اثر منبع DEM بر نسبت رتبه بندی هورتون – استرالر مبتنی بر GIUH برای آبخیزها در دو حوضه رودخانه هندوستان

Effect of DEM source on equivalent Horton–Strahler ratio based GIUH for catchments in two Indian river basins

Abstract

Horton–Strahler (H–S) concept has been extensively used for quantification of characteristics of a stream network since several decades. The quantified values are often sensitive to threshold area specified for initiation of streams to demarcate the network, and to the position of outlet of a catchment. This implies that inferences drawn based on derived characteristics for a stream network are likely to be inconsistent, which is undesirable. To address this, a strategy based on self-similarity properties of channel network was proposed recently by Moussa (2009), which involves estimation of equivalent H–S ratios using catchment shape descriptors that are independent of threshold area. This study investigates effectiveness of the strategy on 42 catchments of various sizes in two Indian river basins (Cauvery and Mahanadi). Effect of digital elevation model (DEM) source on estimates of equivalent H–S ratios and characteristics of Geomorphologic Instantaneous Unit Hydrograph (GIUH) derived based on the same are examined by considering SRTM and ASTER DEMs. Results indicate that self-similarity assumptions are valid for the Indian catchments. Comparison of equivalent GIUH derived for each of the catchments based on real channel network with that derived using different DEM sources indicated differences that could be attributed to DEM-based uncertainty associated with estimates of: (i) equivalent H–S ratios that are functions of the self-similarity properties of channel network, and (ii) equivalent length of highest order stream that depends on self-similarity properties and configuration/characteristics of stream network. This uncertainty cannot be ignored in hydrological studies.

1 Introduction

Distributed hydrological models are often used to model runoff generation mechanism in catchments. They require information on local hillslope profiles and stream network, as hillslopes control production of storm water runoff that is transported through the stream network towards the catchment outlet. Conventional practice is to assimilate information related to stream networks through qualitative classification on the basis of (i) basic patterns such as dendritic, parallel, trellis, rectangular, radial, annular, multi-basinal, contorted, or (ii) modified basic patterns such as subdendritic, pinnate, anastomotic, distributary, subparallel, fault trellis and recurved trellis (Zernitz, 1932; Howard, 1967). However, quantitative analysis of natural stream networks in hydrological systems (catchments) is quintessential for modeling their response. In this perspective, a few attempts were made to quantify stream networks. Among those, Horton (1945) was probably the first work that suggested a procedure involving ordering of streams and laws relating the number and length of streams of various orders. Later Strahler (1952, 1957, 1964) suggested modification to Horton’s stream ordering procedure for avoiding some ambiguities. The resulting procedure is widely referred to as Horton–Strahler (H–S) ordering scheme. Horton’s laws when implemented on a stream network quantified using H–S ordering scheme are referred to as H–S laws. The laws include (i) ‘bifurcation ratio’ relating stream numbers corresponding to streams of consecutive orders, (ii) ‘length ratio’ relating lengths of streams of successive orders, (iii) ‘area ratio’ relating areas drained by streams of successive orders. The ratios find use in establishing relations with the fractal nature of the channel network (e.g., Beer and Borgas, 1993; La Barbera and Rosso, 1989; Tarboton, 1996) and in modeling hydrological response from catchments using geomorphological concepts (e.g., Rodríguez-Iturbe and Valdés, 1979; Gupta et al., 1980). The concept of H–S laws has received some criticism (e.g., Scheidegger, 1965, 1968a,b; Moussa, 2009) owing to factors such as (i) inconsistency in classifying a river network into streams of various orders with change in scale of map and support/threshold area for initiation of first order streams and (ii) sensitivity to the position of outlet of catchment. The factor (i) implies that estimated H–S ratios (e.g., bifurcation ratio, area ratio and length ratio) and inferences drawn for a river network based on those ratios are conditional on support area, which is undesirable (Moussa and Bocquillon, 1996; Moussa, 2008a, 2009). To address this issue, researchers are devoting their efforts to arrive at effective strategies that alleviate the effect of support area on H–S ratios. Moussa and Bocquillon (1996) studied effect of threshold area on morphometric and scaling properties of channel network in three catchments (having areas in the range 75–16,250 km2 ) located in southern France, and developed a strategy based on self-similarity properties of channel network to define new catchment shape descriptors that are independent of threshold area. Those descriptors were subsequently used by Moussa (2009) to develop formulations for equivalent Horton–Strahler (H–S) ratios and GIUH that are independent of the threshold/support area chosen for extraction of stream network. Effectiveness of the equivalent H–S ratios and GIUH was demonstrated by Moussa (2009) through application to seven catchments in France whose areas ranged from 738 km2 to 5346 km2 . There is dearth of further attempts to examine potential of the strategy elsewhere in the world. Further, there is a need to examine: (i) potential of the strategy on catchments of relatively larger size, and (ii) sensitivity of estimates of equivalent H–S ratios to source of DEM. In this perspective, investigations are carried out in this study on a large number of catchments having much wider range in their areas located in Cauvery and Mahanadi river basins, India, with the following two objectives: (1) to test methodology of Moussa (2009) on the catchments, and consequently to verify whether hypothesis of ‘‘self-similarity’’ is valid for channel networks in those catchments, and (2) if the hypothesis of ‘‘self-similarity’’ is valid, then compare (i) morphometric properties of the channel networks, and (ii) equivalent H–S ratios and equivalent GIUHs for the catchments obtained corresponding to two different DEMs (ASTER and SRTM). The developed equivalent GIUHs could prove useful to derive unit hydrographs corresponding to desired durations for target sparsely gauged/ungauged locations in the Cauvery and Mahanadi river basins. The derived unit hydrographs could be used to simulate design flood events at the target locations that find use in hydrological design and risk assessment of water resources systems (e.g., Jain et al., 2000). The subsequent part of this paper is structured as follows. First, background information is provided on H–S laws, their scaling properties, procedure for assessment of self-similarity properties of a channel network, and estimation of equivalent H–S ratios. Following this, case study on catchments in Cauvery and Mahanadi river basins is presented. GIUHs constructed based on the conventional H–S ratios and equivalent GIUHs determined based on derived equivalent H–S ratios are compared for each of the catchments. Finally conclusions drawn based on the investigations are provided.

