# 1. Introduction

## 1.1 Classification of hydrological forecast uncertainty

### 1.1.1 Uncertainty of hydrological phenomena

The hydrological process is affected by many factors in the process of its occurrence, development and evolution, and its state is always reflected as an unstable and fuzzy phenomenon, which is called the uncertainty of hydrological phenomenon, and can be divided into random uncertainty and fuzzy according to the shape. Uncertainty.

### 1.1.2 Hydrological model uncertainty

The structure and parameters of the hydrological model are interdependent and inseparable, and it is neither realistic nor necessary to quantify the errors of the two.

### 1.1.3 Input uncertainty

The input of the hydrological model is divided into deterministic input and uncertain input. The deterministic input is known at the forecast time, and its uncertainty is reflected in the output of the hydrological model, and the resulting uncertainty can be considered as the prior distribution of the uncertainty of the hydrological model. Uncertainty input mainly refers to quantitative precipitation forecast, and the resulting hydrological forecast uncertainty is called quantitative precipitation forecast uncertainty.

## 1.2 Uncertainty of hydrological model

### 1.2.1 Structural error of hydrological model

Hydrological models are the main tools to study the natural laws of hydrology and solve practical problems in hydrology, and are usually divided into system models, conceptual models and physical models. Each model attempts to eliminate hydrological forecast errors by describing complex hydrological physical phenomena at different levels and introducing new theories and methods. Due to the insufficient understanding of hydrological phenomena or hydrological processes, the model structure adopted in reality is not reasonable and cannot truly reflect the actual hydrological process. The parameters of the model are mainly calibrated according to the rainfall and runoff data, and the model parameters obtained in this way must have test statistics and can only reflect the average of the relevant influencing factors (meteorological factors, underlying surface factors, etc.) effect. The deterministic relationship between hydrological models is very complex, and the models are often approximated by a large number of simplified mathematical and physical equations. Most lumped models ignore the randomness of runoff and confluence on the spatial distribution of watersheds. Many models do not take into account the impact of environmental changes (such as global change, the impact of human activities, reservoirs, water diversion, etc.) on the catchment mechanism of the basin. In fact, no matter how reasonable the structure of the model is, the gap between the prediction accuracy of the hydrological model and the actual flood control requirements cannot be eliminated. 1.

### 1.2.2 Optimal error of model parameters

Hydrologists hope to use a hydrological model with fewer parameters to simulate the watershed runoff, which can reduce the uncertainty of the model when the model parameter rate is timed. Theoretically speaking, model parameters can be obtained directly or indirectly from the watershed, but because the hydrological model parameters have both physical meaning and generalization components, most model parameters can only be obtained from measured data (rainfall, evaporation, etc.) The parameter calibration is obtained, which increases the uncertainty of model parameter optimization caused by factors such as the extraction of calibration data, the selection of optimization methods, and the determination and combination of objective functions. Hydrologists have done a lot of research on hydrological model parameter optimization technology, combining mathematical and physical methods with hydrology to analyze the direct relationship between model structure and parameter uncertainty. The influence of the quality of the hydrological data used in the calibration of parameters on the calibration of the model parameters is much greater than the influence of the quantity of the selected hydrological data on the optimization of the model parameters. The quality of hydrological data depends on the amount of information about the hydrological process contained in the data and the existing errors in the data, and the information contained in the data depends on the amplitude of the hydrological process. If it includes high-water, medium-water, and low-water years, it is considered that the data contains more hydrological information, so that the parameters selected for calibration are representative. 1.

## 1.3 Different parameters have the same effect

### 1.3.1 Phenomenon

The current basin hydrological model, because its description and quantitative calculation of the hydrological physical process is too rough and general, actually only has the function of "simulation" and belongs to the "simulation model". This model is easy to simulate or reproduce the hydrological phenomena that have occurred in the past. Using the advantages of computers, as long as the combination of model parameters is continuously adjusted, there may be multiple optimal parameter groups. The output obtained has the same fitting accuracy. And forecasters often just choose a set of parameters that they think are optimal for forecasting. There is bound to be uncertainty.

### 1.3.2 Reasons

1. The objective function is multi-extremal;
2. There is a mutual compensation effect between the parameters included in the model;
3. The model parameters are random.

## 1.4 Sensitivity analysis

### 1.4.1 Concept

On the premise of keeping other parameter values ​​unchanged, change one or several parameter values, and analyze the sensitivity of the parameters to the simulation results through the change of the objective function value. 2.

### 1.4.2 Function

As an effective auxiliary tool for parameter calibration, sensitivity analysis quantitatively identifies important parameters that affect the simulation output of a state variable by studying the changes in model results caused by changes in model input, so as to effectively identify and analyze the corresponding sensitive parameters. Through the sensitivity analysis of the parameters, the parameters in the model that have a greater impact on the simulation results are determined, and the optimization of these parameters can improve the efficiency of parameter calibration and the reliability of model prediction. 3.

