Conditional Independence Test for Weights-of-Evidence Modeling

Weights-of-evidence modeling is a GIS-based technique for relating a point pattern for locations of discrete events with several map layers. In general, the map layers are binary or ternary. Weights for presence, absence or missing data are added to a prior logit. Updating with two or more map layers is allowed only if the map layers are approximately conditionally independent of the point pattern. The final product is a map of posterior probabilities of occurrence of the discrete event within a small unit cell. This paper contains formal proof that conditional independence of map layers implies that T, the sum of the posterior probabilities weighted according to unit cell area, is equal to n, being the total number of discrete events. This result is used in the overall or omnibus test for conditional independence. In practical applications, T generally exceeds n, indicating a possible lack of conditional independence. Estimation of the standard deviation of T allows performance of a one-tailed test to check whether or not T-n is significantly greater than zero. This new test is exact and simpler to use than other tests including the Kolmogorov-Smirnov test and various chi-squared tests adapted from discrete multivariate statistics.     

Geologically Constrained Probabilistic Mapping of Gold Potential, Baguio District, Philippines

Binary predictor patterns of geological features are integrated based on a probabilistic approach known as weights of evidence modeling to predict gold potential. In weights of evidence modeling, the log _e of the posterior odds of a mineral occurrence in a unit cell is obtained by adding a weight, W ⁺ or W ^– for presence of absence of a binary predictor pattern, to the log _e of the prior probability. The weights are calculated as log _e ratios of conditional probabilities. The contrast, C = W ⁺ – W ^–, provides a measure of the spatial association between the occurrences and the binary predictor patterns. Addition of weights of the input binary predictor patterns results in an integrated map of posterior probabilities representing gold potential. Combining the input binary predictor patterns assumes that they are conditionally independent from one another with respect to occurrences.     

Weights of Evidence Modeling of Mineral Potential: A Case Study Using Small Number of Prospects,Abra, Philippines

Weights of evidence (WofE) modeling usually is applied to map mineral potential in areas with large number of deposits/prospects. In this paper, WofE modeling is applied to a case study area measuring about 920 km² with 12 known porphyry copper prospects. A pixel size of 100 m × 100 m was used in the spatial data analyses to represent in a raster-based GIS lateral extents of prospects and of geological features considered as spatial evidence. Predictor maps were created based on (a) estimates of studentized values of positive spatial association between prospects and spatial evidence; (b) proportion of number of prospects in zones where spatial evidence is present; and (c) geological interpretations of positive spatial association between prospects and spatial evidence. Uncertainty because of missing geochemical evidence is shown to have an influence on tests of assumption of conditional independence (CI) among predictor maps with respect to prospects. For the final predictive model, assumption of CI is rejected based on omnibus test but is accepted based on a new omnibus test. The final predictive model, which delineates 30% of study area as zones with potential for porphyry copper, has 83% success rate and 73% prediction rate. The results demonstrate plausibility of WofE modeling of mineral potential in large areas with small number of mineral prospects.     

A Comparative Analysis of Weights of Evidence,Evidential Belief Functions,and Fuzzy Logic for Mineral Potential Mapping Using Incomplete Data at the Scale of Investigation

Large amounts of digital data must be analyzed and integrated to generate mineral potential maps, which can be used for exploration targeting. The quality of the mineral potential maps is dependent on the quality of the data used as inputs, with higher quality inputs producing higher quality outputs. In mineral exploration, particularly in regions with little to no exploration history, datasets are often incomplete at the scale of investigation with data missing due to incomplete mapping or the unavailability of data over certain areas. It is not always clear that datasets are incomplete, and this study examines how mineral potential mapping results may differ in this context. Different methods of mineral potential mapping provide different ways of dealing with analyzing and integrating incomplete data. This study examines the weights of evidence (WofE), evidential belief function and fuzzy logic methods of mineral potential mapping using incomplete data from the Carajás mineral province, Brazil to target for orogenic gold mineralization. Results demonstrate that WofE is the best one able to predict the location of known mineralization within the study area when either complete or unacknowledged incomplete data are used. It is suggested that this is due to the use of Bayes’ rule, which can account for “missing data.” The results indicate the effectiveness of WofE for mineral potential mapping with incomplete data.     