احیای اکولوژیکی در دریای عمیق: دسیدراتا

Ecological restoration in the deep sea: Desiderat

چکیده
1-مقدمه
2- فرصتی برای احیاء در دریای عمیق
3- احیای اکولوژیکی به کار برده شده به دریای عمیق
3-1 خدمات اکوسیستم دریای عمیق و ذی نفعان
3-2 اصول و ویژگی های احیای اکولوژیکی
4- آیا ما بایستی اکوسیستم های دریای عمیق را احیا کنیم؟
4-1 کارگاه احیا: مطالعات موردی و پارامتر های تصمیم
4-2 کارگاه سته: هزینه احیای دریای عمیق
5- نتیجه گیری

چکیده
 عصرحاضر، عصر صنعتی شدن اقیانوسهای عمیق است و سیاست گذاران در حال تدوین چارچوب های نظارتی و حاکمیتی برای مدیریت پایدار منابع دریای عمیق می باشند، در حالی که دانشمندان اطلاعات زیادی را در خصوص ساختار و عملکرد اکولوژیکی بزرگ ترین بیوم بر روی کره زمین هستند. یک موضوعی که در بحث نظارت اقیانوسهای عمیق مغفول مانده است، احیای اکولوژیکی است. در صورتی که فعالیت های موجود در دریای عمیق تداوم یافته و صنایع اقیانوس های عمیق جدید توسعه یابد، نیاز مبرمی به در نظر گرفتن روش های کاهش و ترمیم آسیب های موجود در محیط دریاهای عمیق وجود دارد. به علاوه، میزان آسیب ها نیز باید براورد شود. این مقاله گفتمانی را در خصوص احیای دریای عمیق توسعه داده و دستور العملی در خصوص برنامه ریزی و پیاده سازی پروژه های احیای اکولوژیکی برای اکوسیستم های دریای عمیق ارایه می کند. دو مطالعه موردی احیای دریای عمیق توصیف شد( مرجان های سنگی دریای عمیق در ساحل شرقی اسکاتلند، چاه های هیدروترمال عمیق دریا در حوزه مانوس) و با احیای شوره زار ها در سان فرانسیسکو مقایسه می شوند. برای این مطالعات موردی، مجموعه ای از پارامتر های اقتصادی، اکولوژیکی و فناوری بررسی می شوند. هزینه ها برای سناریو های احیای فرضی در دریای عمیق براورد شده و اولین شواهد نشان می دهد که آن ها دو یا چند برابر در هر هکتار بزرگ تر از هزینه ها برای تلاش های احیا در سیستم های دریایی کم عمق وجود دارد..
1- مقدمه
دریای عمیق در این جا به صورت اقیانوس فراتر از شیب قاره و عمق بیشتر از 200 متر تعریف می شود که یک منطقه حاصلخیز برای صنعتی سازی است. فعالیت های فعلی یا آینده شامل ماهیگیری، دفع زباله، کابل های مربوط به ارتباطات مخابراتی، تحقیقات علمی، توسعه نفت و گاز، زیست شناسی، استخراج مواد معدنی و گردشگری است. از آغاز این قرن گذشته، در حال وقوع و پیش بینی فعالیت ها و تأثیرات انسانی در دریای عمیق بوده است [1-12]. در پاسخ به این تاثیرات تقویتی و بالقوه هم افزایی، خواستار رویکرد پیشگیرانه برای ادامه فعالیت و فعالیت های جدید در دریای عمیق [6]، استفاده از ابزارهای مکان یابی و تطبیقی [7،13،14]، توسعه برنامه های تحقیقاتی برای محاسبه کالاها و خدمات ارائه شده توسط دریای عمیق اکوسیستم [7،15] و ادامه مطالعه اداره اقیانوس ها و حفاظت از محیط زیست دریایی فراتر از صلاحیت های ملی لازم است [16]. علاوه بر این، در مورد نیاز به ایجاد خطوط زیست محیطی، اجماع وجود دارد [8،17] و برای بهبود ابزارها برای پیش بینی، مدیریت و کاهش اثرات انسان شناسی [6،7،18] وجود دارد. مدیریت فضایی دریای عمیق، از جمله ایجاد شبکه های حفاظت از ساحات دریایی، مورد توجه قرار گرفته است [3،11]. تعطیلات منطقه و قوانین حرکت برای شیلات دریایی