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For an accurate and cost-effective assessment, a forest inventory method should match the conditions of the forest stand. Variable-radius plot sampling is commonly used in the southeastern United States because it is viewed as quick and easy. It is best suited to inventories when volume estimates are a priority.

Variable-radius plots (VRPs), also referred to as point sampling or prism cruising, rely on the relationship between the diameter at breast height (DBH) of observed trees and the distance between those trees and an observer who stands at plot center. Like fixed-radius plots, VRPs are often distributed across a forest along a grid and established from a predetermined plot center. Then, using a wedge prism (figure 1) held directly over plot center, the observer rotates in a full circle, identifying and tallying trees &#;in&#; trees (figure 2). When using a wedge prism, &#;in&#; trees are those whose refracted section still overlaps the nonrefracted portion of the main bole, or stem (figure 3). &#;Out&#; trees do not overlap, while borderline trees are those that appear to have the offset portion that is visible in the prism aligned with or barely touching the edge of the main bole of the tree.

Once all in trees are tallied, the results are multiplied by the basal area factor (BAF) of the prism to determine the basal area represented by this sample point. DBH, height, and other tree measurements can be taken in the same manner used for fixed-radius plots.

Basal Area and Basal Area Factor (BAF)

Basal area can refer to the cross section of area that a single tree stem or all trees on an area occupy at a height of 4.5 feet. (breast height). This metric is used to represent the on-the-ground space that one stem is being utilized within a stand. From here, many trees can be summed to calculate basal area within space that one stem covers within a stand. This is typically done on a square feet/acre basis. At this point, basal area can be used with other measurements to provide an index of stand condition.

In many cases, basal area is determined methodically from the diameter at breast height (DBH) of trees measured within a plot. DBH refers to the diameter of a tree, outside of the bark, at a height of 4.5 feet from ground level. It is important to remember that DBH should always be measured on the uphill side of the tree.

Basal area factor (BAF) is a multiplication guide associated with the level of refraction in wedge prisms. By multiplying the number of trees counted in a tally by the BAF, an estimate of basal area is generated; 5, 10, and 20 BAF prisms are most commonly used in the Southeast.

Example:
13 trees tallied with a BAF 10 prism
13 trees * 10 = 130 feet2/acre of basal area

Benefits of Variable-Radius Plots

Variable-radius plots can be completed quickly by one person (table 1). This inventory method provides landowners and land managers with a useful summary of stand stocking in terms of basal area. It is especially well suited to bottomland hardwood forests and other areas where tree diameters are large and understory vegetation is limited. When volume calculations alone are of interest, VRPs may also be used successfully.

Drawbacks of Variable-Radius Plots

Variable-radius plots have several limiting factors that must be considered (table 1). Dense stands, vegetation on trees, and understory brush can make trees hard to see and sampling difficult. Additionally, because tree diameters influence what trees are counted, those trees with large diameters may be some distance away but still in the plot, making them easy to miss.

Other problems arise when borderline trees are encountered (figure 5). Again, borderline trees are those that cannot be classified as in or out of the plot based on visual observations with a prism. Proper determination of borderline trees involves measuring both the diameter of the borderline tree and observer distance from that tree. The diameter is then multiplied by the plot radius factor that corresponds with the BAF of the prism used (table 2). This product is referred to as the limiting distance. If the distance from the midpoint of the tree&#;s bole to the observer is less than or equal to the limiting distance, it is considered to be in and included in the plot; otherwise, it is not tallied.

Due to the increased complexity and time it takes to calculate limiting distance, sampling borderline trees can be subjected to a variety of shortcuts. These include counting every borderline tree, ignoring them, or counting every other borderline tree for that tally. These shortcut methods are not recommended as they may produce higher error rates and can negate the timesaving benefits of using variable-radius plots by increasing sampling error.

As mentioned earlier, the initial tally of a VRP only provides an estimate of the basal area for that plot. Though often used to make management decisions because of their simplicity, basal area estimates alone provide relatively little information about stand condition. For instance, a loblolly pine stand with a basal area of 60 ft2/acre and an average DBH of 6 inches would be the equivalent of about 300 trees per acre. Compare these conditions to a stand that also has 60 feet2/acre, but with an average DBH of 14 inches. This stand would be much more open with fewer trees per acre (about 55 trees per acre). Basal area alone cannot provide this information on stand condition.