A Modified Weights-of-Evidence Method for Mineral Potential Prediction Based on Structural Equation Modeling

Structural equation modeling (SEM) was applied here to modify the ordinary weights-of-evidence (WofE) method for calculating posterior probability to improve conditional independence (CI) in the application of this method mineral potential prediction. The new method attempts to reduce the effect of CI by defining new binary patterns with an optimum combination of cutoff values of patterns. The solution is calculated through SEM, and the goodness of fit between evidence and mineral deposit occurrences is evaluated by a specified target function. The main difference between the new WofE and ordinary WofE is that evidence in the new method maintains a balance between the significance for mineral potential prediction and CI, rather than the significance for mineral prediction only as in ordinary WofE. A case study of prediction of potential for hydrothermal Au mineral deposits in Nova Scotia, Canada, is discussed here. The results indicate that the new method performs better than the ordinary WofE.

     

A Hybrid Fuzzy Weights-of-Evidence Model for Mineral Potential Mapping

This paper describes a hybrid fuzzy weights-of-evidence (WofE) model for mineral potential mapping that generates fuzzy predictor patterns based on (a) knowledge-based fuzzy membership values and (b) data-based conditional probabilities. The fuzzy membership values are calculated using a knowledge-driven logistic membership function, which provides a framework for treating systemic uncertainty and also facilitates the use of multiclass predictor maps in the modeling procedure. The fuzzy predictor patterns are combined using Bayes’ rule in a log-linear form (under an assumption of conditional independence) to update the prior probability of target deposit-type occurrence in every unique combination of predictor patterns. The hybrid fuzzy WofE model is applied to a regional-scale mapping of base-metal deposit potential in the south-central part of the Aravalli metallogenic province (western India). The output map of fuzzy posterior probabilities of base-metal deposit occurrence is classified subsequently to delineate zones with high-favorability, moderate favorability, and low-favorability for occurrence of base-metal deposits. An analysis of the favorability map indicates (a) significant improvement of probability of base-metal deposit occurrence in the high-favorability and moderate-favorability zones and (b) significant deterioration of probability of base-metal deposit occurrence in the low-favorability zones. The results demonstrate usefulness of the hybrid fuzzy WofE model in representation and in integration of evidential features to map relative potential for mineral deposit occurrence.     

Quantitative Prediction and Evaluation of Mineral Resources Based on GIS: A Case Study in Sanjiang Region,Southwestern China

Quantitative prediction and evaluation of mineral resources are one of the important topics of mathematical geology. On the basis of GIS technologies and weights of evidence modeling, MapGIS is integrated with GIS and mineral-resource prediction and evaluation. The final product is a predictor map of posterior probabilities of occurrence of the discrete event within a small unit cell. Predictor layers were created on a digital database that includes 1:200,000 scale geological, and geochemical, and geophysical maps, and remote-sensing images in study area. According to metallogenetic factors extractiont and weights of evidence modeling, there are four main metal ore belts in the study area: (1) the Batang belt; (2) the Lei Wuqi belt; (3) the Basu-Chayu belt; and (4) the Ganzi-Litang belt. The predictor map of posterior probabilities show that 29% of study area as zones with potential for porphyry copper, and 81% known mineral occurrences success rate is circled in the metallogenetic posterior probabilities map. The results demonstrate plausibility of weights-of-evidence modeling of mineral potential in large areas with small number of mineral prospects.     