پایین دریایی توسط سازمان های مدیریت منطقه ای شیلات اجرا شده اند [13،19،20]. سایر ابزارهای حفاظت و مدیریت و اقدامات انجام شده از طریق معاهدات، کنوانسیون ها و توافقنامه های بین المللی عبارتند از شناسایی و حفاظت از سیستم های سازگار با محیط زیست آسیب پذیر و اکولوژیکی یا مناطق زیست شناختی مهم و همچنین یک تماس برای شبکه های مخازن مخلوط [23] برای دریای عمیق سیلاب هیدروترمال و اکوسیستم های عمیق. با این حال، چه چیزی تا به امروز از دست رفته است؟ از دریای عمیق حفاظت، مدیریت و توسعه پایدار، موضوع احیاء است. اکولوژیکی احیاء فرآیند کمک به بازیابی یک اکوسیستم است که تخریب شده، آسیب دیده یا نابود شده است؛ این یک فعالیت عمدی است که مجدد اکولوژیکی فرایندهایی که فعالیت های انسانی را قطع می کنند، را باز می کند [35]. احیاء با هدف بهبود عملکرد زیستی و عملکرد زیست محیطی، سلامتی و سلامت هر دو برای انسان و دیگر موجودات زنده است. اکولوژیکی احیاء به عنوان یک اولویت جهانی در اکوسیستم های زمین و کم عمق شناخته می شود [25-27]. در مقابل، احیاء در دریای عمیق تا به حال توجه زیادی دریافت نکرده است. در کنفرانس 11 (COP11) در اکتبر 2012، کنوانسیون تنوع زیستی (CBD) از 173 کشور عضو کنونی خواسته است تا به شناسایی و احیای حداقل 15 درصد از اکوسیستم های تخریب شده برای هر نوع اکوسیستم در این سیاره کمک کند. 2020، از جمله حفاظت از حداقل 10٪ از مناطق ساحلی و دریایی، به خصوص مناطق دارای اهمیت خاص برای تنوع زیستی و خدمات اکوسیستم می باشد. یک مسئله کلیدی در رابطه با دریای عمیق احیاء به تعهد احزاب مسئول جهت انجام اقداماتی برای تعمیر آسیب های ناشی از فعالیت های تجاری یا دیگری که بر محیط زیست تاثیر می گذارد، تمرکز دارد. صنایع هایی که سیستم های زمینی و ساحلی را تحت تاثیر قرار می دهند مسئولیت آسیب رساندن به منابع طبیعی را بر عهده دارند، باید آسیب هایی را که ایجاد می کنند، اعلام کنند و برای بازیافت زیستگاه پرداخت کنند؛ به همین ترتیب، صنعت باید شامل ارزیابی هزینه های احیا در طرح های خود باشد [28]. دستورالعمل های بین المللی برای مدیریت دریای عمیق ماهیگیری نشان می دهد که این صنعت هنوز مسئولیت احیای اکوسیستم های دریایی را بعد از تأثیر فعالیت های ترال نیست [29]. در مقابل، شواهدی وجود دارد که صنعت استخراج مواد معدنی در دریاها تاثیرات زیست محیطی و نیاز به جبران خسارت را در نظر می گیرد. کد داوطلبانه IMMS برای مدیریت زیست محیطی معدن دریایی که توسط انجمن بین المللی مواد معدنی دریایی [30] تهیه شده است، توصیه می کند که برنامه های استخراج معادن شامل روال های اولیه است که “کمک به بازگشایی استخدام و مهاجرت حیوانات و کمک به مطالعه زیستگاه های قابل مقایسه، پیش از آن، در طی و بعد از عملیات معدن “، از جمله” نظارت بلند مدت در مقیاس های فضایی و زمانی مناسب و تعریف دوره لازم برای اطمینان از اینکه برنامه های تصفیه موثر هستند “. چنین برنامه هایی در بیانیه محیط زیست تاثیر می گذارد که اولین پروژه برای استخراج معادن در یک دریای عمیق سایت است [31]. در این مورد، شرکت درگیر در توسعه، مفهوم سرمایه گذاری در احیاء دریای عمیق را به عنوان مسئولیت شرکتی و جزء مهم فرهنگی نظارت بر محیط زیست شناخته و پذیرفته است.

