Application of an S‐shaped curve model to the temporal development of tafoni of salt‐weathering origin

To describe temporal change in tafone development, an S‐shaped curve equation is proposed: Z = Z_c [1 ? (n + 1) exp (? β t ) + n exp (? (1 + 1/n) β t )] , where Z is observed tafone depth, Z_c is ultimate tafone depth, t is time, and n and β are constants. The applicability of this model is examined using tafone data selected from seven sites, which are categorized into three different salt‐weathering environments: a spray/splash‐dominant (occasionally wave‐affected) supra‐tidal zone, aerosol‐affected coastal regions, and inland desert areas. The results indicate that the equation can well describe tafone development in each of these environments. An investigation based on the values of n and β, determined through a best fit of the equation to the data, suggests that n characterizes site‐specific environmental conditions and β reflects the magnitude of factors controlling the recession mechanism of tafone surfaces. It is found that (1) the maximum rate of tafone growth dramatically decreases from supra‐tidal, through coastal, to desert environments, and (2) the growing mode of tafoni is different depending on the environmental settings. The erosional force to facilitate the development of tafoni at supra‐tidal sites is estimated to be about 400 times greater than that in the general coastal area. Copyright © 2011 John Wiley & Sons, Ltd.     

Rates of tafoni weathering on uplifted shore platforms in Nojima-Zaki,Boso Peninsula,Japan

Many tafoni occur on the faces of marine cliffs on three uplifted shore platforms with different altitudes and with known ages of emergence. The mean value of the ten largest depths of tafoni, D (cm), was obtained and the period of their formation, t (years), was estimated. The relationship between the two variables was found to be D = 20·3 × (1?e^?0.005t).     

Recent seismicity in Northeast India and its adjoining region

Recent seismicity in the northeast India and its adjoining region exhibits different earthquake mechanisms – predominantly thrust faulting on the eastern boundary, normal faulting in the upper Himalaya, and strike slip in the remaining areas. A homogenized catalogue in moment magnitude, M _W, covering a period from 1906 to 2006 is derived from International Seismological Center (ISC) catalogue, and Global Centroid Moment Tensor (GCMT) database. Owing to significant and stable earthquake recordings as seen from 1964 onwards, the seismicity in the region is analyzed for the period with spatial distribution of magnitude of completeness m _t, b value, a value, and correlation fractal dimension D _C. The estimated value of m _t is found to vary between 4.0 and 4.8. The a value is seen to vary from 4.47 to 8.59 while b value ranges from 0.61 to 1.36. Thrust zones are seen to exhibit predominantly lower b value distribution while strike-slip and normal faulting regimes are associated with moderate to higher b value distribution. D _C is found to vary from 0.70 to 1.66. Although the correlation between spatial distribution of b value and D _C is seen predominantly negative, positive correlations can also be observed in some parts of this territory. A major observation is the strikingly negative correlation with low b value in the eastern boundary thrust region implying a possible case of extending asperity. Incidentally, application of box counting method on fault segments of the study region indicates comparatively higher fractal dimension, D, suggesting an inclination towards a planar geometrical coverage in the 2D spatial extent. Finally, four broad seismic source zones are demarcated based on the estimated spatial seismicity patterns in collaboration with the underlying active fault networks. The present work appraises the seismicity scenario in fulfillment of a basic groundwork for seismic hazard assessment in this earthquake province of the country.     

Estimation of seismic demands on isolators in asymmetric buildings using non‐linear analysis

A procedure based on rigorous non‐linear analysis is presented that estimates the peak deformation among all isolators in an asymmetric building due to strong ground motion. The governing equations are reduced to a form such that the median normalized deformation due to an ensemble of ground motions with given corner period T_d depends primarily on four global parameters of the isolation system: the isolation period T_b, the normalized strength η, the torsional‐to‐lateral frequency ratio Ω_θ, and the normalized stiffness eccentricity e_b/r. The median ratio of the deformations of the asymmetric and corresponding symmetric systems is shown to depend only weakly on T_b, η, and Ω_θ, but increases with e_b/r. The equation developed to estimate the largest ratio among all isolators depends only on the stiffness eccentricity and the distance from the center of mass to the outlying isolator. This equation, multiplied by an earlier equation for the deformation of the corresponding symmetric system, provides a design equation to estimate the deformations of asymmetric systems. This design equation conservatively estimates the peak deformation among all isolators, but is generally within 10% of the ‘exact’ value. Relative to the non‐linear procedure presented, the peak isolator deformation is shown to be significantly underestimated by the U.S. building code procedures. Copyright © 2003 John Wiley & Sons, Ltd.     

