2021年美國HiMCM數學建模競賽B題中英文

中文賽題： 應對幹旱

背景：位於(yu) 內(nei) 華達州和亞(ya) 利桑那州邊界的科羅拉多河水庫米德湖是美國最大的水庫（圖 1）。2021 年夏天，米德湖降至了自 1930 年代首次填滿以來的最低水位。 由氣候變化導致的幹旱和米德湖服務的 2500 萬(wan) 人對水不斷增加的需求，使該水庫的容量縮小到其全部容量的 36% 左右。2021年8月16日，墾務局宣布了科羅拉多河有史以來的第一個(ge) 缺水聲明^[1]。最初，這個(ge) 一級缺水聲明導致亞(ya) 利桑那州、內(nei) 華達州和新墨西哥州的供水減少，其中農(nong) 業(ye) 社區是第一個(ge) 感受到削減的。

圖 1: 米德湖概覽（國家公園管理局^[2] ）

隨著世界各地幹旱的加劇，研究人員繼續開發更有效地循環水的方法。 一種方法是回收從(cong) 水槽、廁所和淋浴間流出的廢水。你可能認為(wei) 這是不可能的（甚至可能這很惡心），但這項技術已經存在並且正在改進。 在目前的大多數情況下，這些廢水在排放到當地水道之前要經過幾個(ge) 處理過程。 圖 2 顯示了水源和使用應用的示例。 處理設施使用各種技術來滿足適合用途的規格。 這些規範是將水從(cong) 特定來源引入、處理以使其達到特定目的所需的質量水平，然後使其可用於(yu) 該目的所必需的要求。 用途包括農(nong) 業(ye) 灌溉、生活用水供應（包括飲用水）、工業(ye) 和商業(ye) 用途以及地下水補給。

圖 2: 循環水的水源和用途示例（環保局）^[3]

要求：你的團隊需要使用米德湖來調查和了解更多關(guan) 於(yu) 幹旱對水庫的影響，並考慮將廢水回收作為(wei) 解決(jue) 水資源短缺的方法。 在你開始調查之前，我們(men) 提供以下信息：

• 米德湖的水位是通過以高於平均海平麵的英尺為單位的高程來測量的。
• 米德湖的麵積以英畝為單位。
• 米德湖中的水量以英畝英尺為單位。

表1給出了墾務局2010年測算的米德湖高程、麵積、容積關(guan) 係^[4]。

表 1: 米德湖不同海拔的麵積和體(ti) 積

1. 米德湖體(ti) 積。 米德湖中的水量是流入、流出和損失的函數。 科羅拉多河提供了超過 96% 的流入量、來自其他三個(ge) 支流的額外水量和湖麵的直接降水。 流出是通過釋放水（例如通過大壩）和直接從(cong) 湖中消耗而發生的。 損失通過蒸發發生。

   • 辨認和描述影響米德湖流入、流出和損失的因素。討論這些因素之間的關係及其對米德湖水量和水位的相對影響。
   • 米德湖的形狀非常不規則，深度各不相同。考慮如何驗證表 1 中所示的高度、麵積和體積關係。列出你需要哪些信息和數據，並討論如何使用數學來計算這些度量。請注意：不需要獲取這些數據或計算度量。

2. 米德湖水位。這兩(liang) 個(ge) 問題數據文件提供了有關(guan) 米德湖水位的信息，以高於(yu) 平均海平麵的英尺為(wei) 單位。

   • 模型 1：僅考慮最近幹旱期的數據，並假設最近幹旱期的模式仍在繼續。
   • 模型 2：使用 2005 年至 2020 年的水位數據，並假設這一時期的模式繼續下去。
   • 考慮提供的數據。簡要討論米德湖水位曆史數據的總體模式。定義幹旱期的標準並確定幹旱期的開始和結束。請評論最近的幹旱期與以前的幹旱期相比有什麽差異。
   • 以年為單位，建立米德湖水位的模型。使用每個模型預測 2025、2030 和 2050 年米德湖的水位。比較和評估你的兩個模型及其預測結果。

