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Question 1
ప్రశ్న 1
Which data series is most directly associated with conventional annual-maximum flood-frequency analysis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
An annual maximum series contains one extreme observation from each year. For flood-frequency analysis, this is normally the largest flood peak recorded during that year. Thus a 30-year record generally provides 30 annual-maximum peak values. Smaller floods occurring during the same year are not separately included in this particular series. Therefore Option B is correct.
వివరణ:
Question 2
ప్రశ్న 2
Why is the Gumbel distribution particularly relevant to flood-frequency analysis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The Gumbel distribution is also known as the Extreme Value Type I distribution. It is widely used for statistical modeling of extreme maxima in hydrology. Annual maximum flood peaks and maximum rainfall values are typical applications. Its use does not imply that every flood magnitude has equal probability. Therefore Option B is correct.
వివరణ:
Question 3
ప్రశ్న 3
If p is the annual probability that a specified flood magnitude will be equaled or exceeded, what is its return period T?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Return period is the reciprocal of annual exceedance probability. Thus T = 1/p when the annual exceedance probability is p. A flood with p = 0.01 therefore has a return period of 100 years. The return period is a long-term statistical average rather than a fixed time interval between events. Therefore Option C is correct.
వివరణ:
Question 4
ప్రశ్న 4
What is the annual exceedance probability of a 25-year flood?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Annual exceedance probability is p = 1/T. For T = 25 years, p = 1/25 = 0.04. Expressed as a percentage, this is 4 percent. The complementary non-exceedance probability is 96 percent. Therefore Option B is correct.
వివరణ:
Question 5
ప్రశ్న 5
For a 50-year flood, what is the annual non-exceedance probability F associated with the design magnitude?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The annual exceedance probability is 1/50 = 0.02. The corresponding non-exceedance probability is F = 1 - 0.02. Therefore F = 0.98. This means there is a 98 percent probability in any year that the annual maximum will not exceed that design magnitude under the fitted model. Hence Option C is correct.
వివరణ:
Question 6
ప్రశ్న 6
Which interpretation of a 100-year flood is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A 100-year return period corresponds to an annual exceedance probability of 1/100 or 1 percent. The terminology does not establish a fixed calendar spacing between floods. Two 100-year floods can occur in consecutive years under the usual independence assumption. Similarly, a particular 100-year interval may contain none or more than one such event. Therefore Option C is correct.
వివరణ:
Question 7
ప్రశ్న 7
Assuming independent annual flood occurrences, what is the probability that a 20-year flood will be equaled or exceeded at least once during the next 10 years?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A 20-year flood has annual exceedance probability p = 1/20 = 0.05. The probability of no exceedance in one year is therefore 0.95. Under independence, the probability of no exceedance in 10 years is 0.95^10, approximately 0.5987. Hence the probability of at least one exceedance is 1 - 0.5987 = 0.4013, or about 40.1 percent. Therefore Option C is correct.
వివరణ:
Question 8
ప్రశ్న 8
For the Gumbel distribution, if F is the non-exceedance probability, which expression gives the reduced variate y?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The Gumbel cumulative distribution can be written as F = exp[-exp(-y)]. Solving this relation for the reduced variate gives y = -ln[-ln(F)]. For a return-period calculation, F is generally taken as 1 - 1/T. The reduced variate increases as the return period increases for upper extremes. Therefore Option B is correct.
వివరణ:
Question 9
ప్రశ్న 9
What is the Gumbel reduced variate for a 10-year return period, using yT = -ln[-ln(1 - 1/T)]?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For T = 10 years, the non-exceedance probability is F = 1 - 1/10 = 0.90. Thus yT = -ln[-ln(0.90)]. Evaluating the logarithms gives approximately 2.2504. Rounded to three decimals, the reduced variate is 2.250. Therefore Option C is correct.
వివరణ:
Question 10
ప్రశ్న 10
What is the approximate Gumbel reduced variate for a 50-year return period?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For T = 50 years, F = 1 - 1/50 = 0.98. The reduced variate is yT = -ln[-ln(0.98)]. This evaluates to approximately 3.9019. Therefore the appropriate rounded value is 3.902. Hence Option C is correct.
వివరణ:
Question 11
ప్రశ్న 11
As the return period T increases for upper-tail Gumbel flood estimates, what normally happens to the reduced variate yT?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A larger return period corresponds to a smaller annual exceedance probability. Its non-exceedance probability therefore approaches unity. In the Gumbel transformation, yT = -ln[-ln(F)] increases as F approaches one. Consequently larger return periods are associated with larger reduced variates and generally larger estimated upper-tail floods. Therefore Option C is correct.
వివరణ:
Question 12
ప్రశ్న 12
For the asymptotic standard Gumbel reduced variate used in large-sample frequency analysis, what are its mean and standard deviation approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For the standard Gumbel reduced variate, the theoretical mean is Euler's constant, approximately 0.5772. Its standard deviation is pi divided by square root of 6, approximately 1.2825. These constants lead to the asymptotic Gumbel frequency-factor expression. Finite-sample procedures may instead use sample-size-dependent reduced means and reduced standard deviations. Therefore Option B is correct.
