Empirical scaling relationships between known deepwater siliciclastic submarine fan systems and their linked drainage basins have previously been established for modern to submodern depositional systems and in a few ancient, small-scale basins. Comprehensive mapping in the subsurface Gulf of Mexico basin and geological mapping of the North American drainage network facilitates a more rigorous test of scaling relationships in a continental-size system with multiple mountain source terranes, rivers, deltas, slopes, and abyssal plain fan systems formed over 65 m.y. of geologic time. An immense database of drilled wells and high-quality industry seismic data in this prolific hydrocarbon basin provide the independent measure of deepwater fan distribution and dimensions necessary to test source-to-sink system scaling relationships.Analysis of over 40 documented deepwater fan and apron systems in the Gulf of Mexico, ranging in age from Paleocene to Pleistocene, reveals that submarine-fan system scales vary predictably with catchment length and area. All fan system run-out lengths, as measured from shelf margin to mapped fan termination, fall in a range of 10%-50% of the drainage basin length, and most are comparable in scale to large (Mississippi River-scale) systems although some smaller fans are present (e.g., Oligocene Rio Bravo system). For larger systems such as those of the Paleocene Wilcox depositional episodes, fan runout lengths generally fall in the range of 10%-25% of the longest river length. Submarine fan widths, mapped from both seismic reflection data and well control, appear to scale with fan run-out lengths, though with a lower correlation (R 2 = 0.40) probably due to uncertainty in mapping fan width in some subsalt settings. Catchment area has a high correlation (R 2 = 0.85) with river length, suggesting that fluvial discharge and sediment flux may be primary drivers of ancient fan size.Validation of these first-order source-to-sink scaling relationships provides a predictive tool in frontier basins with less data. Application to less-constrained early Eocene fan systems of the southern Gulf of Mexico demonstrates the utility for exploration as well as paleogeographic reconstructions of ancient drainage systems. This approach has considerable utility in estimating dimensions of known but poorly constrained submarine fans in the subsurface or exposed in outcrop.
In past decades, numerous studies have focused on the alluvial sedimentary record of basin fill. Paleo-drainage basin characteristics, such as drainage area or axial river length, have received little attention, mostly because the paleo-drainage system underwent erosion or bypass, and its record is commonly modified and overprinted by subsequent tectonism or erosional processes. In this work, we estimate the drainage areas of early Miocene systems in the Gulf of Mexico basin by using scaling relationships between drainage area and river channel dimensions (e.g., depth) developed in source-to-sink studies. Channel-belt thickness was used to estimate channel depth and was measured from numerous geophysical well logs. Both lower channel-belt thickness and bankfull thickness were measured to estimate the paleo-water depth at low and bankfull stages. Previous paleogeographic reconstruction using detrital zircon and petrographic provenance analysis and continental geomorphic synthesis constrains independent estimates of drainage basin extent. Comparison of results generated by the two independent approaches indicates that drainage basin areas predicted from channel-belt thickness are reasonable and suggests that bankfull thickness correlates best with drainage basin area. The channel bankfull thickness also correlates with reconstructed submarine fan dimension. This work demonstrates application to the deep-time stratigraphic archive, where records of drainage basin characteristics are commonly modified or lost. Recent advances in source-to-sink (S2S) analysis provide a method for calculating paleo-drainage basin area by examining the dimensions of sedi-mentary strata in the basin. The S2S analysis considers whole-sediment erosional-depositional processes as a contemporaneous and genetically linked system (Allen, 2008a; Sømme et al., 2009a, 2009b; Fig. 1). Among the components of the S2S system, tectonics and climate, acting on the drainage basin area, determine sediment supply and water discharge (Fig. 2A; Matthai, 1990; Milliman and Syvitski, 1992; Syvitski and Milliman, 2007; Allen, 2008b; Sømme et al., 2009a). Although fluvial deposits are volumetrically minor in most basin fills, the majority of the basin fill transits from source to sink through fluvial channels (Galloway, 1981; Hovius, 1998), and channel flow is thus a key link between upstream drainage basin and depositional sinks. Therefore, sediments deposited in fluvial settings should preserve critical signals that could be used to estimate the key parameters of S2S systems (Blum and Törnqvist, 2000; Blum and Womack, 2009). In this study, the term "channel belt" is used to mean both the channels defined by two adjacent river cutbanks and fluvial deposits that preserved in the three-dimensional stratigraphic record. Studies of modern and Quaternary fluvial systems have explored the relationships among channel-belt dimensions, drainage area, and water discharge (e.g.
In asymmetric war scenarios (e.g., counter-terrorism), the adversary usually invests a significant time to learn the system structure and identify vulnerable components, before launching attacks. Traditional game-theoretic defender-attacker models either ignore such learning periods or the entailed costs. This paper fills the gap by analyzing the strategic interactions of the terrorist's costly learning and defender's counter-learning and defense strategies in a game with private defender information. Our model allows six possible attacker strategies: (a) attack immediately; (b) learn and attack; (c) learn and not attack; (d) learn and attack when appearing vulnerable and not attack when appearing invulnerable; (e) learn and not attack when appearing vulnerable and attack when appearing invulnerable; and (f) not attack. Our results show that four of the six strategies (a, d, e, f) are possible at equilibrium and the other two (b, c) are strictly dominated. Interestingly, we find that the counterintuitive strategy (e) could be at equilibrium, especially when the probability that the target appears vulnerable given it is invulnerable is sufficiently high. Our results also show that the attacker's learning cost has a significant impact on both the attacker's best responses and the defender's equilibrium deception and defense strategies. Finally, we study the attacker's values of perfect information and imperfect information, which provide additional insights for defense and counter-learning strategies.
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