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# Variation in the efficacy of remote cameras used to monitor wildlife

## Abstract

Wildlife cameras allow conservation scientists to monitor wildlife. However, there are performance limitations associated with wildlife cameras that must be understood prior to their use. This study compared two wildlife camera models, the Spypoint Solar Trail and the Reconyx Hyperfire 2, on behalf of Calgary Captured, a collaborative project between the Miistakis Institute and the City of Calgary to determine wildlife occupancy in Calgary’s Natural Area Parks. The camera models were set up in pairs at 10 sites. There was no significant difference in detections of white-tailed deer (Odocoileus virginianus) or coyotes (Canis latrans) by either model, but the Reconyx cameras successfully captured two species that the Spypoint model failed to detect: bobcat (Lynx rufus) and deer mouse (Peromyscus maniculatus). The Reconyx cameras had fewer trap days because the Nickel Metal Hydride (NiMH) batteries supplied by Calgary Captured consistently failed in cold weather, whereas the Spypoint cameras’ solar panels continued to function through the duration of the study. Despite having had fewer trap days, the Reconyx cameras captured more species than did the Spypoint cameras. There was no significant difference in the number of false positives and false negatives produced by the two models, but only the Spypoint cameras produced malfunctioned images. For projects like Calgary Captured, the Reconyx Hyperfire 2 is a more effective camera model than the Spypoint Solar Trail because it captures more species and is less prone to malfunction. The results of this study also highlight the importance of choosing appropriate batteries and settings for the camera model in question to ensure successful use.

## Introduction

Determining the distribution of wildlife through wildlife surveys is essential to conservation (Ahumada, Hurtado, & Lizcano, 2013; O’Brien, Baillie, Krueger, & Cuke, 2010). Traditional wildlife survey methods such as scat and track transects can be expensive and time-consuming, and often result in misidentification or failure to detect species, hampering conservation efforts (Mackenzie, 2005). It is therefore important to select appropriate survey methods that avoid these biases (Miller, Nichols, McClintock, Campbell Grant, Bailey, & Weir, 2011). Wildlife cameras overcome some of these shortcomings by collecting data continuously over a long period of time and by easing photographic identification of wildlife, thereby increasing the survey period and reducing the probability of detection error (Vine, Crowther, Lapige, Dickman, Mooney, Piggot, & English, 2009). Vine et al. (2009) compared camera traps, hair traps, scat from bait stations, and spotlighting to determine the best survey method for a low-density red fox (Vulpes vulpes) population and found that camera traps provided the most consistent and univocal results, minimizing bias.

Despite these benefits, wildlife cameras have performance limitations and not all wildlife cameras perform equally (Meek, Ballard, & Fleming, 2015). Broadly speaking, there are performance disparities between “professional” and “recreational” camera models. Professional models are designed for research and incorporate higher-quality components (e.g.: sensors, camera lens, etc.), have more customization options, and are thus more expensive, whereas recreational models are intended for use by the general public and have limited features, resulting in a more affordable and user-friendly product (Newey, Davidson, Nazir, Fairhurst, Verdicchio, Irvine, & van der Wal, 2015). One study completed by Newey et al. (2015) found that recreational camera models are less reliable than professional models and produced a greater number of false positive (blank) images and false negative (missed detection) data. A false positive occurs when a camera takes a photo in the absence of wildlife. False positives can present issues because they consume memory, batteries, and time processing images (Heiniger & Gillespie, 2018). Newey et al. (2015) also note that recreational cameras are prone to malfunction, which also take up memory and drain batteries, possibly leading to data loss. For the purposes of this study, a malfunction is defined as an image that is colour-stained (Figure 1). Sometimes, wildlife can be identified from malfunctioned images, but in more serious malfunctions, the colour stain completely blocks the image, making identification impossible, as seen in Figure 1B.

Researchers also need to consider the specifications of wildlife cameras as differing specifications can result in highly variable data (Meek et al., 2015, p. 3). While countless studies have used wildlife cameras, few have considered the differences between models (Rovero, Zimmermann, Berzi, & Meek, 2013; Trolliet et al., 2013; Meek et al., 2015). A study will typically use only one model, so it is imperative that the selected camera has the minimum specifications required to fulfil the study’s needs. There can also be substantial variation between different units of the same model (Hughson, Darby, & Dungan, 2010), as camera set-up, settings, and the quantity of cameras used can alter the likelihood of detecting certain species (Rovero et al., 2013; Wellington, Bottom, Merrill, & Litvaitis, 2014; Meek, Ballard, & Falzon, 2016). Therefore, it is crucial that researchers consider not only which model, but also which settings to use (Driessen, Jarman, Troy, & Callander, 2017).

The purpose of this study is to examine differences between two wildlife camera models, the Spypoint Solar Trail and Reconyx Hyperfire 2, on behalf of Calgary Captured. Calgary Captured is a collaborative project between the City of Calgary and the Miistakis Institute to determine wildlife occupancy in Calgary’s Natural Area Parks using wildlife cameras (Miistakis Institute, 2017). Initially, Calgary Captured used Spypoint Solar Trail cameras, a recreational model as stated by its manual. However, after a period of two years, Calgary Captured began using the Reconyx Hyperfire 2, a professional model, following concerns from Calgary Captured over potential false negatives produced by the Spypoint models (Nicole Kahal, pers. comm.). It is important to understand the variation in performance between these different camera models to avoid making inaccurate conclusions about their results (Heiniger & Gillespie, 2018). Therefore, this study quantifies the difference in detection efficacy between the two models so that the Calgary Captured project can compare the data collected by the Spypoint cameras with those collected by the Reconyx cameras.

## Methods

### Study area

10 Spypoint and 10 Reconyx wildlife cameras were placed in pairs at 10 sites: four sites in Weaselhead Flats (Figure 2A), one site in North Glenmore Park (Figure 2A), and five sites in Fish Creek Provincial Park (Figure 2B). Weaselhead Flats and North Glenmore Park are adjacent to each other, and Fish Creek Provincial Park is located approximately 15 km southeast of Weaselhead Flats. All the parks used in this study have similar habitats consisting of riparian zones and aspen poplar, balsam poplar, and white spruce forest. The sites used for this experiment are 10 of the existing 69 Calgary Captured camera sites, which are systematically located near game trails and away from human trails in a 1km by 1km grid, except in Weaselhead Flats, where additional cameras were placed by Calgary Captured, resulting in a higher camera density (N. Kahal, pers. comm.).

### Camera set-up

One Spypoint Solar Trail camera and one Reconyx Hyperfire 2 camera were placed together at each site. An ideal setup would place both cameras within 1 m directly next to each other (Hughson et al., 2010; Meek et al., 2015). However, there was rarely a pair of neighbouring trees close enough to allow for this setup. As a consequence, both cameras were placed on the same tree at nine sites, with the Spypoint camera placed above the Reconyx camera so that the Spypoint’s solar panel would not be obstructed by the Reconyx camera (Figure 3A). Site 62 was the exception and the cameras there were placed on neighbouring trees (Figure 3B). Camera height was measured by distance from the centre of the Fresnel lens to the ground (Table 1). Information on the specifications and settings of the cameras used in this study can be found in Table 2.

### Data collection and processing

Memory card collection took place once a week, alternating between the Fish Creek Provincial Park sites and the Weaselhead Flats/North Glenmore Park sites. As a control measure, and to quantify false negatives, “walk-bys” were performed each time a camera was checked, during which the principal researcher walked past each camera three times, each time at three different speeds: slow (0.5m/s-0.6m/s), medium (0.9m/s-1.3m/s), and fast (1.5m/s-3.1m/s). This procedure was not performed at sites 43, 44, and 61 due to safety concerns.

The images captured by the cameras were processed using Microsoft Access and sorted into trap events, which consist of all the photos taken of an animal during its time in front of the cameras (Meek, Ballard, Claridge, Kays, Moseby, O’Brien, O’Connell, …, & Townsend, 2014). The Reconyx cameras captured three pictures for every detection while the Spypoint cameras took one picture for every detection, so the two models’ performances were compared using the number of events instead of the number of detections. Animals were identified by species for each event. Image processing was conducted on a secure computer at the Miistakis Institute office in compliance with the Alberta Freedom of Information and Protection of Privacy (FOIP) Act (Province of Alberta, 2019).

The cameras in Weaselhead Flats/North Glenmore Park wereactive from 15 January 2019 to 3 March 2019, and those in Fish Creek Provincial Park were active from 27 January 2019 to 8 March 2019. Due to the cold weather, the NiMH batteries in some Reconyx units repeatedly lost power, reducing the number of trap nights for those cameras (Table 1). Initially, one Spypoint camera was not working and had to be replaced with a different unit (site 64), reducing its number of trap nights (Table 1). To control for these differences in trap nights, the data were standardized to the number of trap events per 100 trap nights (Kelly & Holub, 2008):

\textit{RAI}=\frac{\textit{no. of events}\times100}{\textit{no. of nights}}

This is the same standard used by the Miistakis Institute and is referred to in their documents as the relative abundance index, or RAI (Miistakis Institute, 2017). Throughout this paper, the number of trap events per 100 trap nights will be abbreviated to RAI.

Microsoft Excel was used to calculate RAI values for each species. To increase sample size, all bird species were placed into one category (“bird”). Because the data were not normally distributed, the non-parametric Wilcoxon signed-rank test was used for all comparisons. JMP (version 14.1.0) was used for statistical analyses and graphing.