3. 根據你在第 2 部分中的模型和水位預測，討論水位變化對未來用水需求的影響，並考慮廢水的回收利用是否可以彌補所有或部分未來的短缺。

   • 確定並描述你將包含在廢水回收計劃中的因素。考慮當地領導人需要做出的決定以及他們可能設定的會影響您的計劃的優先事項。
   • 描述你的計劃以及你將如何衡量實施計劃的影響。

4. 寫(xie) 一篇一頁的非技術新聞文章，報告您調查的關(guan) 鍵要點和建議。

你們(men) 的（PDF格式）論文總頁數不能超過 25 頁並且應該包括：

• 一頁摘要。
• 目錄。
• 你們完整的解決方案。
• 一頁備忘錄。
• 參考文獻。

注意：HiMCM 競賽現在有 25 頁的限製。 你提交的論文所有部分（摘要表、目錄、參考文獻和任何附錄）的總頁數必須控製在25 頁以內(nei) 。

附件：

• 附件 A：2021 HiMCM LakeMead MonthlyElevationData.xlsx 從 1935 年到 2021 年，每年每個月末米德湖的水位高程（以英尺為單位）。墾務局數據見 https://www.usbr.gov/lc/riverops.html。
• 附件 B：2021 HiMCM LakeMead_LowHighElevationData.xlsx 每年米德湖的最高和最低水位（以英尺為單位）。 墾務局數據見 https://www.usbr.gov/lc/riverops.html。

詞匯表：

• 墾務局：管理、開發和保護美國西部 17 個(ge) 州的水資源和相關(guan) 資源的政府機構。

• 高程：以高於(yu) 平均海平麵的英尺數表示的高度量度。

• 適合用途規範：建築設計說明，表明設施（在這種情況下為(wei) 特定類型的水處理設施）滿足其特定預期用途的要求。

• 流入：進入水體(ti) 的水。 平均海平麵：海平麵的平均水平，用作測量地球高度或深度的參考或基準水平。

• 流出：離開水體(ti) 的水。

• 廢水：來自家庭、企業(ye) 、工業(ye) 過程或雨水徑流的受汙染形式的水。 廢水包括家庭用水，如淋浴、洗碗、洗衣，還有衝(chong) 馬桶。

英文賽題：Tackling the Drought

Background：Lake Mead, a Colorado River reservoir on the Nevada-Arizona border, is the largest water reservoir in the United States (Figure 1). In the summer of 2021, Lake Mead registered its lowest level on record since its initial filling in the 1930s. Drought, fueled by climate change, along with increasing demand for water by the 25 million people Lake Mead serves, has shrunk this reservoir to approximately 36 percent of its full capacity. On August 16, 2021, the Bureau of Reclamation announced the first-ever water shortage declaration on the Colorado River^[1], Initially, this Tier 1 water shortage declaration results in reduced water deliveries to the states of Arizona, Nevada, and New Mexico with agricultural communities being the first to feel the cuts.

figure 1: Lake Mead Overview (National Park Service^[2])

As droughts around the world have been increasing, researchers continue to develop means to recycle water more efficiently and effectively. One method is to recycle wastewater that flows out of our sinks, toilets, and showers. You may think this is impossible (and perhaps disgusting), but this technology already exists and is improving. Under most current circumstances, this wastewater goes through several treatment processes prior to being released into local waterways. Figure 2 shows examples of water sources and use applications. A treatment facility uses various techniques to meet fit-for-purpose specifications. These specifications are the requirements necessary to bring water in from a particular source, treat itto get it to the quality level needed for a particular purpose, and then make it available for that purpose. Some purposes include agricultural irrigation, domestic water supply (to include drinking water), industrial and commercial use, and recharging groundwater.

Figure 2. Examples of water sources and use application for recycled water (Environmental Protection Agency^[3]

Requirements：Your team decides to use Lake Mead to investigate and learn more about drought impact on water reservoirs, and to consider the recycling of wastewater as a solution to water shortages. As you begin your investigation, we provide the following information: The level of Lake Mead is measured by elevation of the water surface in feet abovemean sea level. The area of Lake Mead is measured in acres. The volume of water in Lake Mead is measured in acre-feet.