### 1.4.3 Classification

1. Local sensitivity analysis
Only the influence of the change of a single parameter on the model results is tested, and only the average value of other parameters is taken.
2. Global Sensitivity Analysis
Test the total impact of changes in multiple parameters on the model results, and analyze the impact of each parameter and the interaction between parameters on the model results 4.

# 2. Uncertainty Analysis Method

## 2.1 GLUE

### 2.1.1 Concept

The generalized likelihood uncertainty estimation method GLUE(Generalized Likelihood UnCertainty Estimation) is an ensemble forecasting method—the concept of a combination of parameter values ​​with restricted optimality, and is used to analyze the parameter uncertainty of a hydrological mathematical model. The reason why the model simulation results are good or bad is the combination of all model parameters, not a single model parameter.

### 2.1.2 Basic idea

Firstly, the distribution space of model parameters (model parameters that play a decisive role) is set according to the existing knowledge, that is, the prior distribution. Each parameter combination of the model is extracted according to the prior distribution, and the model determined by each parameter group is used to simulate the hydrological process. Select an appropriate likelihood function, calculate the likelihood function value between the model output and the actual measurement, and then normalize the function value as the likelihood value of each parameter group. A critical value of the likelihood function is set, and all likelihood values ​​below this critical value are assigned to zero, indicating that these parameter groups cannot represent the functional characteristics of the model; any likelihood function values ​​higher than this critical value correspond to The parameter groups of , indicate that they can characterize the functional characteristics of the model. Finally, each group of parameters is sampled according to the normalized weight, and each sample of the parameter group is used to simulate a certain hydrological process, and then the uncertainty of the model output under the specified confidence level of the hydrological process is obtained from the simulation results. sexual range 56.

## 2.2 PEST++

### 2.2.1 Concept

The forecast uncertainty is smaller than the potential forecast error, but the forecast error variance is easier to calculate.

In the under-determined parameter estimation context (which is far more representative of the innate complexity of real-world systems), a variety of PEST-suite methodologies can be used for implementation of post-calibration uncertainty analysis. These include the unique and highly efficient null-space Monte Carlo method (see PNULPAR and related utility programs). Linear uncertainty and error analysis can be accomplished very easily through the GENLINPRED utility and/or through the PREDVAR and PREDUNC suite of utility programs7.

▲The underdetermined problem will have the same effect and different parameters. The uncertainty analysis methods include the zero-space Monte Carlo method and the linear analysis method (FOSM first-order second-moment method).

As is described by Doherty (2015), if properly undertaken, the calibration process yields a parameter field of minimized error variance. This is its "passport to uniqueness". The parameter field is not correct; its potential for wrongness (which may be large) is merely minimized. Any prediction that the model makes inherits this status. That is, the prediction is not correct; however its potential for wrongness has been minimized. Hence a prediction made by a calibrated model lies somewhere near the centre of the posterior probability distribution of that prediction. The same concept can be extended to model outputs that describe environmental behaviour to which constraints must be applied8.

▲After the model parameters are calibrated, the output results generated by the operation are not accurate, but only the results with the smallest variance. Hence there is a probability distribution based on the predicted outcome. This is the uncertainty of the model parameters.

PESTPP-GLM-calculated prior and posterior parameter uncertainties are recorded at the end of its run record file. They are also recorded in a comma delimited file named case.par.usum.csv ("usum" stands for "uncertainty summary"). Prior and posterior means, standard deviations and bounds in the latter file pertain to the logs (to base 10) of parameters which are log-transformed in the PEST control file. These upper and lower posterior parameter bounds are calculated as the parameter's estimated or initial value (depending on whether or not inversion has been carried out) plus and minus two standard deviations. Under the Gaussian assumption, bounds calculated in this way approximately span the parameter's 95% posterior confidence interval8.

▲The parameter uncertainty calculated by PEST++ is stored at the end of the operation record file .rec, and is also recorded in the comma-separated file .par.usum.csv. The upper and lower bounds of the parameters are given with 95% confidence intervals. pestpp4.2.1.pdf Pages 91.

As well as calculating parameter uncertainties, PESTPP-GLM can also be asked to calculate the prior and posterior uncertainties of some predictions. This functionality is activated through use of the forecasts() control variable. The values which must be supplied for this variable are the names of predictions whose uncertainties are sought. For example forecasts(ar10,ar11)requests that prior and predictive uncertainties be evaluated for model outputs named "ar10" and "ar11" in the PEST control file on which PESTPP-GLM's operations are based. Despite the fact that these model outputs are predictions, they must be listed in the "observation data" section of the PEST control file; hence sensitivities of these model outputs to parameters are available as rows of the Jacobian matrix which is calculated by PESTPP-GLM. Model predictions should be endowed with weights of zero in a PEST control file; this is because predictions are not used to constrain parameters, and hence do not form part of a calibration dataset. (PESTPP-GLM issues a warning message if this is not the case.) The uncertainties and lower/upper bounds of predictions that are specified in this way are listed at the end of the PESTPP-GLM run record file, and in a comma-delimited file named case.pred.usum.csv. Posterior predictive lower and upper bounds are calculated by subtracting and adding two standard deviations from/to the value of the prediction as calculated by the model using initial or estimated parameter values8.