Although basal area can be determined based on the prism-based tally alone, the determination of in and out trees is only the beginning of adequately assessing a forest. Once in trees are determined, height and diameter information should also be recorded for all in trees. Additional tree and stand information, such as DBH and TPA, are needed to make informed decisions and require more complicated equations to estimate using variable radius plots than with fixed-radius plots.

Finally, variable-radius plots have a higher probability of favoring the sampling of larger trees. The resulting plots may severely over- or underestimate the composition of the stand based on tree diameter, spacing, and distance from the observer.

Base Sampling Methods on Objectives

A study comparing fixed- and variable-radius plots in Georgia found that for board-foot volume estimates, variable radius plots using a BAF-10 wedge prism resulted in error rates greater than 12 percent in second-growth loblolly pine stands. This was higher than in fixed-radius plots completed on the same stand, suggesting that increased error rates are potentially a drawback to VRPs.

Alternatively, the same study also examined uneven-aged bottomland hardwood stands in Georgia and found that BAF-10 cruises were the most efficient measurement technique. Though error rates were slightly higher than those associated with 1/5 and 1/10th acre fixed-radius plots, they were not significant. Additionally, BAF-10 plots were nearly 2.5 minutes faster than 1/10th acre plots. When spread across a day of sampling, that time adds up quickly. However, using VRPs to quantify metrics such as trees per acre or measuring borderline trees can actually increase the time spent at one plot, potentially negating such benefits. This is a drawback to variable-radius plots, not fixed.

Ultimately, the choice of sampling method should be a product of the stand and the measurements needed to assess stand condition. For example, VRPs are often thought of as a more robust sampling and cost-efficient (faster) scheme for determining volume and stand conditions than fixed-radius plots within uneven-aged forest systems, due to the importance placed on large stems. Because larger trees can represent a larger proportion of the product volume within a stand, these trees are more likely to be included in plots even at longer distances from plot center. Those who are interested in diameter distribution or specific product class volumes may lean toward the unbiased approach offered by fixed-radius plots.

By understanding the nuances associated with each sampling technique, managers and landowners have the ability to select the most appropriate sampling scheme for each individual stand. This saves time and money up front by ensuring that samples are correctly measured the first time, resulting in increased profits and appropriate land management decisions.

Calculating Limiting Distance

Assume you are about to start measuring the final variable-radius plot for an inventory when you realize that you have lost your 10 BAF prism. What do you do? You can use the process for calculating limiting distance. Begin by establishing the plot center. Using a tape, measure the distance from plot center to the center of the nearest tree to the right of north. Then, measure the DBH of that tree and multiply it by 2.75 (PRF for BAF-10 prisms) to calculate the limiting distance.

If the number you get from multiplying DBH by 2.75 is greater than the distance you measured from plot center, then the tree would be counted as in your plot, and the equivalent of 10 feet2 of basal area. If the limiting distance is less than the distance you measured from plot center to the tree, then the tree is out and not counted in the tally.

Continue around plot center until all necessary trees are measured. See an example of this process is figure 7.

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Introduction

Previous posts have explained reflective, anamorphic and dispersive  prisms and their use in optical systems.  Today we will explain a prism configuration called Risley Prism.  A Risley prism is actually two wedge prisms close to each other that can be rotated independently.   This configuration was first used in optometry to measure the misalignment of eyes.  However, their current application has moved beyond optometry.

Historical Background

The Risley prism, named after its inventor Captain A. Kingsbury Risley, emerged as a significant innovation in the late 19th century, contributing to the field of optical engineering. Invented in the s, Captain Risley designed the prism with the primary goal of creating a device capable of producing controlled and precise rotations of an optical beam. This invention addressed challenges in applications such as range finding, where the need for accurate measurements and controlled deflections of light beams was paramount.

Historically, Risley prisms found notable application in various fields, particularly in the realm of military optics. During World War I and World War II, Risley prisms were incorporated into range finders and periscopes. Their ability to provide controlled angular deviations allowed military personnel to accurately gauge distances and angles, contributing to the effectiveness of artillery and navigation systems. The Risley prism&#;s role in military applications highlighted its significance in advancing optical technology for practical and strategic purposes.

Beyond the military context, Risley prisms have been employed in scientific instruments, astronomical equipment, and other precision optical devices. The design&#;s simplicity, effectiveness, and ability to achieve precise angular adjustments without the need for complex mechanisms have contributed to its enduring relevance in optical engineering. The invention of the Risley prism by Captain A. Kingsbury Risley stands as a testament to the ingenuity of optical pioneers, providing a practical solution that has endured over a century and continues to find applications in modern optics.