Abstract

An era of expanding deep-ocean industrialization is before us, with policy makers establishing governance frameworks for sustainable management of deep-sea resources while scientists learn more about the ecological structure and functioning of the largest biome on the planet. Missing from discussion of the stewardship of the deep ocean is ecological restoration. If existing activities in the deep sea continue or are expanded and new deep-ocean industries are developed, there is need to consider what is required to minimize or repair resulting damages to the deep-sea environment. In addition, thought should be given as to how any past damage can be rectified. This paper develops the discourse on deep-sea restoration and offers guidance on planning and implementing ecological restoration projects for deep-sea ecosystems that are already, or are at threat of becoming, degraded, damaged or destroyed. Two deep-sea restoration case studies or scenarios are described (deep-sea stony corals on the Darwin Mounds off the west coast of Scotland, deep-sea hydrothermal vents in Manus Basin, Papua New Guinea) and are contrasted with on-going saltmarsh restoration in San Francisco Bay. For these case studies, a set of socio-economic, ecological, and technological decision parameters that might favor (or not) their restoration are examined. Costs for hypothetical restoration scenarios in the deep sea are estimated and first indications suggest they may be two to three orders of magnitude greater per hectare than costs for restoration efforts in shallow-water marine systems..

1- Introduction

The deep-sea—defined here as ocean beyond the shelf break and depths greater than 200 m—is increasingly recognized as a fertile area for offshore industrialization. Current or future activities include fishing, waste disposal, cable lays associated with telecommunications, scientific research, oil and gas development, bioprospecting, mineral extraction, and tourism. Past, on-going, and anticipated human activities and impacts in the deep sea have been increasingly documented since the start of this century [1–12]. In response to these mounting and potentially synergistic impacts, there have been calls for a precautionary approach to continuing and new activities in the deep sea [6], application of spatial and adaptive management tools [7,13,14], development of research programs to quantify goods and services provided by deep-sea ecosystems [7,15] and continuing study of ocean governance and protection of the marine environment beyond national jurisdiction [16]. In addition, there is a consensus on the need to establish environmental baselines [8,17] and to improve tools to predict, manage and mitigate anthropogenic impacts [6,7,18].

Spatial management of the deep sea—including establishment of networks of marine sanctuaries and protected areas—has received considerable attention [3,11]. Area closures and ‘moveon’ rules for High Seas bottom fisheries have been implemented by Regional Fisheries Management Organizations [13,19,20]. Other conservation and management tools and actions implemented through international treaties, conventions, and agreements include identification and protection of Vulnerable Marine Ecosystems (VMEs; UNGA61/105) [13,20] and Ecologically or Biologically Significant Areas (EBSAs) [21,22], as well as a call for networks of Chemosynthetic Ecosystem Reserves [23] for deepsea hydrothermal vent and seep ecosystems.

