Changes

:<math>r_{\text{max}}(x, y) = \frac{(b_s b_D)}{\sqrt{\frac{(b_s^2(x_{\text{origin}} - x)^2) + (b_D^2(y_{\text{origin}} - y)^2)}{(x_{\text{origin}} - x)^2 + (y_{\text{origin}} - y)^2}}}</math>

:<math>r_{\text{max}}(x, y) = \frac{(b_s b_D)}{\sqrt{\frac{(b_s^2(x_{\text{origin}} - x)^2) + (b_D^2(y_{\text{origin}} - y)^2)}{(x_{\text{origin}} - x)^2 + (y_{\text{origin}} - y)^2}}}</math>

To specify highly lobate plan-view clinoform geometry, characteristic of a fluvial-dominated delta ([[:File:BLTN13190fig3.jpg|Figure 3C]]), the user specifies a larger value for the clinoform in the depositional dip direction, BLTN13190eq47, than for the clinoform in the strike direction, BLTN13190eq48. For a highly elongate or near-linear plan-view clinoform geometry, characteristic of a wave-dominated shoreline ([[:File:BLTN13190fig3.jpg|Figure 3A, B]]), the user specifies a much larger value for the clinoform in the depositional strike direction, BLTN13190eq49, than for the clinoform in the dip direction, BLTN13190eq50. Data describing clinoform extent in depositional dip and strike directions can be extracted from published data on the dimensions of ancient shorelines or by analysis of their modern counterparts (e.g., tables 1, 2 in Howell et al., 2008a).

To specify highly lobate plan-view clinoform geometry, characteristic of a fluvial-dominated delta ([[:File:BLTN13190fig3.jpg|Figure 3C]]), the user specifies a larger value for the clinoform in the depositional dip direction, ''t_D'', than for the clinoform in the strike direction, ''t_S.'' For a highly elongate or near-linear plan-view clinoform geometry, characteristic of a wave-dominated shoreline ([[:File:BLTN13190fig3.jpg|Figure 3A, B]]), the user specifies a much larger value for the clinoform in the depositional strike direction, ''t_S'', than for the clinoform in the dip direction, ''t_D''. Data describing clinoform extent in depositional dip and strike directions can be extracted from published data on the dimensions of ancient shorelines or by analysis of their modern counterparts (e.g., tables 1, 2 in Howell et al., 2008a).

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{| class=wikitable

The shape and dip angle of a deltaic or shoreface clinoform in cross section is a function of modal grain size, proportion of mud, and the depositional process regime at the shoreline. In sandy, fluvial-dominated deltas, clinoforms have simple concave-upward geometries and steep dip angles (up to 15°) (Gani and Bhattacharya, 2005) (e.g., [[:File:BLTN13190fig1.jpg|Figure 1]]). Similar geometries have been documented in sandy, tide-influenced deltas (dip angles up to 5°–15°) (Willis et al., 1999). Concave-upward clinoform geometry is also typical of sandy, wave-dominated deltas and strandplains, although the clinoforms have smaller dip angles (typically up to 1°–2°) (Hampson and Storms, 2003; Gani and Bhattacharya, 2005). Clinoforms are consistently inclined paleobasinward down depositional dip; and, along depositional strike, they exhibit bidirectional, concave-upward dips if the delta-front was lobate in plan view (e.g., Willis et al., 1999; Kolla et al., 2000; Roberts et al., 2004) or appear horizontal if the shoreline was approximately linear (e.g., Hampson, 2000). Clinoforms are usually truncated at their tops by a variety of channelized erosion surfaces formed during shoreline advance (e.g., distributary channels, incised valleys) and by channelized and/or planar transgressive erosion surfaces (tide and wave ravinement surfaces sensu Swift, 1968) associated with shoreline retreat. Consequently, most sandy shoreline clinoforms lack a decrease in depositional dip (rollover) near their tops, although this geometry is ubiquitous in larger, shelf-slope margin clinoforms (e.g., Steckler et al., 1999) and in the outer, muddy portion of compound deltaic clinoforms with a broad subaqueous topset that lies seaward of the shoreline (e.g., Pirmez et al., 1998).

The shape and dip angle of a deltaic or shoreface clinoform in cross section is a function of modal grain size, proportion of mud, and the depositional process regime at the shoreline. In sandy, fluvial-dominated deltas, clinoforms have simple concave-upward geometries and steep dip angles (up to 15°) (Gani and Bhattacharya, 2005) (e.g., [[:File:BLTN13190fig1.jpg|Figure 1]]). Similar geometries have been documented in sandy, tide-influenced deltas (dip angles up to 5°–15°) (Willis et al., 1999). Concave-upward clinoform geometry is also typical of sandy, wave-dominated deltas and strandplains, although the clinoforms have smaller dip angles (typically up to 1°–2°) (Hampson and Storms, 2003; Gani and Bhattacharya, 2005). Clinoforms are consistently inclined paleobasinward down depositional dip; and, along depositional strike, they exhibit bidirectional, concave-upward dips if the delta-front was lobate in plan view (e.g., Willis et al., 1999; Kolla et al., 2000; Roberts et al., 2004) or appear horizontal if the shoreline was approximately linear (e.g., Hampson, 2000). Clinoforms are usually truncated at their tops by a variety of channelized erosion surfaces formed during shoreline advance (e.g., distributary channels, incised valleys) and by channelized and/or planar transgressive erosion surfaces (tide and wave ravinement surfaces sensu Swift, 1968) associated with shoreline retreat. Consequently, most sandy shoreline clinoforms lack a decrease in depositional dip (rollover) near their tops, although this geometry is ubiquitous in larger, shelf-slope margin clinoforms (e.g., Steckler et al., 1999) and in the outer, muddy portion of compound deltaic clinoforms with a broad subaqueous topset that lies seaward of the shoreline (e.g., Pirmez et al., 1998).