Principles of Operation

The Risley prism is a pair of wedge-shaped prisms that are mounted together with their flat sides facing each other (A simple schematic is shown in figure 1.) The second prism should be slighter larger than the first one to accommodate the full beam at its larger angle.  The flat surfaces of each of the wedge prisms are parallel to each other and perpendicular to the axis of rotations.  The critical aspect in the design, however, is the mechanical support.  It not only needs to keep the spacing and correct positioning for each prism, but it also needs to allow for rotation of each wedge prism in any direction, independently of each other.  . The key feature of the Risley prism is the ability to rotate one prism relative to the other around a common optical axis. This rotational movement induces a controlled angular deviation in the transmitted light beam. The amount of angular deviation is determined by the angle of the prisms and the degree to which they are rotated relative to each other. By precisely adjusting the rotation of the prisms, users can control the angular deflection of the transmitted beam with high accuracy.

The angular deflection occurs due to the varying thickness of the prisms along their wedge-shaped profiles. As light passes through the prisms, it experiences different optical path lengths, leading to a change in the beam&#;s direction. The relationship between the angular deviation and the rotation of the prisms follows a trigonometric function, providing a predictable and controllable means of adjusting the output angle of the transmitted light.

One of the notable advantages of Risley prisms lies in their simplicity and versatility. The absence of complex mechanical components allows for quick and precise adjustments, making them ideal for applications requiring controlled angular deviations, such as range finders, periscopes, and other optical instruments. The technical workings of Risley prisms showcase their effectiveness in providing a straightforward yet powerful solution for manipulating the direction of light beams in various optical systems.

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Figure 1. Circular patterns created by rotating wedge prisms in a Risley configuration.  Image from Thorlabs: Risley Prism Application Notes

Scanning Pattern

One unique aspect of Risley Prisms is the scanning pattern they can create.  The pattern will change depending on the direction of rotation of each wedge prism and their relative angular velocity with respect to each other.  In general, a Risley Prism will create a series of patterns called Rose Curves.  Two of these patterns are shown in figure 1.  A couple of things are important to note:  one, line and circular pattern are a subset of the Rose curves; and two, there is a gap in the center of the optical axis for almost every pattern.

Figure 2.  Rose Curves created by a Risley Prism.   Image from Thorlabs: Risley Prism Application Notes

Applications and Case Study

Risley prisms have proven to be versatile optical components with applications spanning a range of industries. One notable application is in laser ranging systems, where Risley prisms are employed to precisely steer laser beams for distance measurement. These systems find use in surveying, geodetic measurements, and remote sensing, providing accurate and controlled angular deflections essential for determining distances with high precision.

In astronomy, Risley prisms are utilized in spectrographs and telescopes for fine-tuning the orientation of light entering the instruments. Their ability to perform controlled angular adjustments allows astronomers to optimize observations, compensate for atmospheric effects, and enhance the accuracy of measurements. This application showcases the adaptability of Risley prisms in addressing the complex optical requirements of astronomical instrumentation.

Case Study: Laser Target Designators:

One compelling case study involves the use of Risley prisms in laser target designators. These systems are crucial in military applications for precisely marking targets for guided munitions. Risley prisms enable the controlled steering of laser beams, allowing operators to designate targets with accuracy even at extended distances. The versatility of Risley prisms in these designators enhances the effectiveness of laser-guided munitions, contributing to improved targeting capabilities and minimizing collateral damage.

Case Study: Remote Sensing Lidar Systems:

Risley prisms play a vital role in remote sensing lidar (light detection and ranging) systems, particularly those used in environmental monitoring and forestry. Lidar systems utilize lasers to measure distances and create detailed three-dimensional maps of terrain. Risley prisms enable controlled beam steering, allowing lidar systems to efficiently scan large areas and capture high-resolution data. This application demonstrates how Risley prisms contribute to advancements in environmental research and resource management.

These real-world applications and case studies underscore the versatility and precision of Risley prisms across diverse fields. From laser ranging systems and astronomy instruments to military target designators and lidar technology, Risley prisms continue to play a pivotal role in advancing optical solutions for a wide range of practical challenges.

Comparison of Risley Prisms with Other Optical Beam Steering Devices:

Mechanism of Beam Steering:

• Risley Prisms: Utilize the rotation of two wedge-shaped prisms to induce controlled angular deflection of the transmitted beam. The amount of deflection is determined by the angle of the prisms and the degree of their rotation.
• Galvanometric Mirrors: Rely on the movement of precisely controlled mirrors, typically mounted on galvanometers, to steer the laser beam. The mirrors can be tilted to direct the beam in different directions.