## Results

Throughout the study, nighttime temperatures often fell below the -18° operating limit specified for the NiMH batteries used in the Reconyx models. Consequently, only four of the ten Reconyx cameras functioned for the entire duration of the study. Furthermore, the camera locks were often frozen, which made changing the Reconyx cameras’ batteries impossible, exacerbating the problem. As a result, a substantial amount of Reconyx data were lost, which may have impacted the study.

### Species detected

Nine species were detected during the study (Table 3). White-tailed deer was the most commonly detected species. Bobcat and deer mouse were detected only by the Reconyx cameras. There was no significant difference between the Reconyx and Spypoint cameras with regards to number of coyote detections (W=-7.00, p=0.31; Figure 4A) or deer detections (W=12.50, p=0.44; Figure 4B). Detection of other species did not take place at enough sites to allow for any statistical analysis (Table 3). Taking all species detections into consideration, there was no significant difference between Reconyx and Spypoint models (W=-0.13, p= 0.90; Figure 4C). It should be noted that at every site the camera height of the Spypoint cameras was significantly higher than that of the Reconyx cameras (W=3.44, p=0.0006; Figure 5).

### False positives and malfunctions

All of the cameras in this study collected false positives. There was no significant difference between the number of false positives collected by the Reconyx and Spypoint cameras (W=-15.50, p=0.13; Figure 6). There was less variation in the number of false positives collected by each Reconyx camera compared to the Spypoint cameras (Figure 6). Only the Spypoint cameras produced malfunctions. There was a significant difference between the number of malfunctions in the Spypoint cameras and Reconyx cameras (W=27.00, p=0.0039; Figure 7).

### False negatives

No significant difference was found between the number of pictures taken by the Reconyx and Spypoint cameras during the walk-by tests at slow (W=3.00, p=0.66), medium (W = 0.00, p = 1.00), or fast speeds (W=-1.00, p=0.89; Figure 8). One Reconyx camera failed to take any pictures during one of the fast walk-by tests. There was less variation in the number of pictures taken at each of the three walk-by speeds by the Reconyx cameras than by the Spypoint cameras (Figure 8).

## Discussion

Over the course of the study, the Reconyx cameras had fewer trap nights than did the Spypoint cameras because their batteries failed in the cold weather. Despite this, the Reconyx cameras still detected more species than did the Spypoint cameras. While the RAI allows for comparison of data from cameras with differing numbers of trap nights, the Spypoint cameras should have a greater probability of detecting wildlife because they were functional for a greater period of time (Rovero et al., 2013). Thus, based on the number of trap nights alone, the Spypoint cameras should detect more species than the Reconyx cameras. The fact that the Reconyx cameras detected more species in fewer trap nights suggests that the model has greater detection ability.

The two species detected only by Reconyx cameras were bobcat (Lynx rufus) and deer mouse (Peromyscus maniculatus). Although Reconyx cameras have a reputation for being better able to detect smaller mammals (Kelly & Holub, 2008; Wellington et al., 2014) than the game species (e.g., deer) Spypoint cameras were designed to detect, there is little evidence in the literature to suggest that this is actually the case (Driessen et al., 2014). It was suggested that the reason the Spypoint cameras had lower than expected detection rates of bobcat was because bobcats move too quickly to be effectively detected (N. Kahal, pers. comm.). However, the walk-by tests did not find a significant difference in the number of pictures taken between the two camera models. In fact, the only camera that failed to take any photos during a fast walk-by test was a Reconyx camera.

Camera height has been found to account for variation in detection of species (Meek et al., 2016). To avoid blocking the Spypoint cameras’ solar panels, the Reconyx cameras were placed below the Spypoint cameras at nine sites. Meek et al. (2016) found that cameras placed 300 cm above the ground were less likely to detect species like red foxes and wild dogs (Canis familiaris) than those placed 210 cm lower at 90 cm. The greatest camera height in this study was 118 cm and the greatest difference in height between the two cameras at any site was only 31.25 cm. Moreover, there was no significant difference between the Reconyx and Spypoint cameras in the number of detections of coyote, a species comparable in size to wild dogs. The Spypoint cameras were also able to detect other small mammals such as Eastern grey squirrels and snowshoe hares, which are smaller than bobcats. Therefore, there was little indication that height difference was the reason the Reconyx cameras detected more species than did the Spypoint cameras. Camera placement is affected by many variables such as camera angle, distance from wildlife trails, and slope of the ground between the camera and wildlife trail (Meek et al., 2016). In this study, slope and camera angle may have affected species detections, although data for those variables were not collected so they could not be included in the analysis.

Camera settings such as trigger speed can affect the number of blank (false positive) images taken. If the trigger speed is too slow, it is possible for an animal to move through the field of view before a photo is taken (Wellington et al., 2014). In this study, the Reconyx cameras had a slower trigger speed than the Spypoint cameras, so the Reconyx cameras should have produced more false positives, which was not the case.
Only the Spypoint cameras produced malfunctions. This supports Newey et al.’s (2015) results showing that recreational quality cameras have performance limitations. Malfunctions negatively impact the quality of data because researchers often cannot identify the species that triggered the camera detection. Thus, susceptibility to malfunction should be considered when selecting a camera model for a study.

It was necessary to consider the variation in the number of trap nights in this study to be able to compare the efficacy of the two camera models. There are a variety of methods that have been used in the literature to standardize camera trap data (e.g. Carbone, Christie, Conforti, Franklin, Ginsberg, Griffiths, …, Sharuddin, 2001; O’Brien, Kinnaird, & Wibisono, 2003; O’Brien et al., 2010). This study used the method of Kelly & Holub (2008) to calculate RAIs for consistency with the methodology used by the Miistakis Institute (2017). Sollman, Mohamed, Samejima, & Wilting (2013) criticizes the use of RAIs because factors such as encounter rates, home ranges, trap setup, and repeated surveys all affect data quality, so using RAIs to determine wildlife abundance without considering these factors can return biased results. In this study, the RAI values were not used to extrapolate wildlife abundances, but rather were used to compare data collected by wildlife cameras that operated for different numbers of trap nights. Using the RAI values for this purpose precluded much of the biases described by Sollman et al. (2013).

## Conclusions

Despite operating for fewer trap nights, the Reconyx cameras detected two species that were not detected by the Spypoint cameras. Because the goal of the Calgary Captured project is to document wildlife occupancy in the city, increased species detection is important. Moreover, the Reconyx cameras did not produce malfunctioned images, whereas the Spypoint cameras did. While the Reconyx cameras performed well, the NiMH batteries used did not. Lithium batteries with a specified operating limit of -40° have been used in other winter camera trap studies in Alberta without issue (L. Gould, pers. comm.; K. Anderson, pers. comm.) and may address this problem. Notwithstanding the battery issues in cold weather, Reconyx appears to be the better model in meeting the goals of Calgary Captured because it did not malfunction and was able to detect more species than did the Spypoint cameras despite operating for fewer trap nights

The advantages of using wildlife cameras over traditional survey methods include reduced identification errors and false negatives (Vine et al., 2009). Nonetheless, researchers need to be aware of wildlife cameras’ limitations and ensure the technology they select is suitable for the goals of their projects (Meek et al., 2015; Newey et al., 2015).

## Acknowledgements

The author would like to thank Dr Dorothy Hill “for spending countless hours working on this project with me”, Dr Melanie Rathburn “for helping me run statistical tests”, Nicole Kahal “for helping with camera set-up and for being my contact at the Miistakis Institute”, Calgary Captured “for lending me their sites and equipment and for trusting me to conduct the study”, and Dr Paul Meek (Government of New South Wales), Tyne Baker, and Karen Anderson (Alberta Environment and Parks) “for sharing their knowledge of wildlife cameras and for their advice”.