Assessment of Exploration Bias in Data-Driven Predictive Models and the Estimation of Undiscovered Resources

The spatial distribution of discovered resources may not fully mimic the distribution of all such resources, discovered and undiscovered, because the process of discovery is biased by accessibility factors (e.g., outcrops, roads, and lakes) and by exploration criteria. In data-driven predictive models, the use of training sites (resource occurrences) biased by exploration criteria and accessibility does not necessarily translate to a biased predictive map. However, problems occur when evidence layers correlate with these same exploration factors. These biases then can produce a data-driven model that predicts known occurrences well, but poorly predicts undiscovered resources. Statistical assessment of correlation between evidence layers and map-based exploration factors is difficult because it is difficult to quantify the “degree of exploration.” However, if such a degree-of-exploration map can be produced, the benefits can be enormous. Not only does it become possible to assess this correlation, but it becomes possible to predict undiscovered, instead of discovered, resources. Using geothermal systems in Nevada, USA, as an example, a degree-of-exploration model is created, which then is resolved into purely explored and unexplored equivalents, each occurring within coextensive study areas. A weights-of-evidence (WofE) model is built first without regard to the degree of exploration, and then a revised WofE model is calculated for the “explored fraction” only. Differences in the weights between the two models provide a correlation measure between the evidence and the degree of exploration. The data used to build the geothermal evidence layers are perceived to be independent of degree of exploration. Nevertheless, the evidence layers correlate with exploration because exploration has preferred the same favorable areas identified by the evidence patterns. In this circumstance, however, the weights for the “explored” WofE model minimize this bias. Using these revised weights, posterior probability is extrapolated into unexplored areas to estimate undiscovered deposits.     

Data-Driven Predictive Modeling of Mineral Prospectivity Using Random Forests: A Case Study in Catanduanes Island (Philippines)

The Random Forests (RF) algorithm is a machine learning method that has recently been demonstrated as a viable technique for data-driven predictive modeling of mineral prospectivity, and thus, it is instructive to further examine its usefulness in this particular field. A case study was carried out using data from Catanduanes Island (Philippines) to investigate further (a) if RF modeling can be used for data-driven modeling of mineral prospectivity in areas with few (i.e., <20) mineral occurrences and (b) if RF modeling can handle predictor variables with missing values. We found that RF modeling outperforms evidential belief (EB) modeling of prospectivity for hydrothermal Au–Cu deposits in Catanduanes Island, where 17 hydrothermal Au–Cu prospects are known to exist. Moreover, just like EB modeling, RF modeling allows analysis of the spatial relationships between known prospects and individual layers of predictor data. Furthermore, RF modeling can handle missing values in predictor data through an RF-based imputation technique whereas in EB modeling, missing values are simply represented by maximum uncertainty. Therefore, the RF algorithm is a potentially useful method for data-driven predictive modeling of mineral prospectivity in regions with few (i.e., <20) occurrences of mineral deposits of the type sought. However, further testing of the method in other regions with few mineral occurrences is warranted to fully determine its usefulness in data-driven predictive modeling of mineral prospectivity.     

Modeling the spatial patterns of human wildfire ignition in Yunnan province,China

Despite wildfire being an important regulator of dryland ecosystems, uncontrolled wildfire can be harmful to both forest ecosystems and human society, and wildfire prevention and control continue to raise worldwide concern. Wildfire management depends on knowledge of wildfire ignitions, both for cause and location. The regimes and factors influencing wildfire ignition have been studied at length. Humans have a profound effect on fire regimes and human activity is responsible for igniting the largest number of fires in our study area. Understanding the spatial patterns of ignitions is foremost to achieving efficiency in wildfire prevention. Previous studies mainly concentrate on overall wildfire risk integrating numerous factors simultaneously, yet the importance of human factors on ignition has not received much attention. In this study, we mapped human accessibility to explore the influence of human activity on wildfire ignition in a simple and straightforward way. A Bayesian weights-of-evidence (WofE) method was developed based on fire hotspots in China's Yunnan province extracted from satellite images and verified as known wildfires for the period 2007–2013. We considered a set of factors that impact fire ignition as associated with human accessibility: the locations of settlements, roads, water and farmland susceptible to human wildfire ignition. Known points of likely wildfire ignition were selected as training samples and all suspected thematic maps of the factors were taken as explanatory layers. Next, the weights of each layer in terms of its explanatory power were computed and used to generate evidence based on a threshold to pass a statistical test. The conditional independence (CI) of each layer was checked with the Agterberg-Cheng test. Finally, the posterior probability was calculated and its precision validated using samples of both presence and absence by withheld validation data. A comparison of WofE models was made to test the predictability. Results show proximity to villages, roads and farmland are strongly associated with human wildfire ignition and that wildfire more often occurs at an intermediate distance from high-density human activity. The WofE method proved more powerful than logistic regression, improving predictive accuracy by 10% and was more straightforward in presenting the association of dependence and independence. In addition, WofE with 1000 m buffer bands is more robust in predicting human wildfire ignition risk than binary or 100 m buffers for the ecoregion studied. Our results are significant for advising practical wildfire management and resource allocation, evaluation of human ignition control and also provides a foundation for future efforts toward integrated wildfire prediction.     