Mining soil databases for landscape‐scale patterns in the abundance and size distribution of hillslope rock fragments

Landscape‐scale variation in rock fragments on soil‐mantled hillslopes is poorly understood, despite the potential importance of rock fragments in soil weathering and coarse sediment supply to river networks. We explored the utility of soil survey databases for data mining, with the goals of identifying landscape‐scale patterns in the abundance and size distribution of rock fragments (diameter D > 2 mm) and potential controls on grain size production. We focus on data from three regions: the Hawaiian Islands, and the Sierra Nevada and Cascade Mountains, where elevation transects span a range of environmental conditions. We selected pedons from pits dug on hillslopes with active soil production and transport. For the 27 pedons selected, we constructed depth‐averaged grain size distributions and calculated the mass fraction of rock fragments (F_RF) and the median rock fragment grain size (D_50RF). We also categorized as bimodal, size distributions with a clear ‘breakpoint’ between fine and coarse modes. Several strong patterns emerge from the data. We find rock fragments in 85% of the pedons, primarily in distinct coarse modes within bimodal size distributions. Values of F_RF and D_50RF are strongly correlated, although the best‐fit power law scaling between F_RF and D_50RF differs between the warmer Hawaiian, and colder Sierra Nevada and Cascade Mountain sites. We also find a regional contrast in the variation in F_RF with elevation; F_RF declines with elevation in Hawaii, but increases in the mainland sites. Although this contrast could be an artifact of variable lithology, precipitation may influence many patterns in the data. Lower mean‐annual precipitation correlates with higher F_RF, dominantly bimodal distributions and surface enrichment in the vertical distribution of rock fragments. These observations may be useful in refining models of coarse sediment supply to rivers, and suggest opportunities for future work to test mechanistic hypotheses for rock fragment production on soil‐mantled hillslopes. Copyright © 2011 John Wiley & Sons, Ltd.     

Clarification of regional and local in situ stresses using the compact conical-ended borehole overcoring technique and numerical analysis

Abstract Stress measurement is performed to estimate the states of in situ rock stress at the Torigata open‐pit limestone mine in Japan using the compact conical‐ended borehole overcoring (CCBO) technique. A set of back and forward analyses are then carried out to evaluate the states of regional and local in situ rock stresses and the mine‐induced rock slope stability using a 3‐D finite element model. The maximum horizontal local in situ rock stress measured by the CCBO technique acts in the northeast–southwest direction. The horizontal regional tectonic stresses obtained by the back analysis are in good agreement with those of the horizontal local in situ rock stress measured by the CCBO technique. However, the horizontal regional tectonic stress is more compressive than the horizontal local in situ rock stress. This is because the horizontal regional stress due to gravity is not considered in the back‐analyzed horizontal regional tectonic stress, but it is included in the local in situ rock stress measured by the CCBO technique. The local stress obtained by the forward analysis, especially its horizontal components, is in good agreement with the horizontal local in situ rock stress measured by the CCBO technique, and the magnitude of the vertical normal stress increases more rapidly than those of the horizontal normal stresses with depth. As a result, the ratio of the horizontal normal stress to the vertical normal stress is largest at the nearest excavation level and decreases with depth. This means that the stress field within the mine‐induced rock slope is affected by the horizontal components of the local in situ rock stress.     

Curie Point Depths of Albania Inferred from Ground Total Field Magnetic Data

Ground total magnetic field data of Albania were used to produce estimates of the Curie point isotherm. The strategy followed was to estimate the depth to the bottom of the deepest magnetic sources. Firstly, the average depth to the top of the deepest crustal block, z_t, was computed by linear fitting to the second lowest-frequency segment of the azimuthally averaged power spectrum of the total magnetic field data. Then, the depth to the centroid of the deepest crustal block, z₀, was computed by linear fitting to the lowest-frequency segment of the azimuthally averaged power spectrum of a distribution of magnetic dipoles. Finally, the depth to the bottom, the inferred Curie point depth, z_b, was calculated from z_b=2z₀−z_t. Curie depth estimates for Albania vary from about 17 to 25.5 km (below sea level). These results are consistent with the depths inferred by extrapolating geothermal gradient and heat-flow values, suggesting that the Curie point depth analysis is useful to estimate the regional thermal structure. It also suggests that the approach was valid and that ground total magnetic field data can be used for this purpose.     