## References

1. Ahumada, J. A., Hurtado, J., & Lizcano, D. (2013). Monitoring the trends and status of tropical forest terrestrial vertebrate communities from camera trap data: a tool for conservation. PLoS One, 8(9). https://doi.org/10.1371/journal. pone.0073707
2. Carbone, C., Christie, S., Conforti, K., Coulson, T., Franklin, N., Ginsberg, J. R., Griffiths, M., Holden, J., Kawnanishi, K., Kinnaird, M., Laidlaw, R., Lynam, A., Macdonald, D. W., Martyr, D., McDougal, C., Nath, L., O’Brien, T., Seidensticker, J., Smith, D. J. L., Sunquist, M., Tilson, R., & Wan Shahruddin, W. N. (2001). The use of photographic rates to estimate densities of tigers and other cryptic mammals. Animal Conservation, 4, 75-79. https://doi. org/10.1017/S1367943001001081
3. Driessen, M. M., Jarman, P. J., Troy, S., & Callander, S. (2017). Animal detection vary among commonly used camera trap models. Wildlife Research, 44(4), 291-297. https://doi.org/10.1071/WR16228
4. Heiniger, J. & Gillespie, G. (2018). High variation in camera trap-model sensitivity for surveying mammal species in Northern Australia. Wildlife Research, 45(7), 578-585. https://doi.org/10.1071/WR18078
5. Hughson, D. L., Darby, N. W., & Dungan, J. D. (2010). Comparison of motion-activated cameras for wildlife investigation. California fish and game, 92(2), 101-109.
6. Kelly, M. J. & Holub, E. L. (2008). Camera trapping of carnivores: trap success among camera types and across species, and habitat selection by species, on Salt Pond Mountain, Gile County, Virginia. Northeastern naturalist, 15(2), 249-262. https://doi.org/10.1656/1092-6194(2008)15[249:CTOCTS]2.0. CO;2
7. MacKenzie, D. I. (2005). What are the issues with presence-absence data for wildlife managers? The Journal of Wildlife Management, 69(3), 849-860. https://doi.org/10.2193/0022-541X(2005)069[0849:WATIWP]2.0.CO2
8. Meek, P. D., Ballard, G., Claridge, A., Kays, R., Moseby, K., O’Brien, T., O’Connell, A., Sanderson, J., Swann, D. E., Tobler, M., & Townsend, S. (2014). Recommended principles for reporting on camera trapping research. Biodiversity and Conservation, 23(9), 2321-2343. https://doi.org/10.1007/s10531-014-0712-8
9. Meek, P. D., Ballard, G., & Fleming, P. J. S. (2015). The pitfalls of wildlife camera trapping as a survey tool in Australia. Australian Mammalogy, 37(1), 13-22. https://doi.org/10.1071/AM14023
10. Meek, P. D., Ballard, G., & Falzon, G. (2016). The higher you go the less you will know: placing camera traps high to avoid theft will affect detection. Remote Sensing in Ecology and Conservation, 2(4), 204-211. https://doi.org/10.1002/rse2.28
11. Miistakis Institute. (2017). City of Calgary: wildlife camera monitoring: summary statistic May-August 2017. Calgary, Alberta: Kahal, N., Lee, T., & Rondeau, K.
12. Miller, D. A., Nichols, J. D., McClintock, B. T., Campbell Grant, E. H., Bailey, L. L., & Weir, L. A. (2011). Improving occupancy estimation when two types of observational error occur: non-detection and species misidentification. Ecology, 92(7), 1422-1428. https://doi.org/10.1890/10-1396.1
13. Newey, S., Davidson, P., Nazir, S., Fairhurst, G., Verdicchio, F., Irvine, R. J., & van der Wal, R. (2015). Limitations of recreational camera traps for wildlife management and conservation research: a practitioner’s perspective. Ambio, 44, 625-635. https://doi.org/10.1007/s13280-015-0713-1
14. O’Brien, T. G., Kinnaird, M. F., Wibisono, H. T. (2003). Crouching tigers, hidden prey: Sumatran tiger and prey populations in a tropical forest landscape. Animal Conservation, 6, 131-139. https://doi.org/10.1017/S1367943003003172
15. O’Brien, T. G., Baillie, J. E. M., Krueger, L., Cuke, M. (2010). The wildlife picture index: monitoring top trophic levels. Animal Conservation, 13, 335-343. https://doi.org/doi:10.1111/j.1469-1795.2010.00357.x
16. Reconyx (2018). Hyperfire 2 Outdoor Series Camera: User Manual. Retrieved January 29, 2019 from http://images.reconyx.com/file/HyperFire2PROUserGuide2018_04_24_v3.pdf
17. Rovero, F., Zimmerman, F., Berzi, D., & Meek, P. (2013). “Which camera trap type and how many do I need?” A review of camera features and study designs for a range of wildlife applications. The Italian Journal of Mammalogy, 24(2), 148-156. https://doi.org/10.4404/hystrix-24.2-8789
18. Sollman, R., Mohamed, A., Samejima, H., & Wilting, A. (2013). Risky business or simple solution – relative abundance indices from camera-trapping. Biological Conservation, 159, 405-412. https://doi.org/10.1016/j.biocon.2012.12.025
19. Spypoint. (n.d.). User Manual: Solar Trail Camera. Retrieved January 29, 2019 from https://www.spypoint.com/manuals/Manual_SOLAR_EN.pdf
20. Trolliet, F., Huynen, M., Vermeulen, C., & Hambuckers, A. (2014). Use of camera traps for wildlife studies: a review. BASE, 18(3), 446-454.
21. Vine, S. J., Crowther, M., S., Lapidge, S. J., Dickman, C. R., Mooney, N., Piggott, M. P., & English, A. W. (2009). Comparison of methods to detect rare and cryptic species: a case study using the red fox (Vulpes vulpes). Wildlife Research, 36, 436-446. https://doi.org/10.1071/WR08069
22. Wellington, K., Bottom, C., Merrill, C., & Litvaitis, J. A. (2014). Identifying performance differences among trail cameras used to monitor forest mammals. Wildlife Society Bulletin, 38(3), 634-638. https://doi.org/10.1002/wsb.425


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# Math Help - Linear algebra

1. ## Linear algebra

Let V and W be vector spaces, let T : V--->W be linear, and let {w1, w2,...wk} be a linearly indepemdent subset R(t). Prove that if S={v1,v2,...vk} is chosen so that T(vi)=wi for i=1,2,.....k, then S is lenearly independent.

i've been trying to do that problem for ever.. dont know how to! plz help!

2. Look at this.

format: plain


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## Cryptology ePrint Archive: Report 2016/753

Improved Reduction from the Bounded Distance Decoding Problem to the Unique Shortest Vector Problem in Lattices

Shi Bai and Damien Stehle and Weiqiang Wen

Abstract: We present a probabilistic polynomial-time reduction from the lattice Bounded Distance Decoding (BDD) problem with parameter 1/($\sqrt{2}\cdot\gamma$) to the unique Shortest Vector Problem (uSVP) with parameter $\gamma$ for any $\gamma>1$ that is polynomial in the lattice dimension $n$. It improves the BDD to uSVP reductions of [Lyubashevsky and Micciancio, CRYPTO, 2009] and [Liu, Wang, Xu and Zheng, Inf. Process. Lett., 2014], which rely on Kannan's embedding technique. The main ingredient to the improvement is the use of Khot's lattice sparsification [Khot, FOCS, 2003] before resorting to Kannan's embedding, in order to boost the uSVP parameter.

Category / Keywords: Lattices, Bounded Distance Decoding Problem, Unique Shortest Vector Problem, Sparsification

Original Publication (with minor differences): ICALP 2016

Date: received 4 Aug 2016, last revised 31 Aug 2016

Contact author: weiqiang wen at ens-lyon fr

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2016/753

[ Cryptology ePrint archive ]


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# Inequality

• November 15th 2007, 07:11 AM
james_bond
Inequality
Prove that if $ab+bc+ca=1$ than $\frac{1-a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2}\le\frac{3}{2}$. ( $a,b,c\in\mathbb{R}$)
• November 16th 2007, 11:28 AM
Opalg
Quote:

Originally Posted by james_bond
Prove that if $ab+bc+ca=1$ than $\frac{1-a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2}\le\frac{3}{2}$. ( $a,b,c\in\mathbb{R}$)

Who thinks these horrible things up (and why do I spend hours brooding over them)?

I'll assume that a, b and c are all positive (the problem's hard enough as it is, and I don't want to be bothered with negative quantities).

Notice first that $\frac{1-a^2}{1+a^2} = \frac{2-(1+a^2)}{1+a^2} = \frac2{1+a^2}-1$ (and similarly for b and c), so the inequality can be written

. . . . . $\frac1{1+a^2}+\frac1{1+b^2}+\frac1{1+c^2} \leqslant\frac94$ . . .(1)

If bc+ca+ab=1 then

. . . . . $1+a^2 = (bc+ca+ab)+a^2 = (a+b)(a+c)$ . . .(2)

(with similar inequalities for b and c).

Also, $a+b+c = (a+b+c)(bc+ca+ab) = (b+c)(c+a)(a+b) + abc$ . . .(3)

(just multiply out both sides to verify that).

By the arithmetic-geometric mean inequality,

. . . . . $abc = \sqrt{bc}\sqrt{ca}\sqrt{ab} \leqslant \frac{b+c}2\frac{c+a}2\frac{a+b}2 = {\textstyle\frac18}(b+c)(c+a)(a+b)$ . . .(4)

(that's the only place where I assume that a, b and c are positive).

It follows that

. . . . . $\frac1{1+a^2}+\frac1{1+b^2}+\frac1{1+c^2} = \frac1{(a+b)(a+c)} + \frac1{(b+c)(b+a)} + \frac1{(c+a)(c+b)}$ . . .by (2)

. . . . . . . . . . . . . . . . . . . . . . . $= \frac{2(a+b+c)} {(b+c)(c+a)(a+b)}$

. . . . . . . . . . . . . . . . . . . . . . . $= \frac{2(b+c)(c+a)(a+b) + 2abc}{(b+c)(c+a)(a+b)}$ . . .by (3)

. . . . . . . . . . . . . . . . . . . . . . . $= 2 + \frac{2abc}{(b+c)(c+a)(a+b)}$

. . . . . . . . . . . . . . . . . . . . . . . $\leqslant 2 + \textstyle\frac14 = \frac94$ . . .by (4).

By (1), that was what we wanted to prove.


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# Bounded gaps between primes learning seminar I: An Overview of the Bounded Gaps Between Primes Problem

The first talk in the series is settled! It will be at 4:00 PM, January 15th, in room MC 4062. Abstract and title as below.