Downstream hydraulic geometry of rivers in Victoria, Australia

Downstream variation of hydraulic geometry in rivers, characterized by fine textured banks and low width to depth ratios (7–25), is investigated in Victoria, Australia, with the aim of developing predictive models of channel geometry for large-scale spatial modeling applications. A one-dimensional hydraulic model is used to determine the mean bank-full geometry and discharge (Q_bf) for 93 sites which are investigated in relation to discharge of fixed average recurrence interval (ARI). The median bank-full ARI is estimated at 0.8 years with 75% of sites between 0.5 and 2.5 years. Exponents in the downstream hydraulic geometry relations for width, depth and velocity are respectively 0.43, 0.40 and 0.18 (Q = Q_bf) and 0.44, 0.38 and 0.03 (Q = Q₂, i.e., 2-year ARI), falling near the mode of global values. Q₂ and slope explain 66% of variance in Q_bf, while Q₂ explains 73% and 69% of the variance in width and depth relations, respectively: Q₂ provides a reliable substitute for Q_bf in spatial modeling applications. Spatial variation in hydraulic geometry relations within and between river basins remains largely unexplained. The W/D ratio characteristically decreases with increasing distance along the lower reaches of most rivers and this has contributed to the lower than expected value for the width exponent.     

Representative rainfall thresholds for landslides in the Nepal Himalaya

Measuring some 2400 km in length, the Himalaya accommodate millions of people in northern India and Pakistan, Nepal, Bhutan, and parts of other Asian nations. Every year, especially during monsoon rains, landslides and related natural events in these mountains cause tremendous damage to lives, property, infrastructure, and environment. In the context of the Himalaya, however, the rainfall thresholds for landslide initiation are not well understood. This paper describes regional aspects of rainfall thresholds for landslides in the Himalaya. Some 677 landslides occurring from 1951 to 2006 were studied to analyze rainfall thresholds. Out of the 677 landslides, however, only 193 associated with rainfall data were analyzed to yield a threshold relationship between rainfall intensity, rainfall duration, and landslide initiation. The threshold relationship fitted to the lower boundary of the field defined by landslide-triggering rainfall events is I = 73.90D^− 0.79 (I = rainfall intensity in mm h^− 1 and D = duration in hours), revealing that when the daily precipitation exceeds 144 mm, the risk of landslides on Himalayan mountain slopes is high. Normalized rainfall intensity–duration relationships and landslide initiation thresholds were established from the data after normalizing rainfall-intensity data with respect to mean annual precipitation (MAP) as an index in which N_I = 1.10D^− 0.59 (N_I = normalized intensity in h^− 1). Finally, the role of antecedent rainfall in causing landslides was also investigated by considering daily rainfall during failure and the cumulative rainfall to discover at what point antecedent rainfall plays an important role in Himalayan landslide processes. Rainfall thresholds presented in this paper are generalized so they can be used in landslide warning systems in the Nepal Himalaya.     