వివరణ:
Question 13
ప్రశ్న 13
Which expression gives the asymptotic Gumbel frequency factor KT when yT is the reduced variate?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The frequency factor standardizes the Gumbel quantile relative to the mean and standard deviation. For the asymptotic Gumbel distribution, the reduced-variate mean is approximately 0.5772 and its standard deviation is pi/sqrt(6). Thus KT = (yT - 0.5772)/(pi/sqrt(6)). This is equivalently KT = (sqrt(6)/pi)(yT - 0.5772). Therefore Option B is correct.
వివరణ:
Question 14
ప్రశ్న 14
Using the asymptotic Gumbel method, what is the frequency factor for a 10-year flood if y10 = 2.2504?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The asymptotic frequency factor is K = (sqrt(6)/pi)(yT - 0.5772). Substituting y10 = 2.2504 gives K approximately 0.7797 × 1.6732. The resulting value is approximately 1.3046. Rounded to three decimals, K is 1.305. Therefore Option C is correct.
వివరణ:
Question 15
ప్రశ్న 15
Annual maximum floods at a site have mean 500 m³/s and standard deviation 100 m³/s. Using the asymptotic Gumbel frequency factor K10 = 1.305, what is the estimated 10-year flood?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The frequency-factor equation is QT = Qmean + KT s. Substitution gives Q10 = 500 + 1.305 × 100. This equals approximately 630.5 m³/s. The result is greater than the mean because the 10-year event lies in the upper part of the fitted extreme-value distribution. Therefore Option C is correct.
వివరణ:
Question 16
ప్రశ్న 16
For a flood series with mean 800 m³/s and standard deviation 120 m³/s, estimate the 50-year flood using the asymptotic Gumbel factor K50 = 2.592.
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The Gumbel frequency-factor relation is QT = Qmean + KT s. Hence Q50 = 800 + 2.592 × 120. The increment above the mean is 311.04 m³/s. Thus the estimated design flood is approximately 1111.04 m³/s, or 1111.0 m³/s to the stated precision. Therefore Option C is correct.
వివరణ:
Question 17
ప్రశ్న 17
A Gumbel distribution is written as F(Q) = exp[-exp(-(Q - u)/a)], where u is the location parameter and a is the positive scale parameter. If u = 400 m³/s, a = 100 m³/s and T = 100 years, what is the approximate design flood?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a 100-year flood, the Gumbel reduced variate is approximately y100 = 4.6001. From the stated distribution, Q = u + a y. Thus Q100 = 400 + 100 × 4.6001. This gives approximately 860.0 m³/s. Therefore Option C is correct.
వివరణ:
Question 18
ప్రశ్న 18
Using the asymptotic method-of-moments relations for a Gumbel distribution, a = s sqrt(6)/pi and u = Qmean - 0.5772a. If Qmean = 500 m³/s and s = 100 m³/s, which pair is approximately correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The scale parameter is a = 100 × sqrt(6)/pi. Since sqrt(6)/pi is approximately 0.7797, a is approximately 78.0 m³/s. The location parameter is then u = 500 - 0.5772 × 78.0. This gives approximately 455.0 m³/s. Therefore Option B is correct.
వివరణ:
Question 19
ప్రశ్న 19
For a Gumbel distribution with location parameter u = 455 m³/s and scale parameter a = 78 m³/s, what is the approximate 25-year flood if y25 = 3.1985?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For the stated parameterization, the quantile is QT = u + a yT. Substituting gives Q25 = 455 + 78 × 3.1985. The second term is approximately 249.5 m³/s. Thus Q25 is approximately 704.5 m³/s. Therefore Option C is correct.
వివరణ:
Question 20
ప్రశ్న 20
In a finite-sample version of Gumbel frequency analysis, the frequency factor is written KT = (yT - ybar_n)/Sn. What do ybar_n and Sn represent?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Finite-sample Gumbel procedures can use reduced-variate constants that depend on the number of years in the sample. The quantities ybar_n and Sn are respectively the reduced mean and reduced standard deviation for that sample size. They should not be confused with the mean and standard deviation of the actual discharge observations. The resulting KT is then used with the observed discharge mean and standard deviation. Therefore Option B is correct.
వివరణ:
Question 21
ప్రశ్న 21
For a finite-sample Gumbel calculation, yT = 3.20, ybar_n = 0.53 and Sn = 1.10. If the flood-series mean is 600 m³/s and standard deviation is 80 m³/s, what design flood is estimated?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The frequency factor is K = (3.20 - 0.53)/1.10. This gives K approximately 2.4273. The estimated design flood is then Q = 600 + 2.4273 × 80. This equals approximately 794.2 m³/s. Therefore Option C is correct.