An Overview of the Bounded Gaps Between Primes Problem

It is a long standing conjecture, since antiquity, that there exist infinitely many consecutive prime numbers that are separated by 2, which is of course the closest possible distance. The prime number theorem shows that the gap between $p_n$ and $p_{n+1}$ is on average $\log p_n$. It is surprising then that even proving the existence of infinitely many gaps smaller than some constant multiple of the average has proved difficult for the century that ensued the proof of the prime number theorem. In a breakthrough paper in 2009, Goldston, Pintz, and Yildirim proved that for any small constant $c > 0$, there exist infinitely many primes $p,q$ such that $|p - q| < c \log p$. In doing so they were able to relate the gaps between prime problems with a famous conjecture of Elliott and Halberstam, the first time anyone was able to connect the bounded gap problem to a major and fundamental conjecture in number theory. In May 2013, Yitang Zhang announced a proof of the existence of bounded gaps between primes and just six months later, James Maynard gave a drastically different and technically innovative proof which led to far superior estimates on the size of the gaps. In this talk I will give a brief outline of the arguments of GPY, Zhang, and Maynard.


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## Zimm plot

https://doi.org/10.1351/goldbook.Z06748
A diagrammatic representation of data on scattering from large particles, corresponding to the equation: $\frac{K_{c}}{\Delta R (\theta)} = \frac{1}{\bar{M}_{\text{w}}\ P (\theta)}+2\ A_{2}\ c+\text{ ... }$ and used for the simultaneous evaluation of the mass average molar mass, M ̄ w, the second virial coefficient of the chemical potential, A2, and (usually) the z-average radius of gyration, s 2 z 1 2; c is the mass concentration of the solute, ΔRθ the excess Rayleigh ratio, and P θ the particle scattering function that comprises (usually) the z-average radius of gyration. K depends on the solute, the temperature and the type of radiation employed. Several modifications of the Zimm plot are in frequent use; the most common one uses the excess scattering instead of the excess Rayleigh ratio.
Source:
Purple Book, 1st ed., p. 66 (http://old.iupac.org/publications/books/author/metanomski.html)


text:
# Large Scale Mapping of Fractures and Groundwater Pathways in Crystalline Hardrock By AEM

## Abstract

In hardrocks that cover about 20% of the Earth’s surface, it is difficult to locate steady sources for groundwater due to inadequate understanding of the fracture networks. A comprehensive knowledge of fracture distribution at the regional scale is necessary to delineate sustainable aquifers and manage them efficiently. The resistivity maps derived from the airborne electromagnetic (AEM) survey over the Ankasandra watershed in Karnataka, India, reveal sharp and deep zones of low formation resistivity, which indicate groundwater-bearing zones. It is found that some of these zones are hydrogeologically connected through fracture networks resulting in augmented yield. AEM results in combination with an in-depth understanding of the geological structures successfully map these groundwater-saturated fracture networks (or hydrogeological lineaments) that we term as ‘Hydrolins’. As groundwater occurrence is generally associated with lineaments, we analyzed the drilling and geophysical logs from 21 wells within a 380 sq.km area to study the relationships of various lineaments with ‘Hydrolins’, particularly in respect of their groundwater potential. AEM results, though calibrated and correlated with a limited number of well data, revealed a threshold groundwater horizon (TGWH), found to be at 80 m depth for Ankasandra watershed, beyond which a strong correlation exists between the depth of a well and its yield. While the TGWH may differ for different watersheds, the approach presented here can be readily adopted to map sustainable groundwater sources in hardrocks worldwide.

## Introduction

Groundwater is a prime source of fresh water globally accounting for roughly one-third of the total withdrawals of freshwater1,2. Large aquifers with negligible recharge have been mined for the past several decades3, resulting in a water crisis4,5,6, and estimates show that 21 out of 38 major aquifers in the world are facing groundwater depletion at alarming rates7. The crisis is severe in India, where water security is widely recognized as one of the major challenges to the nation’s economic and social development3,8,9. Increasing demand for water and the growing dependence on groundwater have resulted in dwindling of the groundwater resources and drying up of aquifer systems10,11. In this context, targeting groundwater in the Precambrian shields that mostly comprise the granitoid hardrocks is even more challenging as the groundwater occurs within narrow and often isolated fracture zones. Large parts of the tropical countries, most of them developing, are occupied by the Precambrian shields and are facing acute water shortage problems. Efficient location of narrow fracture zones in hardrocks is crucial to successful groundwater targeting. In hardrock terrains groundwater occurs within a limited depth column of secondary porosity caused by weathering and fracturing processes12. As weathered zones have a relatively higher porosity than underlying fractured bedrock, they have traditionally been the most important horizon for groundwater resource development. However, in India, over-exploitation has caused the weathered zones to dry up in many cases and groundwater has to be withdrawn from the underlying fissured and fractured bedrock that has limited porosity. The fractures provide all or most of the porosity, and effective permeability, which commonly decreases with depth12. Thus, the successful mapping and management of groundwater resources in hard rock aquifers entails delineating fracture networks and recharge zones, and in optimal siting of sustainable wells in the fractured aquifer system.

Groundwater pathways in hard rocks are usually controlled by fractures, joints, geological contacts, shear zones, faults, vugs, and other discontinuities. Their multifaceted interrelationship controls overall aquifer dynamics13. The fractures in hard rock are generated by several processes such as cooling stresses in the magma, tectonic activity14, lithostatic decompression15,16, and the weathering process17,18. In highly foliated gneisses or schists, the orientation of the fractures can also be controlled by the rock structure2,19. The bedrock thus usually consists of dense horizontal as well as crosscutting vertical to sub-vertical fractures with depth-decreasing density. Despite the fact that bedrock fractures are thin, associated weathering widens their effective aperture and enhances the effective porosity. Saturated fractures and weathered rocks are therefore good electrically conductive targets for AEM methods.

The classical approach to fracture mapping is to use interpolation between limited numbers of sparsely located borehole observations, which often yields an oversimplified assessment of the complex aquifer dynamics. In this context, fast and inexpensive regional geophysical surveys like aeromagnetic (AM) and/or AEM are effective for characterizing the subsurface from micro-to-mega scale. Initially the AM methods were used as a regional mapping tool and to aid hydrocarbon and mineral exploration. Subsequently their combination with the AEM measurements provided an efficient tool to explore for conductive sulphide minerals. Refinements in the instrumentation, processing and modeling techniques further established the singular or joint applicability of the AM and AEM methods for groundwater exploration.

Applications of the AM methods for groundwater exploration are based on the premise that fractures, lineaments and geological contacts are the favourable zones for the occurrence of groundwater. State of the art developments in the AM methodology facilitate effective mapping of these features and successful targeting of groundwater resources20,21,22,23,24,25.

Historical reviews and latest developments of the AEM methods, application of helicopter borne EM (HEM) methods to hydrological exploration for both the frequency domain (FEM) and time domain (TEM) methods as well as successful applications of multi-frequency HEM for groundwater exploration can be found in various publications26,27,28,29,30,31,32,33. More recently, these methods have been applied for a variety of problems like: saltwater intrusions34,35, freshwater pockets in tsunami-affected regions, pollution threat to aquifers36, mapping faults associated with aquifers34, hydrogeophysical problems37. Applications of time domain AEM methods covering various aspects of groundwater studies are also reported9,38,39,40,41,42,43,44.

The above mentioned applications elucidate the applicability of the AEM and AM methods in studying various aspects of groundwater exploration. Many of them deal with the delineation of structures, lineaments and fractures primarily with the objective of locating associated groundwater resources. However, a systematic study of the depth-wise distribution of fracture networks facilitating hydrological connectivity, the main thrust of the present work, has not been reported so far. Many of the recent AM studies clearly demonstrate how detailed mapping of magnetic lineaments can provide clues to target potential groundwater resources. Yet all lineaments and fractures may not necessarily be associated with groundwater. Also, there may be hydrogeological settings with fractured bedrock even in the absence of a lineament45 as shown in the present study.

A variety of electrical resistivity methods has been employed to locate water-bearing fracture zones in hardrocks46,47,48,49,50,51. As fractures are hard to detect, attempts have also been made to map them employing a combination of geophysical methods45,52,53. These approaches have been successful to a certain extent in mapping fractures on a local scale. However, they are cumbersome and expensive for mapping bedrock fracture networks at regional scale. The distribution of fractures and their groundwater potential in hardrocks is random and warrant a high density of measurements.

In the present study, we have used AEM data in combination with magnetic and borehole data to map the network of fractures in a crystalline hard rock area of 380 sq. km in Ankasandra micro-Watershed, Karnataka. The AEM results provide the key to understanding the large-scale fracture network and help in locating sustainable groundwater sources in the area.

## Study Site

The Ankasandra watershed mainly comprises Archaean Peninsular gneiss (GNS) belonging to Dharwar Super Group54 (Fig. 1). A NW-SE oriented meta ultramafic schist belt of the Sargur Group bisects the area into two nearly equal parts. The figure also features agglomerate, quartz sericite (QS), banded ferruginous quartzite (BFQ), banded iron formation (BIF) within the schist belt and dolerite dykes in the watershed mapped during the present study. The schist belts in the Dharwar craton are known to have sulfide mineralization (auriferous, at places) due to hydrothermal solutions and associated alterations in the QS, BIF and BFQ55,56, and movement and migration of mineral-rich fluids through the shear zones57. The mineralization continues up to a few kilometers with varying concentration along the strike of bands and shear zones within the schist belt58. The micro-watershed also has a few quartz veins in the central part and younger basic intrusives in the southern part. The gently undulating topography ranges from 720 m to 940 m above sea level with an average elevation of around 850 m. The area receives erratic rainfall with mean annual precipitation around 680 mm, of which around 50% occurs during the monsoon season. Groundwater is the only available water source and has been over-exploited by as much as 158% of the annual recharge59,60.