Plant water relations of thornscrub shrub species, north-eastern Mexico

As an approach to understand how diurnal and seasonal plant water potentials (Ψ) are related to soil water-content and evaporative demand components, the responses of six thornscrub shrubs species (Havardia pallens, Acacia rigidula, Eysenhardtia texana, Diospyros texana, Randia rhagocarpa, and Bernardia myricaefolia) of the north-eastern region of Mexico to drought stress were investigated during the course of 1 year. All study species showed the typical diurnal pattern of variation in Ψ. That is, Ψ decreased gradually from predawn (Ψ_pd) maximal values to reach minima at midday (Ψ_md) and began to recover in the late afternoon. On a diurnal basis and with adequate soil water-content (>0.20 kg kg⁻¹), diurnal Ψ values differed among shrub species and were negatively and significantly (p<0.001) correlated with air temperature (r=−0.741 to −0.883) and vapor pressure deficit (r=−0.750 to −0.817); in contrast, a positive and significant (p<0.001) relationship was found to exist with relative humidity (r=0.758–0.842). On a seasonal basis, during the wettest period (soil water-content>0.20 kg kg⁻¹), higher Ψ_pd (−0.10 MPa) and Ψ_md (−1.16 MPa) values were observed in R. rhagocarpa, whereas lower figures (−0.26 and −2.73 MPa, respectively) were detected in A. rigidula. On the other hand, during the driest period (soil water-content<0.15 kg kg⁻¹), Ψ_pd and Ψ_md values were below −7.3 MPa; i.e. when shrubs species faced severe water deficit. Soil water-content at different soil layers, monthly mean relative humidity and monthly precipitation were significantly correlated with both Ψ_pd (r=0.538–0.953; p<0.01) and Ψ_md (r=0.431–0.906; p<0.05). Average soil water-content in the 0–50 cm soil depth profile explained between 70% and 87% of the variation in Ψ_pd. Results have shown that when gravimetric soil water-content values were above 0.15 kg kg⁻¹, Ψ_pd values were high and constant; below this threshold value, Ψ declined gradually. Among all shrub species, A. rigidula appeared to be the most drought tolerant of the six species since during dry periods it tends to sustain significantly higher Ψ_pd in relation to B. myricaefolia. The remaining species showed an intermediate pattern. It is concluded that the ability of shrub species to cope with drought stress depends on the pattern of water uptake and the extent to control water loss through the transpirational flux.     

Failure Mechanism and Deformation Characteristics of Gob-Side Entry Retaining in Solid Backfill Mining: A Case Study

In solid backfilling mining, gangue and other solid materials are backfilled into the gob to restrain the roof subsidence. As a result, instead of being subjected to periodic collapse, the overlying strata will be able to maintain its integrity and bend or sink as a whole with partial fracture and failure, thus facilitating the gob-side entry to retain. To study the mechanical deformation characteristics and failure mechanism of such gob-side supporting wall, this research analyzes the structure features of the surrounding rock in gob-side entry and builds mechanical damage model for gob-side supporting based on the hypothesis of planar slip surface in the theory of Coulomb earth pressure. The relationship among normal force acting on the contact face N, the angle between the slip surface of gangue and vertical face η and other relevant factors are discussed. The result suggests that η is mainly affected by δ, the friction angle between the backfill body and the supporting wall interface, and φ, the internal friction angle of backfill materials; η increases as δ increases and decreases as φ increases. N, mainly affected by δ, φ and uniformly distributed load q, has positive correlation with q and negative correlation with δ and φ. The research also reveals that compressive stress exists only when the width of wall exceeds 1.741 m; such stress will decrease sharply when the width of wall is between 2 m and 3 m, and become stable when the width goes beyond 3 m. According to numerical simulation and field test, the maximum values of roof-to-floor longitudinal deformation and lateral deformation between two sidewalls are 350 mm and 210 mm, respectively. The research results proved that the model employed can accurately predict and evaluate the deformation and damage of the gob-side entry.