Dynamic damage and its physical meaning as a seismicity status parameter

Whentherearemicroormacrofissuresinrock,itisconsideredthattherockseveresdamage.Becausethesefissuresarenotsimplifiedintooneormorecracks.Intheconditionoffatigueandofcreepstrain,manymicrofissurescandevelopintothemacrofractureintherockwhere\/ecallitadamagedregion.Recentlythemechanicsfunctionofadamagedregionisconsideredacontinuedvariancefieldoradamagedfield.Withthestatisticalmechanicsmethod,thedamagedrockresearchescanmakeusseethewholeprocessofbreaking.lDynamicdamageThedamageisaresultfrommanyfissure…     

Equilibrium bank geometry and the width of shallow sandbed streams

Cross-sections of 16 straight sandbed streams in Minnesota, Iowa, and Nebraska were surveyed. Two stratigraphic horizons were found in the banks at each site, an upper cohesive unit usually composed of silt and clay and a lower unit composed of sand. Bank erosion on these rivers occurs when the upper cohesive unit is undercut by scour at bends. The overhanging cohesive block fails by beam or cantilever failure. As upper bank failure is a direct result of undercutting, the stability and rate of retreat of the bank are largely determined by erosion of the sandy part of the bank. The cohesive layer has little influence on bank retreat and width adjustment on the rivers studied here. A quantitative lateral sediment transport model developed by Parker (1978a) is used to calculate the steady-state geometry of the sandy part of the bank. Results are obtained for the shape, length, and height of the sandy part of the bank. The model predicts the length of the bank fairly well, and the theoretical equation for the height of the bank is of the correct form. The model, however, overestimates the slope of the bank. The height of the sandy part of the bank (D_b) is approximately equal to the depth of the mean annual flow. Since D_b is determined by the lateral sediment transport model, the width (W) may be obtained from the equation of continuity (Q = WD_bV), published flow (Q) data, and a resistance equation for the mean velocity, V. The calculated widths are similar to those measured in the field.     

A multi‐method investigation of temperature,moisture and salt dynamics in tafoni (Tafraoute,Morocco)

Despite numerous investigations and theoretical models, tafoni weathering is still not fully understood largely because of limited data available on temperature, moisture and salt regimes. We investigated tafoni developed in granite in the Tafraoute region, Morocco, through an exploratory, two‐week multi‐method field campaign. Temperatures were measured with iButtons and by means of infrared thermography; moisture distribution and progress were captured with handheld moisture sensors and with drilled‐in iButtons. Salts were analysed in drill dust samples from different positions and rock depths. The results derived from very different techniques mutually support one another. Salts and moisture are concentrated near the base of the investigated tafoni, probably due to a saturated pore water body around the base of rock tors. Salts are accumulated close to the rock surface in tafoni, but not on the surrounding rock surfaces. A clear correlation was found between moisture and salt contents. Within a tafone, areas of higher humidity also display increased salt concentration near the surface. The temperature/humidity records allied with ionic analyses suggest that sodium sulfate dominates and is likely to undergo frequent phase changes from thenardite to mirabilite and vice versa. Two pathways of salt transport in and around tafoni are assumed based on the data: infiltration with rainfall on the top and around tors and boulders, and capillary rise from saturated pore water bodies to the surface. Copyright © 2015 John Wiley & Sons, Ltd.     

Flow threads in surface run‐off: implications for the assessment of flow properties and friction coefficients in soil erosion and hydraulics investigations