## Methods

We employed an AEM survey using a dual-moment SkyTEM304 system61, which facilitates measurement of the transient electromagnetic (TEM) response of the subsurface from very early times (~10 microsecond) to late times (up to 10 milliseconds), providing shallow to very deep subsurface information simultaneously. The system also recorded the magnetic field variations. Since the area is complex with significant geological and hydrogeological variations, closely spaced flight lines with 145 m separation were chosen and a total of 2,840 line km were flown to cover the watershed. The average flight speed was 60 km/hour while the average height of the sensor was at 35 m from the ground surface. The resulting data sampling after processing is approximately 30 m along the profiles.

The magnetic data was processed as follows: (1) stacking to a sampling of 10 Hz; (2) shifting the measurement positions of the moving magnetometer mounted on the frame to the frame center point; (3) corrections for diurnal variations, heading effect, and international geomagnetic reference field (IGRF); (4) data leveling; (5) reduction to pole and calculation of vertical and horizontal first derivatives. Finally, lineaments were delineated from maps using total magnetic intensity (TMI) and derivative data.

The AEM data was processed using the following major steps62: (1) importing of the stacked raw data for both the moments (low and high) and applying moving average filters, correction for transmitter altitude and tilt, processing system geometry, etc.; (2) removal of noisy data accounting for visible geomorphological and geological setup and power line distribution; (3) laterally-constrained inversion (LCI)63,64 along the flight lines and filtering of the noisy data with high data residual. The AEM data, after laterally constrained inversion (LCI), was inverted employing the scheme of spatially-constrained inversion (SCI)50,54,65 using a smooth model description with 30 layers and finally, the depth of investigation (DOI) was calculated66. Since the present study is significantly based on the concept of DOI, its behavior in different geoelectrical settings is briefly described below.

### Depth of Investigation

Following the diffusion principle, the skin-depth (δ) for a TEM signal due to a uniform primary field67 is given by

$$\delta =\sqrt{\frac{2\rho t}{{\mu }_{0}}}$$
(1)

where ρ is the resistivity of the earth, μ0 is the permeability of free space and t is the delay time. The above equation implies that the penetration is very deep in highly resistive earth, while it is shallow in conductive formations. This equation provides a qualitative estimate of the depth of penetration as it is based on uniform field approximation and does not take into account the actual measurement system, layering of the subsurface, sensitivity of the equipment and ambient electromagnetic noise, etc. For this reason, a measure of the DOI was developed based on the sensitivity matrix of the final output model of the inversion66. The sensitivity matrix takes in to account factors such as the layered conductivity structure of the subsurface, the system transfer function, the flight height of the sensor and the noise estimates of the data. In addition, the DOI is purely data driven as model regularization is removed before the calculation. DOI is highly dependent on the signal and noise level, but for a constant noise level it can, in general terms, be assessed as (1) limited in highly resistive terrains where the signal is low and decays very fast to “drown” in noise at very early times, (2) good in moderate resistivities, and (3) limited in very conductive terrains where the signal is high but the skin-depth is very moderate.

### Major Lineaments

Lineaments are important as the groundwater occurrence is closely associated with them. In this study the following four types of lineaments are considered: (1) Satellite lineaments, (2) Magnetic lineaments, (3) Hydrological lineaments and (4) Mineralogical lineaments. The satellite lineaments were obtained from the satellite imageries and are termed as ‘Satlins’ in this paper. The magnetic lineaments were derived from the total magnetic intensity (TMI) map of the region that revealed a number of linear dipolar anomalies. These are termed as ‘Maglins’ here. The groundwater saturated networks that provide hydrogeological connectivity to DOI patches are already described as ‘Hydrolins’. The AEM results also showed a number of linear conducting features coinciding with the sulfide mineralization in the schist belt. These are termed as mineralogical lineaments or ‘Minlins’. Relationships among various lineaments and their potential in delineating sustainable groundwater sources are studied in detail.

The AEM results were interpreted in conjunction with the lithology and the geophysical logs from 21 wells in the watershed. Fourteen wells had water quality data based on geochemical studies. Also a few (5) of them had isotope and age data. These were analyzed to study the connectivity of water from select wells with the shallow surface water. Finally, correlation between yield of wells with depth was studied that elucidated the most significant aspect of the study dealing with location of sustainable wells in hardrock terrains.

## Results

The TMI values range from 41095 nT to 41295 nT, showing good correspondence with major geological features such as dikes, schist belts, quartz veins, etc. The contact of gneiss and schist is clearly seen as dipolar anomalies in Fig. 2a. The data also reveals a number of dipolar linear features striking in NW-SE, NE-SW, EW and NS directions, which are shown by the white dotted lines denoted as ‘Maglins’. They are usually associated with structure, tectonics, and/or abrupt variations in lithology and are significant indicators of groundwater storage68.

Only a few Maglins were known prior to the survey. The lineaments striking in N-S, NNW-SSE directions reveal a discontinuity with a noticeable shift at the point of intersection with other Maglins (for instance at points marked as A, B and C) and hence they are interpreted as concealed faults that are potential groundwater flow zones. This is validated by high-yielding wells such as KW-7, 8 and 15. High-yielding well KW9 is located at the contact of the dolerite dikes and gneisses and KW12 is close to the schist-gneiss contact. However, other wells located over Maglin and geological contact viz., KW-2, 4, 11 and 13 yield moderate or low groundwater discharge. The well KW11, though located over a major Maglin attributed as fault, yields only moderately. Thus, knowledge of Maglin and geological contacts are not always adequate to locate a high-yielding well. Possibly, they do not necessarily have the same potential all along their path. Similarly, out of 5 wells located over or close to ‘Satlins’ (KW4, 5, 7, 10 and 18), 2 wells yield high, 1 moderate and 2 low groundwater discharge (Table 1). Thus characterization with other parameters like electrical resistivity is necessary to improve the location of wells with high yield.

Formation resistivity is found varying widely from 10 Ωm to 10,000 Ωm showing good contrast between various hydrogeological units. The range of resistivity derived from AEM and well-lithologs for various litho-units are listed as: (i) 15–1,500 Ωm for the top soil and unconsolidated weathered material, (ii) 10–120 Ωm for the weathered layer, (iii) 100–3000 Ωm for the fractured layer, and (iv) ≥3,000 Ωm for the compact bedrock. The upper weathered zone varies in thickness from 5 m to 30 m below the ground surface. At greater depths, resistivities are as high as 10,000 Ωm between 25 m to 250 m, indicating properties of hard and compact bedrock. This is corroborated by the borehole data from the study area as well as by published research articles on similar geological settings40.

The resistivity of subsurface geologic materials depends largely on weathering, fracturing, moisture content, mineralogy and compactness of the rock matrix, as well as overburden material. The analysis of AEM results brought out two zones with distinct resistivity behavior: (1) Low resistivity patches with deep DOI values in the gneisses over the water-filled fracture zones and (2) Linearly-oriented mineralized bands in the schist belt showing low resistivity and deep DOI values (Minlins).

The high yielding wells, KW7, 8 and 9, are located on Maglins and show resistivities of about 400, 700 and 5,000 Ωm respectively at 100 m depth (Fig. 2b). The background resistivity in the compact rock in these locations is more than ~10,000 Ωm. The well KW11 located on a Maglin and with a resistivity of ~700 Ωm has only moderate yield (Fig. 2b). Although wells KW10 and KW12 do not fall on any Maglin and are not in significant low resistivity zones, they still have high water yield. This suggests that there are some additional factors–most likely, a network of fractures–that contribute significantly to the hardrock hydrogeology.

## Discussions

Figure 3 shows the geology along the profile line (PP′) and the corresponding AEM resistivity section crosscutting the area and passing through wells KW1, 13 and 7. The resistivity models were masked below the DOI. In general, the weathered zone is found to be varying between 20 to 40 m in thickness. The underlying bedrock shows several sharp vertical resistivity zones (steep conical troughs in isolated patches) extending to depths up to 300 m. These patches are interpreted as electrically conductive water-saturated fracture zones in gneiss. The location of high-yielding well KW7 coincides with one of these patches (formation resistivity ~370 Ωm at 100 m depth). The KW13 located in the schist zone has low yield.

Poorly yielding KW1 is located at a low DOI patch in gneiss. Figure 3(c) shows a few AEM sounding responses in the form of db/dt vs. time and depth vs. resistivity curves obtained by smooth inversion along with DOI values corresponding to wells KW1, 13 and 7. Since KW1 is located in shallow bedrock with no connectivity to deeper fractures, only low moment data was available after processing, and DOI is limited to 40 m. In contrast, both low and high moments data are available for wells KW13 and 7 due to the deep conductive subsurface setup, as they fall in the schist and deep water saturated fracture zones respectively.

As mentioned earlier, a deep DOI indicates a high degree of model sensitivity extending to greater depths. We interpret deep DOI in gneiss areas as deep-seated water-saturated fracture zones. Analysis of DOI patterns at different depth levels reveals that some of the water saturated fracture zones are hydrogeologically connected (Hydrolins) and thus have enhanced potential for groundwater.