     

Storm rainfall conditions for floods and debris flows from recently burned areas in southwestern Colorado and southern California

Debris flows generated during rain storms on recently burned areas have destroyed lives and property throughout the Western U.S. Field evidence indicate that unlike landslide-triggered debris flows, these events have no identifiable initiation source and can occur with little or no antecedent moisture. Using rain gage and response data from five fires in Colorado and southern California, we document the rainfall conditions that have triggered post-fire debris flows and develop empirical rainfall intensity–duration thresholds for the occurrence of debris flows and floods following wildfires in these settings. This information can provide guidance for warning systems and planning for emergency response in similar settings.Debris flows were produced from 25 recently burned basins in Colorado in response to 13 short-duration, high-intensity convective storms. Debris flows were triggered after as little as six to 10 min of storm rainfall. About 80% of the storms that generated debris flows lasted less than 3 h, with most of the rain falling in less than 1 h. The storms triggering debris flows ranged in average intensity between 1.0 and 32.0 mm/h, and had recurrence intervals of two years or less. Threshold rainfall conditions for floods and debris flows sufficiently large to pose threats to life and property from recently burned areas in south-central, and southwestern, Colorado are defined by: I = 6.5D^− 0.7 and I = 9.5D^− 0.7, respectively, where I = rainfall intensity (in mm/h) and D = duration (in hours).Debris flows were generated from 68 recently burned areas in southern California in response to long-duration frontal storms. The flows occurred after as little as two hours, and up to 16 h, of low-intensity (2–10 mm/h) rainfall. The storms lasted between 5.5 and 33 h, with average intensities between 1.3 and 20.4 mm/h, and had recurrence intervals of two years or less. Threshold rainfall conditions for life- and property-threatening floods and debris flows during the first winter season following fires in Ventura County, and in the San Bernardino, San Gabriel and San Jacinto Mountains of southern California are defined by I = 12.5D^−0.4, and I = 7.2D^−0.4, respectively. A threshold defined for flood and debris-flow conditions following a year of vegetative recovery and sediment removal for the San Bernardino, San Gabriel and San Jacinto Mountains of I = 14.0D^−0.5 is approximately 25 mm/h higher than that developed for the first year following fires.The thresholds defined here are significantly lower than most identified for unburned settings, perhaps because of the difference between extremely rapid, runoff-dominated processes acting in burned areas and longer-term, infiltration-dominated processes on unburned hillslopes.     

Combining indicator patterns in weights of evidence modeling for resource evaluation

The weights of evidence model for combining indicator patterns in mineral resource evaluation is briefly explained with emphasis on the effect of undiscovered deposits on the estimation of the weights and posterior probabilities. A group of six statistical tests is proposed for analyzing the interaction of two or three indicator patterns with the point pattern for mineral deposits. A distinction is made between statistics that depend on choice of unit cell size and those that are approximately or completely independent of it. Finally, weights of evidence are compared to regression coefficients obtained by means of the logistic model.     

The role of climate and vegetation in weathering and clay mineral formation in late Quaternary soils of the Swiss and Italian Alps

Interactions between climate and soil remain ambiguous, particularly when silicate weathering and clay mineral formation and transformation rates are considered in relation to global climate changes. Recent studies suggest that climate affects weathering rates much less than previously thought. Here we show that the climate in the central European Alps has a significant, but indirect, influence on the weathering of soils through vegetation. The pattern of element leaching and mineral transformations is not only due to precipitation and temperature. Element leaching was greatest in subalpine forests near the timberline; weathering is lessened at higher and lower altitudes. Vegetation, therefore, contributes significantly to weathering processes. The highest accumulation of organic matter was found in climatically cooler sites (subalpine range) where the production of organic ligands, which enhance weathering, is greatest. Patterns of smectite formation and distribution had strong similarities to that of the elemental losses of Fe and Al (R = 0.69; P < 0.01) or base cations (R = 0.58; P < 0.05). Higher precipitation rates and the production of organic chelating compounds in the soil promoted the appearance of smectites. The relationship between climate, element leaching (Fe, Al, Ca, Mg, K, Na), and smectite formation is strongly nonlinear and driven by the podzolisation process, which is more pronounced near the timberline because of the bioclimatic constellation. Climate warming will probably, in the future, lead to a decrease in SOM stocks in the subalpine to alpine range because of more favourable conditions for biodegradation that would also affect weathering processes.     