Soil surface microtopography produces non‐uniform surface run‐off, in which narrow threads of relatively deep and fast ?ow move within broader, shallower, slower‐moving regions. This kind of ?ow is probably widespread, given that microtopography is itself common. Methods used to record the properties of surface run‐off include grid‐ or transect‐based depth observations, with a single mean ?ow speed derived by calculation from V = Q/WD, and the use of dye timing to estimate velocity, with an effective mean depth calculated from D = Q/WV. Because these methods allow only single, ?ow‐?eld mean values to be derived for V or D, neither is well suited to non‐uniform ?ows. The use of depth data to derive a ?ow‐?eld mean V furthermore implicitly applies area weighting to the depth data; likewise, the use of dye speeds for V inherently overestimates mean V because dye dominantly follows the faster ?ow threads. The associated errors in derived parameters such as friction coef?cients are not readily quanti?ed and appear not to have been addressed previously. New ?eld experiments made on untilled soil surfaces in arid western NSW, Australia, explore these circumstances and the implications for deriving meaningful measures of ?ow properties, including friction coef?cients. On surfaces deliberately chosen for their very subtle microtopography, average thread velocities are shown to be commonly 2·5 times greater than the ?ow‐?eld mean, and locally 6–7 times greater. On the other hand, non‐thread ?ow speeds lie below the ?ow‐?eld mean, on average reaching only 84 per cent of this value, and often considerably less. Flow‐?eld means conceal the existence of regions of the ?ow ?eld whose properties are statistically distinct. Results con?rm that a reliance on ?ow‐?eld average depths yields estimates of friction coef?cients that are biased toward the shallower, high‐roughness parts of the ?ow, while if dye speeds are relied upon the results are biased toward the deeper, smoother threads of ?ow. A new approach to the evaluation of friction coef?cients in non‐uniform ?ows is advanced, involving the determination of separate coef?cients for threads and non‐thread zones of the ?ow ?eld. In contrast, ?ow‐?eld friction coef?cients as they are customarily derived in run‐off plot experiments subsume these distinct coef?cients in proportions that are generally unknown. The value of such coef?cients is therefore questionable. Copyright © 2004 John Wiley & Sons, Ltd.     

Ray equation migration of wide-angle reflections in Dabie orogenic zone

Ray equation migration of wide-angle reflections in 2-D medium is one kind of Kirchhoff prestack depth migration method. Based on ray theory, this method can be used extensively in 2-D inhomogeneous medium, and shows its advantage in wide-angle reflection study. After calculating ray fields, we can get the wave fields of sources and receivers by interpolation, and the intensity (or amplitude) A _p corresponding to the depth point P at travel-time t _P =t _SP +t _PR can be found in seismograms. Ray equation migration is completed by transforming A _P according to the specific relation, migrating it to the depth point, then calculating the ray fields of lots of sources and receivers in the same way and finally stacking the outcomes. Numerical calculation has yielded satisfactory results. Data processing of the Zhuangmu-Yuexi-Huangmei-Zhanggondu profile passing through the Dabie orogenic zone provides a structural form of M-discontinuity along the profile. The result shows that the high pressure metamorphic rock zone of South Dabie rock mass is related to the uplifting of M-discontinuity here. "Mountain root" exists under North Dabie rock mass, which conforms with gravity isostasy theory. The Xiaotian-Mozitan fault is a suture zone as a result of the collision of North China plate with Yangtze plate, and extends through M-discontinuity. The abyssal fault near Liu’an is the extended eastern section of the Luonan-Minggang fault, which is also confirmed here. Contribution No. RCEG 96012, Research Center of Exploration Geophysics, SSB, China. This project was supported by the National Natural Science Foundation of China and the Chinese Joint Seismological Science Foundation.     

Effects of rock fragment size and cover on overland flow hydraulics,local turbulence and sediment yield on an erodible soil surface

The interactions between overland flow hydraulics and sediment yield were studied in flume experiments on erodible soil surfaces covered by rock fragments. The high erodibility of a non-cohesive fine sediment (D₅₀ + 0·09mm) permitted the effects of local turbulence and scour on sediment yield to be examined. Overland flow hydraulics and sediment yield were compared for experiments with pebble (D₅₀ + 1·5cm) and cobble (D₅₀ + 8·6cm) rock fragment covers. Cover percentages range from 0 to 99 per cent. Rock fragment size strongly affects the relations between flow hydraulics and rock fragment cover. For pebbles spatially-averaged hydraulic parameters (flow velocity, flow depth, effective flow width, unit discharge, total shear stress, Darcy-Weisbach friction factor, percentage grain friction and grain shear stress) vary most rapidly within cover percentages at low covers (power functions). In contrast, for cobbles these parameters vary most rapidly within cover percentages at high covers (exponential functions). As the type of the function that describes the relation between flow hydraulics and cover percentage can be deduced from the ratio of rock fragment height to flow depth, the continuity equation can be employed to determine the actual coefficients of the functions, provided the regression of one hydraulic parameter (e.g. flow velocity) with cover percentage is known and a good estimate exists for two values of another hydraulic variable for a low and a high cover percentage. The variation of sediment yield with cover percentage is also strongly dependent on rock fragment size, but neither the convex-upward relation for pebbles, nor the positive relation for cobbles can be solely attributed to the spatially averaged hydraulics of sheet-flow. Rock fragments induce local turbulence that leads to scour hole development on the stoss side of the rock fragments while deposition commonly occurs in the wake. This local scour and deposition substantially affects sediment yield. However, scour dimensions cannot be predicted by spatially averaged flow hydraulics. An adjustment of existing scour formulas that predict scour around bridge piers is suggested. Sediment yield from non-cohesive soils might then be estimated by a combination of sediment transport and scour formulas.     