The DOI contours, representing the groundwater filled fracture zones and Hydrolins in gneiss and mineralization bands in schist, are found shrinking with depth in both the formations. The areas covered by these contours for both the formations are digitized at every 10 m from ground level to 50 m depth, at every 25 m down to 150 m and at every 50 m up to 250 m. Depth dependence of these areas, expressed as normalized percentage of total area of granite gneiss and schist respectively, are shown in Fig. 4b. The top weathered and altered zones for both the lithologies are about 25 to 30 m thick and show very little (<10%) areal reduction at the base. For deeper levels the general decrease can be described as shown  in eq. (2) below:

$$A(d)=k.{e}^{-\propto .d}$$
(2)

where A(d) is the normalized DOI area at depth d, while k and are constants specific to the survey area and lithology that can be derived by fitting an exponential decay curve to the observed data. For Ankasandra watershed these values for gneiss are:

k = 260 and  = − 0.026 while for schist the values are: k = 140 and  = − 0.012. Figure 4b clearly shows that the areal reduction of DOI patches in gneiss is faster as compared to that in schist. The reduction in schist can be ascribed to thinning and/or discontinuation of mineralized bands. However, from the view point of groundwater targeting, rate of reduction of the DOI area, that practically represents the groundwater potential, assumes significance. Both the parameters, k and , are important. While k relates to the quantum of groundwater filled fractures at the base of the weathered zone, determines how fast or slowly these fracture zones and associated Hydrolins reduce with depth. A high value of k and low value of , for example, indicates an area with very good groundwater potential.

To understand any possible relationship between the Hydrolins and other lineaments, we have plotted all the lineaments together and studied their directional characteristics. Fig. 5a shows all the lineaments and wells. The Minlins are very distinct in the schist belt. The Satlins and Maglins are mostly linear and cut across geological formations displaying a regional nature. Hydrolins display curvilinear features due to inherent heterogeneities and crystal foliations in magma69, and anisotropic steering70.

The directions and lengths of various lineaments are plotted in the form of polar bar charts (Fig. 5b). Cumulative length of lineaments are calculated for various azimuths and then analyzed for their maximum, minimum and average lengths (Table 2) and matrix of their correlation coefficients is computed (Table 3).

The Minlins show strong directionality with maximum length among all. Whereas other lineaments have relatively more directional distribution. The Hydrolins show maximum directional distributions and have highest cumulative length. Their largest number (206) implies that they also have a good areal distribution (Table 2). The correlation matrix shows very poor correlation ranging from 0.32 to 0.01 for  correlation coefficients (CC). Out of these, Hydrolins show highest CC (i.e. 0.32) with Minlins and lowest (i.e. 0.12) with Satlins. Based on the directional characteristics and length, it is evident that Hydrolins are not influenced by any particular regional structural lineaments in the study area.

The Hydrolins describe the hydrogeological connectivity among groundwater bearing fracture zones. In contrast, the Satlins represent only the first few centimeters of the geomorphology, while the Maglins do not necessarily correlate with the high yielding wells. Therefore, we believe that Hydrolins qualify as more effective indicators of groundwater occurrence than Maglins or Satlins. Order of potentiality of these lineaments for groundwater prospects is shown in eq. (3) as follows:

$${\rm{Hydrolin}} > {\rm{Maglin}} > {\rm{Satlin}} > {\rm{Minlin}}$$
(3)

Yield data from a limited number of 21 bore wells in the watershed were analyzed and correlated with the AEM DOI. While additional wells are available in the watershed, we selected only those wells with authentic and verifiable records to ensure the veracity of results. Initially correlation coefficient is determined for all the well yields with corresponding AEM DOI and then successively by removing shallower DOI wells i.e. lowering the depth ranges viz., 20–250 m, 25–250 m, …, 200–250 m. The correlation coefficient for wells with all the DOI is found to be 0.48. It reduces to 0.43 and 0.41 for correlation over wells with range of DOI values 25–250 m and 30–250 m, respectively. The decreasing trend of correlation coefficient continues till the depth range 90–250 m. Thereafter correlation coefficient starts increasing to finally reach 0.9 for wells with DOI 170–250 m. A second order polynomial trend line fit shows the minimum at 80 m DOI, which can be considered as threshold depth beyond which deep DOI wells have better fracture connectivity and hence high yield potential. We term this depth as the ‘Threshold Groundwater Horizon’ (TGWH). In case of zones deeper than TGWH, chances of isolated fractures are significantly reduced and a clear positive correlation is achieved. The shallow (<80 m) zones comprise many hydrogeological complexities such as differential weathering and fracturing, reduction in the lateral connectivity, increased aquifer compartmentalization, etc. Some zones may have very good fracture connectivity, whereas others may be isolated fractured zones. Thus the wells at connected fractured pathways will experience increased yield with depth. Whereas isolated fractures may experience lowering of yield with depth.

Figure 6b shows correlation of yield of wells with DOIs in two parts i.e. shallower wells above the TGWH and deeper wells. The correlation coefficient for shallower DOI-wells is found to be 0.04 that shows almost no correlation. However, the deeper DOI-wells reveal strong correlation (R2 = 0.64). The deeper zones beyond the TGWH are dominated by fractures with regional connectivity. While this result has useful implications, the trend beyond TGWH is based only on 7 data points. More data indeed would be useful to establish the exact nature of this trend.

Geochemical and isotope data are utilized to understand how Hydrolins control the hardrock hydrodynamics. Fig. 6c shows the regression relation of well-yield with groundwater age derived from C-14 dating and also with DOI. Well yield shows positive correlation (R2 = 0.68) with the DOI and negative correlation (R2 = 0.84) with the water age.

The bedrock fractures, in general, may have varying depth depending on local geological condition. In such conditions, groundwater occurrences are most likely expected to be more at deeper horizons under the influence of gravity. A deeper water saturated fracture will also increase the DOI. However, deeper well with strong interconnectivity and high transmissivity of fractures will facilitate movement of groundwater and hence will show lower age. In addition, C-14 dated age of water samples of the highest yielding well (i.e. KW12, Sasalu village) collected at the start of pumping and after 200 and 300 minute intervals have all shown modern age. This establishes the strong interconnectivity of fractures carrying fresh water recharged from the latest monsoon precipitation.

Although both natural and anthropogenic sources introduce NO3 contents into the groundwater, anthropogenic sources are recognized to be potential and most probable source causing increased nitrate content in groundwater. In such a case NO3 concentration ideally decreases with depth. However, because of strong lateral and vertical fracture interconnectivity and high transmissivity, no significant relationship of NO3 is seen with well yield and DOI (Fig. 6d). Even other major anions and cations do not have any noticeable correlations. Thus the above observation validates AEM derived Hydrolin as a potential groundwater pathway in crystalline hardrock.

The DOI provides the maximum depth limit of deep-seated conductive zones. This may mislead the interpreter to believe that there are continuous vertical fractures from the top to the maximum DOI. In fact, multiple sets of fractures could occur in a single well due to sets of horizontal to sub-horizontal and vertical to sub-vertical fractures. For example, in the existing 21 wells, the number of fracture sets varying from single (in KW3, 5, 11 & 13) to a maximum of seven (in KW12) have been encountered within a depth of 200 m (Table 1).

While DOI can be an effective indicator of the presence of water in deep fractures, it may be noted that a higher transmitter moment leads to a stronger S/N ratio and hence a deeper DOI. Thus a deep saturated fracture zone will produce a deeper DOI with an increase in the transmitter moment, indicating the presence of water at still deeper levels.

The results obtained by the SCI can be used in principle to create 3D images of subsurface electrical structures. However, we have used constrained inversions that assume the earth structures to be spatially coherent and the regularization concept results in an overall improvement of the resolution of geological structures that may not be well resolved by individual soundings14. A rigorous 3D inversion simultaneously considering the total survey area71,72 may improve the results. However, it would require extremely large computational resources in terms of memory, time and parallelization, particularly because fracture zones need very fine meshing. There is no software package currently available commercially to accomplish this objective.

## Conclusion

The present work emphasizes the importance of regional surveys in finding sustainable groundwater sources in hardrock terrains where most of the shallow wells located in the weathered zone have gone dry due to over exploitation and the challenge lies in locating sustainable groundwater sources in bedrock where limited secondary porosity is provided by sporadically distributed fracture zones.

The results demonstrate that the AEM surveys in combination with borehole data and geological information provide a potential tool to achieve this goal as they are capable of characterizing crystalline hardrocks and lineaments, mapping the spatial extent of fracture networks and associated hydrogeological pathways, termed as Hydrolins, that are derived from the AEM data. The DOI information efficiently demarcates the deep groundwater bearing fracture zones. Their yield potential is determined by the connectivity provided by the Hydrolins. The study also brings out a threshold depth, TGWH, below which the well-connected fracture zones are likely to provide high yielding wells. The AEM data also helps in studying the reduction in the groundwater bearing fracture zones with depth. The groundwater potential of a given watershed thus can be assessed by determining the availability of fracture zones below the TGWH. Lineaments are important in locating groundwater and the study reveals that Hydrolins provide the most significant clues followed by Maglins and Satlins. Additionally, a precise knowledge of Hydrolins can also be helpful in identifying suitable recharge zones. Knowledge of the fracture network in hardrock terrains is useful to understand regional hydrogeology and provide crucial inputs to simulate the groundwater flow system, and develop sustainable groundwater management plans. Since the study deals with the important problem of locating sustainable sources of groundwater in hardrocks, it is important that the present results supported by a limited number of groundwater data are further confirmed by expanding the analysis to larger groundwater data sets73.

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Sharma, A. & Rajamani, V. Weathering of gneissic rocks in the upper reaches of Cauvery river, south India: implications to neotectonics of the region. Chem. Geol. 166, 203–233 (2000).