Here, a geometric approach is used to represent the depositional dip cross-section shape of a clinoform with a dimensionless shape function, BLTN13190eq62 ([[:File:BLTN13190fig2.jpg|Figure 2E]]), such as a power law for concave-upward, sandy, shoreline clinoforms:  

Here, a geometric approach is used to represent the depositional dip cross-section shape of a clinoform with a dimensionless shape function, ''s(r_c)'' ([[:File:BLTN13190fig2.jpg|Figure 2E]]), such as a power law for concave-upward, sandy, shoreline clinoforms:  

:<math>s(r_c) = \left ( \frac{(r_{\text{max}}(x, y) - r_c(x, y))^P}{r_{\text{max}}(x, y) - r_{\text{min}}(x, y))^P} \right )</math>

:<math>s(r_c) = \left ( \frac{(r_{\text{max}}(x, y) - r_c(x, y))^P}{r_{\text{max}}(x, y) - r_{\text{min}}(x, y))^P} \right )</math>

However, as the algorithm is generic, the mathematical expression of the dimensionless shape function is interchangeable so that other clinoform geometries can be represented; for example, a sigmoid function can be used to represent clinoforms in a larger, shelf-slope margin settings (e.g., Steckler et al., 1999). By combining the height function (equation 1), with the shape function (equation 7), the clinoform shape function, BLTN13190eq63, is used to construct the shape of a clinoform surface:  

However, as the algorithm is generic, the mathematical expression of the dimensionless shape function is interchangeable so that other clinoform geometries can be represented; for example, a sigmoid function can be used to represent clinoforms in a larger, shelf-slope margin settings (e.g., Steckler et al., 1999). By combining the height function (equation 1), with the shape function (equation 7), the clinoform shape function, ''c(r_c)'', is used to construct the shape of a clinoform surface:  

:<math>c(r_c) = h_{\text{min}}(r_c) + \left ( \frac{(r_{\text{max}}(x, y) - r_c(x, y))^P}{r_{\text{max}}(x, y) - r_{\text{min}}(x, y))^P} h(r_c) \right )</math>

:<math>c(r_c) = h_{\text{min}}(r_c) + \left ( \frac{(r_{\text{max}}(x, y) - r_c(x, y))^P}{r_{\text{max}}(x, y) - r_{\text{min}}(x, y))^P} h(r_c) \right )</math>

By varying the exponent in the clinoform shape function, BLTN13190eq64, the user can increase or decrease the dip angle and change the shape of the clinoform ([[:File:BLTN13190fig2.jpg|Figure 2E]], Table 1). If a similar geometry is interpreted for each clinoform within a parasequence, because they are inferred to have formed under the influence of similar hydrodynamic and sedimentologic processes, then the same value of BLTN13190eq65 (equation 7) can be applied to each clinoform modeled in the parasequence. Different values of BLTN13190eq66 can be applied to distinct geographic regions of a parasequence in which clinoforms are interpreted to have different geometries (e.g., on different flanks of an asymmetric wave-dominated delta; Bhattacharya and Giosan, 2003; Charvin et al., 2010), provided that the bounding surfaces of these geographic regions have been defined (in the initial step of the method).

By varying the exponent in the clinoform shape function, ''P'', the user can increase or decrease the dip angle and change the shape of the clinoform ([[:File:BLTN13190fig2.jpg|Figure 2E]], Table 1). If a similar geometry is interpreted for each clinoform within a parasequence, because they are inferred to have formed under the influence of similar hydrodynamic and sedimentologic processes, then the same value of ''P'' (equation 7) can be applied to each clinoform modeled in the parasequence. Different values of ''P'' can be applied to distinct geographic regions of a parasequence in which clinoforms are interpreted to have different geometries (e.g., on different flanks of an asymmetric wave-dominated delta; Bhattacharya and Giosan, 2003; Charvin et al., 2010), provided that the bounding surfaces of these geographic regions have been defined (in the initial step of the method).