What has been missing to date, however, from the deep-sea conservation, management, and sustainable development discourse is the topic of restoration. Ecological restoration is the process of assisting the recovery of an ecosystem that has been degraded, damaged, or destroyed; it is an intentional activity that reinitiates ecological processes that were interrupted by human activities [35]. Restoration aims to recover biodiversity and ecosystem functioning, health, and integrity, both for humans and for other living organisms [24]. Ecological restoration is increasingly recognized as a global priority in terrestrial and shallow-water ecosystems [25–27]. In contrast, restoration in the deep sea has yet to receive much attention. At its 11th Conference of the Parties (COP11) in October 2012, the Convention on Biological Diversity (CBD) called on its 173 Contracting Parties to commit to helping identify and restore at least 15% of degraded ecosystems for every ecosystem type on the planet by 2020, including the conservation of at least 10% of coastal and marine areas, especially areas of particular importance for biodiversity and ecosystem services (CBD COP11 Decision XI/16).

A key issue regarding deep-sea restoration focuses on the obligation of responsible parties to undertake steps to repair damage that result from commercial or other activities that affect the environment. Industries that impact terrestrial and coastal systems are liable for injuries to natural resources, must declare the damage they cause, and pay for habitat recovery; as such, industry needs to include an assessment of restoration costs in their project plans [28]. International guidelines for management of deep-sea fisheries indicate that this industry does not yet take responsibility for restoring seabed ecosystems after impacts of trawling activities [29]. In contrast, there is evidence that the seafloor minerals extraction industry does consider environmental impacts and the need for offsets. The voluntary IMMS Code for Environmental Management of Marine Mining developed by the International Marine Minerals Society [30] recommends that plans for mining include at the outset procedures that “aid in the recruitment, re-establishment and migration of biota and to assist in the study of undisturbed, comparable habitats before, during, and after mining operation”, including “long-term monitoring at suitable spatial and temporal scales and definition of the period necessary to ensure remediation plans are effective”. Such plans are incorporated into the Environmental Impact Statement of the first project to propose mineral extraction at a deep-sea site [31]. In this case, the company involved with the development recognized and embraced the concept of investing in restoration of the deep sea as a corporate responsibility and an important component of a culture of environmental stewardship..

احیای اکولوژیکی در دریای عمیق: دسیدراتا

Ecological restoration in the deep sea: Desiderat

قانون گسترده و تعمیم یافته هوتون برای بررسی الگوریتم های تعبیه شده شبکه های توپولوژیکی رودخانه

Extending generalized Horton laws to test embedding algorithms for topologic river networks

Abstract

River networks in the landscape can be described as topologic rooted trees embedded in a three-dimensional surface. We examine the problem of embedding topologic binary rooted trees (BRTs) by investigating two space-filling embedding procedures: Top-Down, previously developed in the context of random selfsimilar networks (RSNs), and Bottom-Up, a new procedure developed here. We extend the concept of generalized Horton laws to interior sub catchments and create a new set of scaling laws that are used to test the embedding algorithms. We compare the two embedding strategies with respect to the scaling properties of the distribution of accumulated areas Aω and network magnitude Mω for complete order streams ω. The Bottom-Up procedure preserves the equality of distributions Aw/E[Aw]=dMw/E[Mw] ; a feature observed in real basins. The Top-Down embedded networks fail to preserve this equality because of strong correlations of tile areas in the final tessellation. We conclude that the presence or absence of this equality is a useful test to diagnose river network models that describe the topology/geometry of natural drainage systems. We present some examples of applying the embedding algorithms to self similar trees (SSTs) and to RSNs. Finally, a technique is presented to map the resulting tiled region into a three-dimensional surface that corresponds to a landscape drained by the chosen network. Our results are a significant first step toward the goal of creating realistic embedded topologic trees, which are also required for the study of peak flow scaling in river networks in the presence of spatially variable rainfall and flood-generating processes.