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Ejepu, J. S., Olasehinde, P., Okhimamhe, A. A. & Okunlola, I. Investigation of Hydrogeological Structures of Paiko Region, North-Central Nigeria Using Integrated Geophysical and Remote Sensing. Techniques, Geosciences 7, 122 (2017).

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Grauch, V. J. S. & Bankey. Aeromagnetic Interpretations for Understanding the Hydrogeologic Framework of the Southern Española Basin, New Mexico, U.S. Geological Survey, MS 964, Federal Center, Denver, CO 80225-0046, Open-File Report 03–124 (2003).

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Joel, E. S., Olasehinde, P. I., De, D. K. & Maxwell, O. Regional Groundwater Studies Using Aeromagnetic Technique, Adapted from extended abstract based on oral presentation given at AAPG/SEG International Conference & Exhibition, Cancun, Mexico, September 6–9, 17p (2016)

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Bedrosian, P. A., Ball, L. B. & Bloss, B. R. Airborne electromagnetic data and processing within Leach Lake Basin, Fort Irwin, California. In: Geology and Geophysics Applied to Groundwater Hydrology at Fort Irwin, California, Chap. G (ed. Buesch, D. C.). U.S. Geological Survey Open File Report 2013–1024, 20 (2014).

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## Acknowledgements

We are thankful to the Director, CSIR-NGRI, Hyderabad for his support, and according permission to publish this paper. The help extended by Dr. D.V. Reddy, Dr. D. Muralidharan, Dr. R. Rangarajan, Dr. E. Nagaiah, Dr. N.C. Mondal, Dr. S. Sonkamble, Mr. N. Veerababu, Ms. Ankita Chatterjee, and Joy Choudhury, CSIR-NGRI, and Jesper Bjergsted Pedersen, HydroGeophysics Group, Aarhus University Denmark, during the project work is sincerely acknowledged. This project was sponsored by the Ministry of Water Resources, RD & GR, Government of India. Stimulating interactions with Dr. P.C. Chandra Ex-Regional Director, CGWB and scientists of CGWB, Karnataka are duly acknowledged.

## Author information

Authors

### Contributions

S.C., S.A. and S.K.V. conceived and designed the study. S.C. E.A., P.K.M. S.A. and S.K.V. performed the experiments. S.C. and P.K.M. processed and analyzed the data. S.C. interpreted the data and drafted the manuscript. E.A., S.K.V. and S.A. provided critical review.

### Corresponding author

Correspondence to Subash Chandra.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Chandra, S., Auken, E., Maurya, P.K. et al. Large Scale Mapping of Fractures and Groundwater Pathways in Crystalline Hardrock By AEM. Sci Rep 9, 398 (2019). https://doi.org/10.1038/s41598-018-36153-1

• Accepted:

• Published:

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text:
# EPS

EPS, or Encapsulated PostScript is the programming language designed for the description of the graphical information and bidimensional objects; usually it refers to some pattern (text, pictures) printed on a paper or shown at the screen of the monitor or projector.

The program, written at EPS is called "EPS document". Usually, the file name of such a document finishes with string ".eps" or ".EPS" .

Some commands of the Postscript are specified in the Tutorial [1].

## Structure of the EPS documents

At the head, the EPS document specifies the paper size in the following way:

%!PS-Adobe-2.0 EPSF-2.0
%%BoundingBox: 0 0 $A$ $B$

where $A$ and $B$ should be substituted to the numbers represented in the standard decimal positional system. The standard paper sizes have names, but in the EPS document, the paper size is specified in points. Usually, the first number $A$ refers to the width of a document, and $B$ refers to its height (for example, at the screen). In the most of cases, the scaling of the resulting image is assumed, so, the units of length do not have their usual sense, as they have in physics.

The EPS document usually contains some drawing, paintings, texts and so on. The special commands are used to describe them.

The EPS document finishes with lines

showpage
%%Trailer

No any text is expected after these lines. Even word %%Trailer happen to be not necessary; it does not affect the view of the image, nor on the conversion into the PDF file.

## EPS commands

The EPS commands are basically the same as in the PostScript; some of them are mentioned below. The EPS command either define some object (used in other commands), or draw some lines, of fill some places with certain colors, or put some texts, or put some paintings.

### Line commands

The basic line commands are moveto, lineto and stroke. The arguments of the moveto and lineto are specified before the command, they should be numbers. The command stroke draws the lines defined previously with moveto and lineto.

In TORI, for the graphics, the EPS file is often written in C++; and the beginning, header of the EPS file is written with the routine ado(FILE*, int, int) stored at ado.cin . In that header, the short abbreviations are defined for commands moveto, lineto and stroke, namely,

/M {moveto} bind def
/L {lineto} bind def
/S {stroke} bind def

After such definitions, letter "M" (separated with spacebars or carriage returns) does the same as moveto. In the similar way, letter "L" does the same as lineto, and "S" does the same as stroke.

## Roster graphics

The black and white roster graphics in EPS document can be realized with the command image.

Let the roster is stored in the bi-dimensional array $a$ of sizes $M$, $N$. Then, this roster can be placed at the page in the following way:

gsave
$M$ $N$ scale
$M$ $N$ 4 [$M$ 0 0 -$N$ 0 $N$]
<
$a$
>
image
grestore

where $a$ should be represented in the hexagonal form.

Such a code paints the matrix of squares of gray level, specified with values of array $a$. The size of each square is unity. (By default this corresponds to size one pt at the printing or the displaying). The coordinates of the left down corner of the matrix are (0,0). Such a convention allows to reduce the number of parameters at the calling. However, the position and the scale of the resulting matrix can be changed (customized) with commands translate and scale.

The example of the EPS program with the roster graphic is suggested below; its result result is shown at right.

#### Code, Roster case

Example of the EPS roster graphics
Example of the EPS vector graphics

%%BoundingBox: 0 0 322 222

/RGB {setrgbcolor} bind def
/W {setlinewidth} bind def
/M {moveto} bind def
/L {lineto} bind def
/S {stroke} bind def
1 1 translate
10 10 scale
gsave
32 22 scale
32 22 4 [32 0 0 -22 0 22]
<
fffffffffffd73100137dfffffffffff
fffffffffe500000000005efffffffff
ffffffffc20026acca62002cffffffff
fffffffc1018effffffe8101cfffffff
ffffffe202cffffffffffc202effffff
ffffff501cffffffffffffc105ffffff
fffffd008ffffffffffffff800dfffff
fffff702effffffffffffffe207fffff
fffff306ffffffffffffffff603fffff
fffff10affffffffffffffffa01fffff
ffffe00cffffffffffffffffc00effff
ffffe00cffffffffffffffffc00effff
fffff10affffffffffffffffa01fffff
fffff306ffffffffffffffff603fffff
fffff702effffffffffffffe207fffff
fffffd008ffffffffffffff800dfffff
ffffff501cffffffffffffc105ffffff
ffffffe202cffffffffffc202effffff
fffffffc1018effffffe8101cfffffff
ffffffffc20026acca62002cffffffff
fffffffffe500000000005efffffffff
fffffffffffd73100137dfffffffffff
>
image
grestore
16.000 -1.000 M
16.000 23.000 L
-1.000 11.000 M
33.000 11.000 L
1 0 0 RGB .3 W S
showpage

%%Trailer

The empty lines and the spacebar at the beginning of lines are introduced to improve the formatting at TORI and simplify the loading, they do not affect the view and conversion of this file to other formats.

It worth to compare the roster graphics to the vector graphics of a similar picture; such a vector graphics is suggesed at the second figure, produced with the code below.

#### Code, Vector case

%!PS-Adobe-2.0 EPSF-2.0
%%BoundingBox: 0 0 322 222
/RGB {setrgbcolor} bind def
/W {setlinewidth} bind def
/M {moveto} bind def
/L {lineto} bind def
/C {closepath} bind def
/S {stroke} bind def
1 1 translate
10 10 scale
16.000 11.000 10 0 360 arc C 2 W S
16.000 -1.000 M
16.000 23.000 L
-1.000 11.000 M
33.000 11.000 L
1 0 0 RGB .3 W S
showpage

The examples shows, that EPS allow to combine both, roster and vector graphics at the same page. However, for the addition of formulas, the more suitable is to combine various EPS pictures in the latex document.

While the EPS graphics are not supported at TORI, both examples above are converted to the JPG format. For this reason, at the strong zooming-in, the "vector" graphics also shows some defects of the circuar shape of the ring painted. However, the original eps documents above, being displayed, show no defects, even at very strong zooming-in. The same refers to the PDF versions of the files, made of these EPS.

## Text commands

In principle, any text can be defined using the EPS syntax. However, the programming and debugging of such document may be boring. For this reason, in TORI, the labels at the pictures are drawn using the latex language.

However, few graphical commands are still useful.

The string

/times-Roman findfont $p$ scalefont setfont

determines the Roman font of size $p$ for the placing text in the resulting picture. as usually, the default unit of length is pt.

The text can be placed with command

$x$ $y$ moveto ($s$) show

where $x$ and $y$ are coordinates of the left down cornet of the text, and $s$ is the ascii string.