===Spacing and Progradation Direction of Clinoforms===

===Spacing and Progradation Direction of Clinoforms===

The clinoform-modeling algorithm allows the user to specify the main progradation direction of the clinoforms and to define the intervals along the progradation path at which clinoforms are generated (i.e., the clinoform spacing). The user specifies a progradation direction relative to north, BLTN13190eq67 ([[:File:BLTN13190fig4.jpg|Figure 4C]], Table 1), along which successive clinoforms are generated, which corresponds to the progradation path of the shoreline during clinoform deposition (plan-view shoreline trajectory of Helland-Hansen and Hampson, 2009). The user also specifies the initial insertion point for the clinoforms, BLTN13190eq68 ([[:File:BLTN13190fig4.jpg|Figure 4C]]). This provides flexibility in determining where to place the initial clinoform relative to the proximal model boundary. The spacing between each clinoform surface, BLTN13190eq69 (Table 1), is also designated by the user. Clinoform spacing is defined as the distance between the top-truncation points of two successive clinoforms, and it determines the origin position, BLTN13190eq70, of successive clinoforms ([[:File:BLTN13190fig4.jpg|Figure 4D]]).

The clinoform-modeling algorithm allows the user to specify the main progradation direction of the clinoforms and to define the intervals along the progradation path at which clinoforms are generated (i.e., the clinoform spacing). The user specifies a progradation direction relative to north, ''θ'' ([[:File:BLTN13190fig4.jpg|Figure 4C]], Table 1), along which successive clinoforms are generated, which corresponds to the progradation path of the shoreline during clinoform deposition (plan-view shoreline trajectory of Helland-Hansen and Hampson, 2009). The user also specifies the initial insertion point for the clinoforms, ''P_o'' ([[:File:BLTN13190fig4.jpg|Figure 4C]]). This provides flexibility in determining where to place the initial clinoform relative to the proximal model boundary. The spacing between each clinoform surface, ''S'' (Table 1), is also designated by the user. Clinoform spacing is defined as the distance between the top-truncation points of two successive clinoforms, and it determines the origin position, ''(x<sub>origin>/sub>, y<sub>origin>/sub>)'', of successive clinoforms ([[:File:BLTN13190fig4.jpg|Figure 4D]]).

===Stochastic Modeling of Clinoforms===

===Stochastic Modeling of Clinoforms===

Because many of the input parameters can be defined stochastically, one of the consequences of this aspect of the clinoform-modeling algorithm is that it is possible to generate complex geometries, such as cases in which clinoforms are observed to onlap against older clinoforms in the same parasequence. A combination of three factors is postulated to cause subtle changes in clinoform geometry and position, which combine to produce onlap in depositional-dip-oriented cross sections: (1) in fluvial-dominated deltas, distributary mouth bars and bar complexes have complex 3-D geometries that can shift along depositional strike as well as down depositional dip (e.g., Olariu et al., 2005; Wellner et al., 2005); (2) riverine sediment supply to delta-front clinoforms exhibits temporal and spatial variability that is related, at least in part, to downstream branching and switching of distributary channels as deltas advance (e.g., Wellner et al., 2005; Ahmed et al., 2014); and (3) clinoform geometries are locally modified by basinal processes such as waves and tides (e.g., Gani and Bhattacharya, 2007).

Because many of the input parameters can be defined stochastically, one of the consequences of this aspect of the clinoform-modeling algorithm is that it is possible to generate complex geometries, such as cases in which clinoforms are observed to onlap against older clinoforms in the same parasequence. A combination of three factors is postulated to cause subtle changes in clinoform geometry and position, which combine to produce onlap in depositional-dip-oriented cross sections: (1) in fluvial-dominated deltas, distributary mouth bars and bar complexes have complex 3-D geometries that can shift along depositional strike as well as down depositional dip (e.g., Olariu et al., 2005; Wellner et al., 2005); (2) riverine sediment supply to delta-front clinoforms exhibits temporal and spatial variability that is related, at least in part, to downstream branching and switching of distributary channels as deltas advance (e.g., Wellner et al., 2005; Ahmed et al., 2014); and (3) clinoform geometries are locally modified by basinal processes such as waves and tides (e.g., Gani and Bhattacharya, 2007).

To produce onlap and other subtle geometric features between successive clinoforms, the user can use the stochastic component of the clinoform-modeling algorithm to generate small variations in the parameter values of either or all of the following: progradation direction, BLTN13190eq71; clinoform spacing, S; and clinoform length, BLTN13190eq72. If the parameters that define a clinoform cause it to be present below an earlier surface, it is truncated by the earlier surface to produce onlap. Application of the algorithm to (1) a rich, outcrop data set and (2) a sparse, subsurface data set is described in the examples in the following two sections.

To produce onlap and other subtle geometric features between successive clinoforms, the user can use the stochastic component of the clinoform-modeling algorithm to generate small variations in the parameter values of either or all of the following: progradation direction, ''θ''; clinoform spacing, ''S''; and clinoform length, ''L''. If the parameters that define a clinoform cause it to be present below an earlier surface, it is truncated by the earlier surface to produce onlap. Application of the algorithm to (1) a rich, outcrop data set and (2) a sparse, subsurface data set is described in the examples in the following two sections.

==Example 1: Ferron sandstone reservoir analog==

==Example 1: Ferron sandstone reservoir analog==