2018 MCM Problem A notes
- 2019 年 10 月 8 日
- 筆記
每年一見的美賽賽題點評又見面啦~
這次工作太忙，有點遲到，實在抱歉~
還是老規矩，先走三類信息標註法，再看點評哦~
本篇是A題，後兩題會延後發佈，敬請期待~
2018 MCM Problem A: Multi-hop HF Radio Propagation(bk1)
Background: On high frequencies (HF, defined to be 3 – 30 mHz), radio waves can travel long distances (from one point on the earth』s surface to another distant point on the earth』s surface) by multiple reflections off the ionosphere and off the earth.(bk1』) For frequencies below the maximum usable frequency (MUF), HF radio waves from a ground source reflect off the ionosphere back to the earth, where they may reflect again back to the ionosphere, where they may reflect again back to the earth, and so on, travelling further with each successive hop.(bk1』』) Among other factors, the characteristics of the reflecting surface determine the strength of the reflected wave and how far the signal will ultimately travel while maintaining useful signal integrity. (imp1)Also, the MUF varies with the season, time of day, and solar conditions.(imp2) Frequencies above the MUF are not reflected/refracted, but pass through the ionosphere into space. In this problem, the focus is particularly on reflections off the ocean surface. It has been found empirically that reflections off a turbulent ocean are attenuated more than reflections off a calm ocean.(imp3) Ocean turbulence will affect the electromagnetic gradient of seawater, altering the local permittivity and permeability of the ocean, and changing the height and angle of the reflection surface.(imp3』) A turbulent ocean is one in which wave heights, shapes, and frequencies change rapidly, and the direction of wave travel may also change.(imp3』』)
Problem:
Part I: Develop a mathematical model for this signal reflection off the ocean.(spm1) For a 100-watt HF constant-carrier signal, below the MUF, from a point source on land, determine the strength of the first reflection off a turbulent ocean and compare it with the strength of a first reflection off a calm ocean.(rsc1) (Note that this means that there has been one reflection of this signal off the ionosphere.) If additional reflections (2 through n) take place off calm oceans, what is the maximum number of hops the signal can take before its strength falls below a usable signal-to-noise ratio (SNR) threshold of 10 dB? (mss1)
Part II: How do your findings from Part I compare with HF reflections off mountainous or rugged terrain versus smooth terrain? (spm2)
Part III: A ship travelling across the ocean will use HF for communications and to receive weather and traffic reports.(bk2) How does your model change to accommodate a shipboard receiver moving on a turbulent ocean?(spm3) How long can the ship remain in communication using the same multi-hop path? (mss2)
Part IV: Prepare a short (1 to 2 pages) synopsis of your results suitable for publication as a short note in IEEE Communications Magazine. (mss3)
Your submission should consist of:
· One-page Summary Sheet,
· Two-page synopsis,
· Your solution of no more than 20 pages, for a maximum of 23 pages with your summary and synopsis.
· Note: Reference list and any appendices do not count toward the 23-page limit and should appear after your completed solution.
這道題屬於「經典物理學」範疇，設定在「無線電傳播」這一經典領域，考察的是仔細理清題意所指明的條件和問題以及查閱應用已知模型公式的能力，這一切都依賴於對題目信息的準確定位和解讀，用三類信息標註發標註的相關信息標註已經列在題干中。
通過bk1s的信息和資料查閱，我們基本可以確定可以使用的無線電傳播模型（任選一經典公式作為參考），後面的imp1,2,3則切實地給出了本題需要給定公式的自變量，參數，以及可能的函數形式和構成要素：
A. 我們關注的wave strength(WS) = f(surface characteristic(SC))；
B. SC = (permittivity, permeability, height, angle)
C. MUF = f(season, time of day, solar conditions)
影響傳播的關鍵現象turbulence也逃不過上面的要素，現在spm1中的模型需要建立的就是這樣一套WSn = f(WSn – 1, SC)模型，即給定傳播條件下，強度的變化情況，這裡又涉及三種情況：
1. 空間傳播衰減；
2. 在電離層削弱；
3. 在海洋表面削弱；
其中1,2均可假設以固定衰減係數分別建立連續和離散的模型，3則需要把海洋的特殊情況一併考慮進去，對海洋表面要素對無線電強度減小方式細緻分析建模；
關於信噪比SNR，可以假設噪音以一定能力混入原信號，並以同樣的機理和有用信號一同傳播，最後可算出SNR值隨n次海面傳播的結果，題目中mss1解決，注意對未知參數帶入已知值給出參考值並作靈敏度分析；
Spm2要求比較崎嶇和光滑地形的傳播強度影響，同樣從SC的四個參數角度入手建模即可；
Spm3和mss2要求對移動船隻和turbulence建模並回答問題，在前面模型基礎上這些都是計算問題而已；
Mss3是摘要，對完整模型以及特點，適用範圍，靈敏度，魯棒性都需要有重點地闡述。
綜合看，本題審題難度不大，關鍵是要完成每一步合理而可行的假設恰好把題目提到的影響參數納入其中，即為所求，對物理原理理解及公式推導有着一定的要求，有此方面特長的同學可選此題。
