sizes of colluvial wedges along normal faults. As an example,2-D and 3-D seismic surveys were conducted across the Oquirrh fault, Utah with the purpose of imaging the normal-fault structure to a depth of about 10  m. Results show that the 3-D tomogram clearly delineates the fault zone and a colluvial wedge, both of which correlate extremely well with the geological cross-section interpreted from an adjacent trench. The thickness of the colluvial wedge image is used in conjunction with a seismic section to compute an estimate of a 6.8 moment magnitude earthquake for the most recent event on this fault, which is in close agreement with the 7.0 estimate based on a nearby trenching study. These tomographic results demonstrate, for the first time, that seismic imaging methods can be used in some cases to estimate unambiguously the shapes of colluvial wedges and the sizes of prehistoric earthquakes. Thus, seismic tomography has the possibility of providing cheaper, deeper and wider, but less resolved, images of fault systems than the intrusive excavation of trenches across faults.     

A resampling approach to test stress-field uniformity from fault data

Several methods have been proposed to constrain the stress field from fault plane orientations and slip directions within a crustal volume characterized by brittle deformation. All the methods are based on the assumption that the stress field is uniform in the volume considered. If this hypothesis is not checked in advance, however, the methodology may lead to misleading conclusions. In this work, a procedure is defined to check stress-field uniformity by a statistical analysis of the available fault data. Since, in most cases, the statistical features of the uncertainties that affect such data are not well known, a distribution-free approach is proposed. It is based on a simple search algorithm, devoted to selecting stress configurations compatible with available data, combined with a bootstrap resampling approach. The test results are more conservative than the ones so far proposed in the literature. When the test allows stress heterogeneities to be safely excluded, approximate confidence intervals for the principal stress directions can be obtained; otherwise, the level of stress heterogeneity present in the volume under study can be assessed. An application of the proposed procedure to a sample of fault data deduced from seismological data is presented.     

Evolution of the commercial blocks in ancient Beijing city from the street network perspective

The synergistic relationship between urban functions and street networks has always been a focus in the field of urban research and practice. From the perspective of street networks, by adopting space syntax, this study analyzed the deep structural characteristics and potential evolution rules of commercial blocks attached to street networks in different periods, as well as the corresponding economic, political, and cultural characteristics of ancient Beijing city over the past 800 years. By combining these with changes in the street network, we further explained the function mechanism of layout and level adjustment in commercial blocks, and the influence of the street network on commercial blocks in the process of historical change. The main conclusions included the following: (1) The urban centripetal-centrifugal siphon effect: the layout form, topological structure, and traffic mode changes in the street network had corresponding guidance for the layout and hierarchical system of commercial blocks, while the centripetal development of the street network could guide the agglomeration effect of commercial blocks, although centrifugal development caused commercial blocks to display outward evacuation. (2) Stage transformation from mutation node to smooth development: the layout of commercial blocks came to depend on the ability to cross the commuting flow center, which originally relied on the accessibility of transportation nodes as local centers. Changes in traffic modes mainly affected the adjustment of the first-level commercial blocks, which easily led to overall layout mutation. Traffic levels have an obvious positive hierarchical relation with the second- and third-level commercial blocks. (3) The adaptation of traditional commercial blocks to the needs of a modern city: affected by the different emerging times and unevenness of the original commercial foundation, commercial blocks have formed various developmental models that meet the needs of modernization, and reach a balance between cultural continuity and functional adaptation.     

High-resolution structures of the Landers fault zone inferred from aftershock waveform data

High-frequency body waves recorded by a temporary seismic array across the surface rupture trace of the 1992 Landers, California, earthquake were used to determine fault-zone structures down to the seismogenic depth. We first developed a technique to use generalized ray theory to compute synthetic seismograms for arbitrarily oriented tabular low-velocity fault-zone models. We then generated synthetic waveform record sections of a linear array across a vertical fault zone. They show that both arrival times and waveforms of P and S waves vary systematically across the fault due to transmissions and reflections from boundaries of the low-velocity fault zone. The waveform characteristics and arrival-time patterns in the record sections allow us to locate the boundaries of the fault zone and to determine its P - and S -wave velocities independently as well as its depth extent. Therefore, the trade-off between the fault-zone width and velocities can be avoided. Applying the method to the Landers waveform data reveals a low-velocity zone with a width of 270–360 m and a 35–60 per cent reduction in P and S velocities relative to the host rock. The analysis suggests that the low-velocity zone extends to a depth of ∼7 km. The western boundary of the low-velocity zone coincides with the observed main surface rupture trace.     

Invariant expressions for elastic radiation from a fault with prescribed slip

Summary New formulae are obtained for the displacement potentials and displacements due to a point source with moment tensor, and for a fault with prescribed slip. These formulae, unlike previous formulae, are invariant, i.e. they are valid in any coordinate system, not just Cartesian coordinates or orthogonal curvilinear coordinates.
Apart from their invariance, these formulae have other advantages: they are exceedingly simple; the expressions for P -motion are nearly the same as those for S -motion; and the separation of far field motion from final static displacement is automatic.