Some extensions allow to put also the non–ascii characters, but the result depends on the implementation of the EPS interpreter. Especially difficult is representation of the Russian characters. During the USSR, the soviet veterans used to suppress various branches of science; in particular, they fought against cybernetics. (The official Soviet doctrine qualified the Cybernetics as pseudo-science; then, the prohibition of the Russian characters in computers was used to fight the computer–supported samizdat.) As the result, no unique system for encoding of the Russian characters (analogy of the English ascii) has been established, and the Russian texts prepared at some computer may be, by default, unreadable on another computer; in many cases, the special recoding is necessary [2]. For this reason, the English is strongly recommended for any strings introduced from PostSctipt. The complicated strings should be added with latex.

## Gallery

The EPS format is efficient tool for Reproduction of some famous paintings; in particular, those by Kazimir Malevic. Three examples are suggested below.

$~$ $~$ $~$

## References

1. http://paulbourke.net/dataformats/postscript/ Paul Bourke. PostScript Tutorial. (1990 - 1998)
2. http://www.rcom.ru/new/coders.htm Существующие кодировки русских букв. In Russian, cited for the state for year 2011: .. русскому языку не повезло - существует множество используемых в настоящее время различных кодовых таблиц для русских символов:. .. Существуют программы, которые сами пытаются подобрать вариант последовательного перекодирования, автоматически анализирую степень читабельности полученного сообщения. Но это у них не всегда получается.


text:
# 帶有數字的棋盤

The smallest sum that can be achieved is

Because

Reasoning

If the summation cannot be a negative number, the minimum sum that can be achieved is 0. If the summation can be a negative number, then, the minimum that can be achieved is -107 (if the greatest numbers in each row is negated)

This is one of the first on puzzling.SE that I can actually do! :)

There are many ways to get $$0$$. Note that each row has precise $$4$$ or each $$+/-$$, so we can translate so that it's just eight copies of $$\{1, ..., 8\}$$. By symmetry, we have $$1 + 8 - 2 - 7 = 0 \quad\text{and}\quad 3+6 - 4-5 = 0.$$ So we can easily make each row sum to $$0$$. Other combinations work. Simply alternate these down each row, eg $$\begin{matrix} 1 & -2 & -3 & 4 & 5 & -6 & -7 & 8 \\ -1 & 2 & 3 & -4 & -5 & 6 & 7 & -8 \\ 1 & -2 & -3 & 4 & 5 & -6 & -7 & 8 \\ -1 & 2 & 3 & -4 & -5 & 6 & 7 & -8 \\ 1 & -2 & -3 & 4 & 5 & -6 & -7 & 8 \\ -1 & 2 & 3 & -4 & -5 & 6 & 7 & -8 \\ 1 & -2 & -3 & 4 & 5 & -6 & -7 & 8 \\ -1 & 2 & 3 & -4 & -5 & 6 & 7 & -8 \end{matrix}$$ There are loads of similar combinations.


text:
## An old talk on mediation analysis

I talk I gave at KCL in 2006, obviously hoping that the visual allusion to both William Blake and Isaac Newton would rub off on my own reputation.

## Why change scores are generally a bad idea for PROMs data: the maths part

So, thanks to Math for the Masses I can now do equations using LaTeX, a typesetting language I have avoided learning to date, if only because so many people have told me that I simply must learn LaTeX.

To return to the problem at hand, why do we get a negative correlation between preoperative score & health gain?

The reason is that ‘health gain’ is a change score, calculated from the postoperative score minus preoperative score:

Health Gain = Postoperative Score – Preoperative Score

I haven’t worked out how to make LaTeX do ‘friendly’ equations so we will have to simplify this to:

c = b – a

The correlation between preoperative score (a) and health gain (c) is clearly the same as the correlation of a with b-a. We can write the equation for this as follows:

$r_{ac}= r_{a(b-a)}= \frac{r_{ab}\times sd_b-sd_a}{\sqrt{sd_{a}^{2}+sd_b^2-2r_{ab}\times sd_a\times sd_b}}$

The important thing about this equation the top part. The standard deviations of pre- and postoperative scores are likely to be very similar: we have the same people filling in the same questionnaire on both occasions. Similarly, rab is likely to be positive because it’s the correlation between pre- and postoperative scores, and less than 1.0.

This means that rab × sdb will be almost always be smaller than  sda so the expression will be negative, making the correlation between a and c negative.

Note also that the corrleation between pre- and postoperative scores might be quite small if the questionnaire is unreliable so the greater the measurement error, the greater the negative bias.

## Why change scores are generally a bad idea for PROMs data: a worked example

Let’s imagine we have a Patient-Reported Outcome Measure (PROM) for surgical outcomes and we give it to patients pre- and post- operatively to see if health status improves after surgery. In common with many questionnaire measures, our PROM has reliability of 0.7.

Let’s also suppose that the surgery is equally effective for everyone, increasing health status by 3 points. In this example, that’s an effect size of d=0.36.

We calculate ‘health gain’ as the difference between pre- and post- operative score:

Health Gain = Postoperative Score – Preoperative Score

We know this should be around 3 points for each person, give or take a bit due to measurement error.

What happens when we plot health gain against preoperative score?

The correlation between preoperative score and health gain is negative and significant: r = -0.46, p<0.05: can we conclude that surgery is more effective for patients with the lowest initial health status?

NO: because we know that everyone improved by the same amount, 3 points. The correlation should be zero.

Why this happens is quite easy to explain but will have to wait until I work out how to do equations in WordPress.

UPDATE: just in case you were wondering if this sort of thing happens in real life, this graph is taken from a BMJ article on PROMs (1) in which it is concluded that “better preoperative health tends to be associated with smaller, not larger, health gains “.

Look familiar?

1. J. Appleby, “Patient reported outcome measures: how are we feeling today?” BMJ 344, no. jan11 2 (January 11, 2012): d8191-d8191.

## Why change scores are generally a bad idea for PROMs data

Patient-reported outcome measures (PROMs) are essentially questionnaire scales designed to measure the treatment outcomes that matter to patients. These are usually characterised as ‘health status’ or ‘health-related quality of life’, but really there’s no reason to limit the scope of PROMs to these domains: you could argue, for example, that the numerous measures of severity of depression (PHQ-9, BDI etc.) are PROMs in their own right.

The NHS Information Centre states that:

“The health status information collected from patients by way of PROMs questionnaires before and after an intervention provides an indication of the outcomes or quality of care delivered to NHS Patients”

A key feature of PROMs analysis is the reliability of the data: the degree of measurement error entailed in collecting data using a PROM. Concerns about reliability can be dismissed because reliability is an index of unbiased measurement error and, well, that just cancels out with large samples, doesn’t it? Well, not always. And especially not if you use ‘change scores’.

The natural thing to do with a set of before and after scores is to subtract the ‘before’ from the ‘after’ and call that a ‘change score’ or ‘health gain’:

##### change score = health gain = [post-treatment score] – [pre-treatment score]

There are two related problems when you do this with PROMs data:

1. The reliability of the change score is usually much lower than the reliability of the pre- and post- scores
2. The correlation between pre-score and change score is usually strongly biased in the negative direction

What this means in practice is:

1. Change scores tend to have greater measurement error than the scores they were derived from
2. They tend also to have moderate (but spurious) negative correlations with pre-scores

Point (2) is interesting because it can be misinterpreted as an interaction effect: the treatment appears most effective for patients with the lowest pre-treatment scores. But the effect is entirely spurious.

What’s also interesting is that the size of the (spurious) negative correlation between pre-treatment score and change score increases as the reliability of the PROM decreases: the greater the measurement error, the larger the (spurious) correlation, and the more likely it is be significant.

So, to conclude: change scores are generally a bad idea for PROMs data, and the more unreliable the PROMs data, the worse the problem is.

## How sample size calculations work

What the researchers say:

“The sample size of 102 participants will be sufficient to detect an effect size of d=0.5 or greater with 80% power and significance of 0.05”

What the researchers mean:

“We have cut & pasted this from our last application as we were short of time. The effect size of d=0.5 was chosen because Cohen said something about it being a ‘medium’ effect size and it doesn’t really relate to the hypothesis of this study at all. Even if it did, 0.5 would probably be a point estimate, so really we should have chosen the lower bounds of this estimate for the sample size calculation. This would have made the sample size very large, however, so we didn’t. Likewise, we chose 80% power and 0.05 significance because everyone else does. Hopefully no-one on the funding committee will notice the similarity of this sample size calculation to every other sample size calculation.”

## Defining cluster centres in SPSS K-means cluster

A student asked how to define initial cluster centres in SPSS K-means clustering and it proved surprisingly hard to find a reference to this online. It turns out to be very easy but I’m posting here to save everyone else the trouble of working it out from scratch.

SPSS offers Hierarchical cluster and K-means clustering. K-means clustering is often used to ‘fine tune’ the results of Hierarchical clustering, taking the cluster solution from Hierarchical clustering as its inputs.

The easiest way to set this up is to read the cluster centres in from an external SPSS datafile: the problem is finding out how this data file should be formatted.

The answer is that that SPSS requires one row of data for each cluster, and one column of cluster means for each variable. The first column must be called CLUSTER_ and is simply the cluster number for each row. So for a two-cluster solution with five variables it should look like this

 CLUSTER_ Var_A Var_B Var_C Var_D Var_E 1 2.99 3.00 2.99 2.83 2.87 2 2.15 2.72 2.13 1.87 2.52

The K-means clustering procedure can then be pointed to this file by ticking the Cluster Centers ‘Read initial’ option and telling SPSS where the ‘External data file’ is saved. Note that the ‘Number of Clusters’ also has to be set to the same number as defined in the data file.

See Jane Clatworthy’s paper here for further details on different clustering methods.