వివరణ:
Question 22
ప్రశ్న 22
Thirty years of annual maximum floods are arranged from largest to smallest. Using the Weibull plotting position T = (N + 1)/m, what return period is assigned to the third-largest flood?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The record length is N = 30 and the third-largest value has rank m = 3. The Weibull plotting position gives T = (30 + 1)/3. Thus T = 31/3 = approximately 10.33 years. This is an empirical plotting-position return period for the observed ranked value, not a guarantee of future timing. Therefore Option C is correct.
వివరణ:
Question 23
ప్రశ్న 23
For the same fitted Gumbel distribution, how should the estimated 100-year flood generally compare with the estimated 20-year flood?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For an upper-extreme Gumbel distribution, increasing return period reduces annual exceedance probability. The corresponding non-exceedance probability and reduced variate both increase. Since the scale parameter is positive, the estimated flood magnitude therefore increases with return period. Hence a 100-year design flood is generally greater than a 20-year flood from the same fitted distribution. Therefore Option C is correct.
వివరణ:
Question 24
ప్రశ్న 24
Which situation most directly weakens the validity of a conventional flood-frequency analysis based on a stationary annual-maximum record?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Classical flood-frequency analysis assumes that the available record is representative of a consistent underlying population. A major reservoir, urbanization, land-use change or other systematic alteration can change the statistical characteristics of annual floods. Combining distinctly different regimes without appropriate treatment can therefore bias frequency estimates. Independence and consistency of the annual maxima are important assumptions rather than problems. Therefore Option B is correct.
వివరణ:
Question 25
ప్రశ్న 25
Consider the following statements about Gumbel flood-frequency analysis: I. Annual exceedance probability is the reciprocal of return period. II. A 100-year flood can occur in two successive years. III. The Gumbel reduced variate for return period T may be written yT = -ln[-ln(1 - 1/T)]. IV. Estimation far beyond the length of the observed record generally carries increasing statistical uncertainty. Which option is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Statement I correctly expresses the relationship between annual exceedance probability and return period. Statement II is correct because return period is not a mandatory waiting time between independent floods. Statement III gives the standard Gumbel reduced-variate expression for an upper-tail return-period estimate. Statement IV is correct because large-return-period estimates require increasing extrapolation beyond the available sample. Therefore all four statements are correct and Option D is the valid answer.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: B. The single largest flood peak observed in each year సమాధానం: B.
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Question 2 ప్రశ్న 2Answer: B. It is an extreme-value distribution suitable for modeling maxima such as annual flood peaks సమాధానం: B.
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Question 3 ప్రశ్న 3Answer: C. T = 1/p సమాధానం: C.
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Question 4 ప్రశ్న 4Answer: B. 4% సమాధానం: B.
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Question 5 ప్రశ్న 5Answer: C. 0.98 సమాధానం: C.
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Question 6 ప్రశ్న 6Answer: C. It has approximately a 1% chance of being equaled or exceeded in any individual year సమాధానం: C.
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Question 7 ప్రశ్న 7Answer: C. 40.1% సమాధానం: C.
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Question 8 ప్రశ్న 8Answer: B. y = -ln[-ln(F)] సమాధానం: B.
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Question 9 ప్రశ్న 9Answer: C. 2.250 సమాధానం: C.
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Question 10 ప్రశ్న 10Answer: C. 3.902 సమాధానం: C.
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Question 11 ప్రశ్న 11Answer: C. It increases సమాధానం: C.
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Question 12 ప్రశ్న 12Answer: B. Mean = 0.5772 and standard deviation = 1.2825 సమాధానం: B.
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Question 13 ప్రశ్న 13Answer: B. KT = (sqrt(6)/pi)(yT - 0.5772) సమాధానం: B.
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Question 14 ప్రశ్న 14Answer: C. 1.305 సమాధానం: C.
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Question 15 ప్రశ్న 15Answer: C. 630.5 m³/s సమాధానం: C.
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Question 16 ప్రశ్న 16Answer: C. 1111.0 m³/s సమాధానం: C.
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Question 17 ప్రశ్న 17Answer: C. 860 m³/s సమాధానం: C.
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Question 18 ప్రశ్న 18Answer: B. a = 78.0 m³/s and u = 455.0 m³/s సమాధానం: B.
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Question 19 ప్రశ్న 19Answer: C. 704.5 m³/s సమాధానం: C.
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Question 20 ప్రశ్న 20Answer: B. Sample-size-dependent reduced mean and reduced standard deviation of the Gumbel variate సమాధానం: B.
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Question 21 ప్రశ్న 21Answer: C. 794.2 m³/s సమాధానం: C.
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Question 22 ప్రశ్న 22Answer: C. 10.33 years సమాధానం: C.
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Question 23 ప్రశ్న 23Answer: C. The 100-year flood should generally be larger సమాధానం: C.
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Question 24 ప్రశ్న 24Answer: B. A major reservoir or land-use change introduces a systematic shift in the flood regime during the record సమాధానం: B.
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Question 25 ప్రశ్న 25Answer: D. I, II, III and IV సమాధానం: D.