Transport and dispersal of organic debris (peat blocks) in upland fluvial systems

This paper assesses the mechanisms and pathways by which peat blocks are eroded and transported in upland fluvial systems. Observations and experiments from the north Pennines (UK) have been carried out on two contrasting river systems. Mapping of peat block distributions and appraisal of reach‐based sediment budgets clearly demonstrates that macro‐size peat is an important stream load component. In small streams block sizes can approximate the channel width and much of the peat is transported overbank. Local ‘peat jams’ and associated mineral deposition may provide an important component of channel storage. In larger systems peat blocks rapidly move down‐channel and undergo frequent exchanges between bed and bank storage. Results of peat block tracing using painted blocks indicate that once submerged, blocks of all sizes are easily transported and blocks break down rapidly by abrasion. Vegetation and bars play an important role in trapping mobile peat. In smaller streams large block transport is limited by channel jams. Smaller blocks are transported overbank but exhibit little evidence of downstream fining. In larger rivers peat blocks are more actively sorted and show downstream reduction in size from source. A simple model relating peat block diameter (D_p) to average flow depth (d) suggests three limiting transport conditions: flotation (D_p < d), rolling (d < D_p > d/2) and deposition (D_p > d/2). Experiments demonstrate that peat block transport occurs largely by rolling and floating and the transport mechanism is probably controlled by relative flow depth (d/D_p ratio). Transport velocity varies with transport mechanism (rolling is the slowest mode) and transport lengths increase as flow depth increases. Abrasion rates vary with the transport mechanism. Rolling produces greater abrasion rates and more rounded blocks. Abrasion rates vary from 0 to 10 g m^?1 for blocks ranging in mass from 10 to 6000 g. Copyright © 2001 John Wiley & Sons, Ltd.     

Remote sensing of volumetric storage changes in lakes

Three‐dimensional remote sensing promises a giant leap forward for surface‐water hydrology in much the same way that radar altimetry transformed physical oceanography. However, the complex geometries of small terrestrial water bodies introduce difficulties, particularly with respect to trade‐offs between changing water depth and inundation area. We use in situ measurements of water‐surface stage (ΔH/dt) and remotely‐sensed area (A) to compute time varying storage changes (ΔS) in nine lakes of the Peace‐Athabasca Delta, Canada. Despite their identical geomorphic setting, regression slopes between ΔH and A vary significantly between lakes, primarily from a predictable ‘area‐effect’ but also small bathymetric variations between basins. On average, lateral contraction/expansion (versus stage adjustment) contributes as little as 7% (versus 93%) to as much as 76% (versus 24%) of overall storage change ΔS. We conclude that both surface‐area and ΔH/dt, rather than just either alone, must be measured to confidently estimate ΔS from space. Copyright © 2009 John Wiley & Sons, Ltd.     

Controls on at‐a‐station hydraulic geometry in steep headwater streams,Colorado, USA

Detailed hydraulic measurements were made in nine step‐pool, five cascade and one plane‐bed reach in Fraser Experimental Forest, Colorado to better understand at‐a‐station hydraulic geometry (AHG) relations in these channel types. Average values for AHG exponents, m (0·49), f (0·39), and b (0·16), were well within the range found by other researchers working in steep gradient channels. A principal component analysis (PCA) was used to compare the combined variations in all three exponents against five potential control variables: wood, D₈₄, grain‐size distribution (σ), coefficient of variation of pool volume, average roughness‐area (projected wetted area) and bed gradient. The gradient and average roughness‐area were found to be significantly related to the PCA axis scores, indicating that both driving and resisting forces influence the rates of change of velocity, depth and width with discharge. Further analysis of the exponents showed that reaches with m > b + f are most likely dominated by grain resistance and reaches below this value (m < b + f) are dominated by form resistance. Copyright © 2010 John Wiley & Sons, Ltd.     