Table 1 shows the relationship of elevation, area, and volume of Lake Mead as calculated by the Bureau of Reclamation in 2010^[4]

Table 1. Area and Volume of Lake Mead by Elevation Level

1. Lake Mead Volume. The amount of water in Lake Mead is a function of inflow, outflow, and loss. The Colorado River provides over 96% of the inflow with additional water coming from three other tributaries, as well as direct precipitation on the lake surface. Outflow occurs through releasing water (for example, through a dam) and consumption directly from the lake. Loss occurs through evaporation.

   • Identify and describe factors that impact inflow, outflow, and loss in Lake Mead. Discuss the relationship of these factors and their relative influences on the volume and water level of Lake Mead.
   • Lake Mead has a very irregular shape and varying depths throughout. Consider how you would verify the elevation, area, and volume relationships shown in Table 1. Describe what information and data you would need and discuss how you would use mathematics to calculate these measures. Note: You are NOT required to obtain these data or calculate the measures.

2. Lake Mead Water Level. The two problem data files provide information about Lake Mead water levels given by elevation in feet above mean sea level.

   • Model 1: Consider data from only the most recent drought period and assume the most recent drought period's pattern continues.
   • Model 2: Use water level data from 2005 - 2020 and assume this period's pattern continues.
   • Consider the data provided. Briefly discuss overall patterns in the historical data for Lake Mead water levels. Define your criteria for drought periods and identify the beginnings and ends of periods of drought. Comment on how the most recent drought period compares to earlier ones.
   • Develop two models for the water level in Lake Mead as a function of the year. Use each model to predict the water level in Lake Mead in the years 2025, 2030, and 2050. Compare and evaluate your two models and their predictions.

3. Based on your models and water level predictions in Part 2, address the impact on future water usage demands and consider if the recycling of wastewater could make up all or a part of any future shortfalls.

   • Identify and describe the factors you would include in a plan to recycle wastewater. Consider the decisions local leaders would need to make and the priorities they might set that would impact your plan.
   • Describe your plan and how you would measure the impact of implementing your plan.

4. Write a one-page non-technical news article reporting the key takeaways and recommendations from your investigation.

Your PDF solution of no more than 25 total pages should include:

• One-page Summary Sheet.
• Table of Contents.
• Your complete solution.
• One-page Article.
• References list.

Note: The HiMCM Contest now has a 25-page limit. All aspects of your submission count toward the 25-page limit (Summary Sheet, Table of Contents, Reference List, and any Appendices).

Attachments：

• Attachment A: 2021 HiMCM LakeMead MonthlyElevationData.xlsx Elevation of water at Lake Mead (in feet above sea level) at the end of each month by year from 1935 to 2021. Bureau of Reclamation data found at https://www.usbr.gov/lc/riverops.html.
• Attachment B: 2021 HiMCM LakeMead_LowHighElevationData.xlsx The highest and lowest water elevation at Lake Mead (in feet above sea level) by year. Bureau of Reclamation data found at https://www.usbr.gov/lc/riverops.html.

Glossary：

• Bureau of Reclamation: a government agency that manages, develops, and protects waterand related resources in the seventeen western states of the Unites States.

• Elevation: a measure of height indicated by the number of feet above mean sea level.

• Fit For Purpose Specifications: a description of construction design that indicates the facility (in this case a particular type of water treatment facility) meets the requirements for its specific intended use.

• Inflow: the water entering a body of water. Mean Sea Level: an average level of the surface of the sea used as a reference or base level to measure height or depth on Earth.

• Outflow: the water leaving a body of water.

• Wastewater: the polluted form of water that comes from homes, businesses, industrial processes, or rainwater runoff. Wastewater includes water from household uses such as showering, dishwashing, laundry and, of course, flushing the toilet.

參考資料

[1]Bureau of Reclamation. Reclamation Announces 2022 Operating Conditions for Lake Powell and Lake Mead, 2021: https://www.usbr.gov/newsroom/#/news-release/3950

[2]National Park Service. Overview of Lake Mead, 2020: https://www.nps.gov/lake/learn/nature/overview-of-lake-mead.htm

[3]United States Environmental Protection Agency. 《Basic Information about Water Reuse, 2021: https://www.epa.gov/waterreuse/basic-information-about-water-reuse

[4]National Park Service. Storage Capacity of Lake Mead., 2019: https://www.nps.gov/lake/learn/nature/storage-capacity-of-lake-mead.htm