Growth response of three globose cacti to radiation and soil moisture: An experimental test of the mechanism behind the nurse effect

Cactus seedlings often establish under nurse plants which modify environmental conditions by increasing moisture and decreasing solar radiation, which may cause beneficial and detrimental effects, respectively, on seedling growth. Three soil moisture treatments (5%, 25% and 60%) and two solar radiation levels (100% exposure=243 μmol m⁻² s⁻¹, and 40%=102 μmol m⁻² s⁻¹) were used in a factorial design to analyze seedling growth response of three rare cactus species (Mammillaria pectinifera, Obregonia denegrii and Coryphantha werdermannii). The variables evaluated were relative growth rate (RGR), root/shoot ratio (R/S), and K (RGR_roots/RGR_shoot), obtained from an initial seedling harvest (6-month-old seedlings) and a final harvest 6 months after treatment application. All three species had slow RGRs (0.002–0.012 g g⁻¹ day⁻¹). O. denegrii had the lowest RGR values, but was the only species for which R/S and K varied with soil moisture. While all seedlings responded markedly to soil moisture, no response was observed to radiation treatments. The latter might have been related to the relatively low solar radiation levels present in the greenhouse. Yet, our results suggest that the main benefit nurse plants offer to seedlings is the increase in soil moisture.     

Probabilistic Estimates of Number of Undiscovered Deposits and Their Total Tonnages in Permissive Tracts Using Deposit Densities

Empirical evidence indicates that processes affecting number and quantity of resources in geologic settings are very general across deposit types. Sizes of permissive tracts that geologically could contain the deposits are excellent predictors of numbers of deposits. In addition, total ore tonnage of mineral deposits of a particular type in a tract is proportional to the type’s median tonnage in a tract. Regressions using size of permissive tracts and median tonnage allow estimation of number of deposits and of total tonnage of mineralization. These powerful estimators, based on 10 different deposit types from 109 permissive worldwide control tracts, generalize across deposit types. Estimates of number of deposits and of total tonnage of mineral deposits are made by regressing permissive area, and mean (in logs) tons in deposits of the type, against number of deposits and total tonnage of deposits in the tract for the 50th percentile estimates. The regression equations (R ² = 0.91 and 0.95) can be used for all deposit types just by inserting logarithmic values of permissive area in square kilometers, and mean tons in deposits in millions of metric tons. The regression equations provide estimates at the 50th percentile, and other equations are provided for 90% confidence limits for lower estimates and 10% confidence limits for upper estimates of number of deposits and total tonnage. Equations for these percentile estimates along with expected value estimates are presented here along with comparisons with independent expert estimates. Also provided are the equations for correcting for the known well-explored deposits in a tract. These deposit-density models require internally consistent grade and tonnage models and delineations for arriving at unbiased estimates.     

The grade index model as a rationale for autogenic nonequilibrium responses of deltaic clinoform to relative sea-level rise

Grade index (G_index) is a dimensionless number given as the volume-in-unit-time ratio of subaerial allocation to both subaerial and subaqueous allocations of sediment supplied to a delta from upstream. It was originally proposed for understanding the effect of basin water depth on the morphodynamics of delta distributary channels under stationary relative sea level. We here examine how rising relative sea level modulates the G_index, using geometrical reasoning and numerical simulations. We find that the grade index model can account for autoretreat of the deltaic shoreline, autodrowning of the whole system, and autobreak of the deltaic sedimentation, all of which are the consequences of autogenic nonequilibrium responses to steadily rising relative sea level. The regressive-to-transgressive threshold (i.e. the onset of autoretreat) is crossed when the delta plain's dimensionless basal area (A_t*) encounters a critical value that is expressed in terms of G_index: regression and transgression are sustained when A_t* is below and above the threshold, respectively. The mode of transgression depends on the slope conditions. If the hinterland slope (γ) is steeper than the foreset slope (β), both A_t* and G_index decrease as the relative sea-level rises. Eventually, the depositional system experiences autodrowning when A_t* = G_index = 0. If γ < β; on the other hand, both A_t* and G_index increase. This latter slope condition eventually causes autobreak of the deltaic sedimentation, afterward of which A_t* = G_index = 1. The grade index model is useful for interpreting and predicting the stratigraphic responses of natural deltaic clinoforms in conditions of rising relative sea level.