Origin and development of tafoni in Tunnel Spring Tuff,Crystal Peak,Utah, USA

Bedding‐parallel tafoni are well developed over much of the surface of the Tunnel Spring Tuff (Oligocene) exposed in 300‐m‐high Crystal Peak, an inselberg. The Tunnel Spring Tuff is a crudely stratified, non‐welded rhyolite ash‐flow tuff with > 30 per cent porosity. Clasts of Palaeozoic dolomite, limestone and quartzite make up 10 per cent of the tuff. The tafoni are remarkable because of their size (up to 20 m wide but rarely wider than 4 m), shape of the openings (spherical, arch‐like or crescent‐shaped) and abundance (up to 50 per cent of an outcrop face). They are actively forming today. Calcite is abundant (10 to 40 per cent by weight) in tafoni as an efflorescence in spalling flakes of tuff on their roofs and walls. Halite and gypsum generally make up less than 0·01 per cent of the efflorescence. The absence of corroded quartz and feldspar grains in spall fragments indicates that chemical weathering is unimportant in development of the tafoni. Calcite, aragonite, halite and gypsum dust from modern salt pans less than 20 km from Crystal Peak are potential sources of salt in the tuff, but the prevailing winds are in the wrong direction for significant amounts of these evaporite minerals to reach the inselberg. Calcite is the only evaporite mineral present in the tafoni in more than trace amounts, and this mineral is readily available within the tuff itself as a result of rock weathering. We propose that meteoric water containing carbonic acid infiltrates the tuff, dissolves carbonate clasts, and migrates to the steep flanks (>20°) of the peak through abundant megapores and micropores. There it evaporates and precipitates calcite. Crystallization pressure spalls off grains and sheets as the physical manifestation of salt weathering. The quasi‐uniform spacing of tafoni suggests that a self‐organization process is active in the water flow. Copyright © 2000 John Wiley & Sons, Ltd.     

Control strategy for variable damping element considering near‐future excitation influence

A control strategy is proposed for variable damping elements (VDEs) used together with auxiliary stiffness elements (ASEs) that compose a time‐varying non‐linear Maxwell (NMW) element, considering near‐future excitation influence. The strategy first composes a state equation for the structural dynamics and the mechanical balance in the NMW elements. Next, it establishes a cost function for estimating future responses by the weighted quadratic norms of the state vector, the controlled force and the VDEs' damping coefficients. Then, the Euler equations for the optimum values are introduced, and also approximated by the first‐order terms under the autoregressive (AR) model of excitation information. Thus, at each moment t_k, the strategy conducts the following steps: (1) identify the obtained seismic excitation information to an AR model, and convert it to a state equation; and (2) determine VDEs' damping coefficients under the initial conditions at t_k and the final state at t_k+L, using the first‐order approximation of the Euler equations. The control effects are examined by numerical experiments. Copyright © 2000 John Wiley & Sons, Ltd.     

Determination of Forchheimer equation coefficients a and b

This study focuses on the determination of the Forchheimer equation coefficients a and b for non‐Darcian flow in porous media. Original theoretical equations are evaluated and empirical relations are proposed based on an investigation of available data in the literature. The validity of these equations is checked using existing experimental data, and their accuracy versus existing approaches is studied. On the basis of this analysis, some insight into the physical background of the phenomenon is also provided. The dependence of the coefficients a and b on the Reynolds number is also detected, and potential future research areas, e.g. investigation of inertial effects for consolidated porous media, are pointed out. Copyright © 2006 John Wiley & Sons, Ltd.     

Theoretical investigation of the time variation of drainage density

A theoretical equation was developed to express the time variation of drainage density in a basin or geomorphic surface: D_i(t, T) is the drainage density at time T on the i-th basin or geomorphic surface, which was formed at time t; β(τ) is a factor related to the erosional force causing the development of the rivers of the basin or surface at time τ; δ_i is the maximum drainage density; and D_i is the initial drainage density on the i-th geomorphic surface or basin. The equation is based on the assumption that the drainage density increases with time until it reaches a specific upper limit δ_i(t)), the maximum drainage density, which is related to certain physical properties of the basin. The equations for various dated basins or geomorphic surfaces can be combined into one modified equation if the same relative erosional forces have acted on those basins or surfaces (β(t) = β(t) and if the basins or surfaces have the same physical properties δ_i(t) = δ_i(t), (D_i = D₀). The application of this equation to coastal terraces and glacial tills shows that the model is compatible with observed drainage densities on various dated basins or surfaces.