1 Introduction

The study of river network topology is an active area of research in geomorphology (Meakin et al., 1991; Maritan et al., 1996; Dodds and Rothman, 2000; Molnar, 2005). Several mathematical models have been introduced in the literature to describe, using simple principles, the complex network topology of river networks (Tokunaga, 1966; Scheidegger, 1967; Shreve, 1967; Veitzer and Gupta, 2000). These topologic models have helped put their mathematical foundations on a firm footing, and many of them have been successful in explaining major geomorphic features observed in natural river networks. Topologic river network models, by definition, only describe how nodes of the network are connected with each other. These models do not provide a description of the spatial embedding of the topology in the three-dimensional landscape of a river basin. As a result, they are unsuitable to study the interaction between spatially correlated rainfall and runoff generation processes and its impact on the transport of flows through a river network. Some network models, such as optimal channel networks (Rigon et al., 1993; Maritan et al., 1996; Rinaldo et al., 2006), Gibbsian networks (Troutman and Karlinger, 1994, 1998), and networks generated by random walks (Leopold and Langbein, 1962; Meakin et al., 1991) include explicitly the spatial geometry of networks by embedding them on a twodimensional lattice. However, two disadvantages of lattice models relative to topologic models are, first, that generation of lattice models is typically more computationally demanding and, second, that it is more difficult to obtain analytic results for lattice models. Self-similar river network models play a fundamental role in understanding observed scaling in the magnitudes of peak flows (Gupta et al., 2010). For example, Gupta et al. (1996), Menabde and Sivapalan (2001), and Troutman and Over (2001) considered idealized self-similar networks embedded in a two-dimensional space to understand how the interaction between multifractal rainfall and self-similar river networks determines the magnitude and scaling characteristics of peak flows. Gupta et al. (1996) assumed that rainfall follows spatially correlated beta random cascade that is deposited on a Peano channel network. Likewise, Menabde and Sivapalan (2001) assumed that rainfall follows a log-normal cascade on a Mandelbrot–Viscek network. Troutman and Over (2001) made more general assumptions regarding spatial variability of rainfall and self-similar network structure to study scaling in peak flows. In all these studies, however, the spatial cascade structure of rainfall was assumed to be aligned with the topology of the network. Mantilla (2007) conducted simulation of scaling in peak flows on RSNs. In order to generalize this approach to spatially correlated rainfall embedding realistic1 topological networks in a three-dimensional space is necessary. The embedding process is schematically illustrated in Fig. 1 for a very simple network topology. We begin with, first, a network for which it is assumed that only the connectivity structure (a topologic tree) is known (Fig. 1a) and, second, a two-dimensional region with a fixed shape (Fig. 1b) into which the network will be embedded. The goal is to develop an algorithm, which yields the embedded network, as shown in Fig. 1c. The embedding process, as defined in this paper, must therefore fulfill two minimal requirements: (i) it must be space filling, which means that every point in the given region has to be assigned to, or drain into, a unique link of the binary rooted tree (BRT); the process is known as tiling, and (ii) the topologic structure of the network needs to be preserved. These two minimal requirements of the embedding process do not uniquely define an algorithm for obtaining the embedded network, so clearly any proposed procedure needs to be further tailored so as to preserve, to the extent possible, important geometrical properties of real drainage basins. Examination of Fig. 1c makes clear that the distribution of the size of areas assigned to the individual links, and the resulting size of accumulated areas of sub basins is a fundamental characteristic of embedded networks. It is the focus here. A key contribution of this paper is to apply generalized Horton’s laws (Peckham and Gupta, 1999; Veitzer and Gupta, 2000; Troutman, 2005) to obtain a sensitive measure of how well the area distribution of embedded networks conforms with behavior observed in real networks. Generalized Horton’s laws are based on the idea of statistical selfsimilarity (SSS) of the distribution of basin variables defined for streams of different Strahler orders (Strahler, 1957). Strahler ordering of streams is determined by the following rules: first-order streams are those with no upstream inflows and the stream immediately downstream from the junction of two streams of order ω1 and ω2 is ω1+ 1 if ω1=ω2 and the maximum between ω1 and ω2 if ω1≠ω2. The two basin variables of interest here are drainage area, A, and network magnitude, M, defined as the number of first-order streams in the network. A comparison of the statistical distribution of these two variables provides a test of embedding algorithms. In Section 2 we provide some background on the problem of embedding BRTs and also present data from real networks. In Section 3, the concept of basin decomposition into hillslopes is used to introduce an extension of Horton’s laws (Horton, 1945; Strahler, 1957). It is used to test the properties of the tessellations generated by our embedding strategies. In Section 4 we present two embedding strategies. First, in Section 4.1 we present the embedding procedure that Veitzer (1999) developed that we call the top-down embedding (TDE) algorithm. We explain the deficiencies of this algorithm with respect to the generalized Horton’s laws. In Section 4.2 the bottom-up embedding (BUE) algorithm is described, and advantages of using this approach are explained. Some examples of the BUE are presented using self similar trees (SSTs) and RSNs. In Section 5 we develop a method to generate three-dimensional landscapes using the tiled region obtained by the BUE. Finally, in Section 6 our conclusions are presented and areas of future research are highlighted.

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Role of the River Danube in the Spatial Development of Central- and South-East Europe

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Role of the River Danube in the Spatial Development of Central- and South-East Europe

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