Precision and Accuracy:

• Risley Prisms: Offer high precision and accuracy in beam steering, especially when fine adjustments are required. The angular deviation is directly related to the rotation angle, providing predictable and reliable control.
• Acousto-Optic Devices: Use acoustic waves to create periodic variations in the refractive index, deflecting the laser beam. While acousto-optic devices can achieve fast beam steering, the precision may be limited compared to Risley prisms in certain applications.

Speed of Operation:

• Risley Prisms: Provide relatively slower beam steering compared to some other devices, making them well-suited for applications where precise control is more critical than rapid movement.
• Electro-Optic Modulators: Alter the polarization state of the light beam in response to an applied electric field, allowing for rapid changes. This technology is suitable for high-speed beam steering requirements.

Complexity and Size:

• Risley Prisms: Characterized by a simple and compact design, making them easy to integrate into optical systems without adding significant bulk. The absence of complex mechanical components enhances their reliability.
• Liquid Crystal Spatial Light Modulators: Utilize liquid crystal arrays to manipulate the phase of the incident light, allowing for beam steering. These devices can be more complex and may have larger footprints compared to Risley prisms.

Applications:

• Risley Prisms: Versatile and well-suited for applications requiring controlled angular deviations, such as laser ranging systems, astronomical instruments, and laser target designators.
• Micro-Electro-Mechanical Systems (MEMS): Employ tiny, movable mirrors to steer laser beams rapidly, making them suitable for applications like laser displays and microprojectors.

Recent Advancements in Risley Prism Technology:

Recent advancements in Risley prism technology have focused on improving precision, versatility, and integration with modern optical systems. Notable developments include the incorporation of advanced materials and manufacturing techniques to enhance the performance of Risley prisms. Metamaterials and nanostructuring have been explored to tailor the optical properties of prisms, potentially enabling more efficient beam steering and reducing aberrations.

Integration with adaptive optics systems represents another significant advancement. By combining Risley prisms with adaptive optics, which dynamically correct optical aberrations, researchers aim to achieve real-time adjustments and improved performance in changing environmental conditions. This integration holds promise for applications in astronomy, laser communication, and remote sensing where environmental factors can impact optical performance.

Potential Future Developments:

The future of Risley prism technology holds exciting possibilities, with researchers exploring avenues for further innovation:

1. Enhanced Precision and Resolution: Ongoing research seeks to enhance the precision and resolution of Risley prism systems. Advances in manufacturing techniques, such as precision machining and nanofabrication, may contribute to the development of prisms with reduced tolerances and improved angular control.
2. Multi-Wavelength Capabilities: Future developments may focus on designing Risley prism systems capable of steering multiple wavelengths simultaneously. This could broaden their applications in areas such as telecommunications, where the ability to control different wavelengths independently is valuable
3. Adoption in Augmented and Virtual Reality: As augmented and virtual reality technologies advance, Risley prisms could find new applications in these domains. Their ability to steer laser beams with precision may contribute to improved optical displays and imaging systems, enhancing user experiences.
4. Miniaturization and Compact Designs: Future developments may focus on miniaturizing Risley prism systems for integration into compact and portable devices. This could lead to the creation of lightweight and agile optical systems for applications such as wearable optics and drone-based sensing.

Conclusions

In conclusion, Risley prisms stand as versatile and reliable components in the realm of optical engineering, offering precise control over the direction of light beams. Recent advancements have propelled Risley prism technology forward, enhancing precision, exploring novel materials, and integrating with adaptive optics for improved performance in dynamic environments. These developments have expanded the range of applications, from laser ranging systems and astronomical instruments to augmented reality displays and quantum communication systems. The potential for further innovation, including enhanced multi-wavelength capabilities and miniaturization, promises a continued evolution of Risley prism technology, making it a valuable asset for addressing diverse optical challenges. As optical engineering continues to progress, Risley prisms remain at the forefront, providing a robust and adaptable solution for controlled beam steering in various fields.

Risley Prisms Explained: FAQs 

What is a Risley prism and how does it work?

A Risley prism is an optical device used for precise beam steering, consisting of two rotating wedge prisms that can direct light in various directions.

What are the primary applications of Risley prisms?

They are commonly used in laser targeting, beam alignment, and precise light control in various optical systems.

How does the rotation of Risley prisms affect light direction?

The rotation of the two prisms changes the refraction angle of the light, allowing for precise control over the beam&#;s direction.

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