▲The uncertainty of prediction results calculated by PEST++ is stored at the end of the operation record file .rec and also recorded in the comma-separated file .pred.usum.csv. The upper and lower bounds of the forecast results are given with 95% confidence intervals. pestpp4.2.1.pdf Pages 91.

### 2.2.2 Application

Take the Xin'anjiang model as an example. For specific data, see Automatic calibration of Xin'anjiang model parameters (PEST++).

#### 2.2.2.1 Command Line

After preparing the corresponding files, execute the following commands in sequence on the command line to start parameter automatic calibration and parameter and prediction result uncertainty analysis.

```xaj.exe
tsproc.exe tsproc.dat record.txt
echo ++ forecasts(i_mod_test_1,i_mod_test_2,i_mod_test_3,i_mod_test_4,i_mod_test_5,i_mod_test_6,i_mod_test_7,i_mod_test_8,i_mod_test_9,i_mod_test_10,i_mod_test_11,i_mod_test_12,i_mod_test_13,i_mod_test_14,i_mod_test_15,i_mod_test_16,i_mod_test_17,i_mod_test_18,i_mod_test_19,i_mod_test_20,i_mod_test_21,i_mod_test_22,i_mod_test_23,i_mod_test_24,i_mod_test_25,i_mod_test_26,i_mod_test_27,i_mod_test_28,i_mod_test_29,i_mod_test_30,i_mod_test_31,i_mod_test_32,i_mod_test_33,i_mod_test_34,i_mod_test_35,i_mod_test_36,i_mod_test_37,i_mod_test_38,i_mod_test_39,i_mod_test_40,i_mod_test_41,i_mod_test_42,i_mod_test_43,i_mod_test_44,i_mod_test_45,i_mod_test_46,i_mod_test_47,i_mod_test_48,i_mod_test_49,i_mod_test_50,i_mod_test_51,i_mod_test_52,i_mod_test_53,i_mod_test_54,i_mod_test_55,i_mod_test_56,i_mod_test_57,i_mod_test_58,i_mod_test_59,i_mod_test_60,i_mod_test_61,i_mod_test_62,i_mod_test_63,i_mod_test_64,i_mod_test_65,i_mod_test_66,i_mod_test_67,i_mod_test_68,i_mod_test_69,i_mod_test_70,i_mod_test_71,i_mod_test_72,i_mod_test_73,i_mod_test_74,i_mod_test_75,i_mod_test_76,i_mod_test_77,i_mod_test_78,i_mod_test_79,i_mod_test_80,i_mod_test_81,i_mod_test_82,i_mod_test_83,i_mod_test_84,i_mod_test_85,i_mod_test_86,i_mod_test_87,i_mod_test_88,i_mod_test_89,i_mod_test_90,i_mod_test_91,i_mod_test_92,i_mod_test_93,i_mod_test_94,i_mod_test_95,i_mod_test_96,i_mod_test_97,i_mod_test_98,i_mod_test_99,i_mod_test_100,i_mod_test_101,i_mod_test_102,i_mod_test_103,i_mod_test_104,i_mod_test_105,i_mod_test_106,i_mod_test_107,i_mod_test_108,i_mod_test_109,i_mod_test_110,i_mod_test_111,i_mod_test_112,i_mod_test_113,i_mod_test_114,i_mod_test_115,i_mod_test_116,i_mod_test_117,i_mod_test_118,i_mod_test_119,i_mod_test_120,i_mod_test_121,i_mod_test_122,i_mod_test_123,i_mod_test_124,i_mod_test_125,i_mod_test_126,i_mod_test_127,i_mod_test_128,i_mod_test_129,i_mod_test_130,i_mod_test_131,i_mod_test_132,i_mod_test_133,i_mod_test_134,i_mod_test_135,i_mod_test_136,i_mod_test_137,i_mod_test_138,i_mod_test_139,i_mod_test_140,i_mod_test_141,i_mod_test_142,i_mod_test_143,i_mod_test_144,i_mod_test_145,i_mod_test_146,i_mod_test_147,i_mod_test_148,i_mod_test_149,i_mod_test_150,i_mod_test_151,i_mod_test_152,i_mod_test_153,i_mod_test_154,i_mod_test_155,i_mod_test_156,i_mod_test_157,i_mod_test_158,i_mod_test_159,i_mod_test_160,i_mod_test_161,i_mod_test_162,i_mod_test_163,i_mod_test_164,i_mod_test_165,i_mod_test_166,i_mod_test_167,i_mod_test_168,i_mod_test_169,i_mod_test_170,i_mod_test_171,i_mod_test_172,i_mod_test_173,i_mod_test_174,i_mod_test_175,i_mod_test_176,i_mod_test_177,i_mod_test_178,i_mod_test_179,i_mod_test_180,i_mod_test_181,i_mod_test_182,i_mod_test_183,i_mod_test_184,i_mod_test_185,i_mod_test_186,i_mod_test_187,i_mod_test_188,i_mod_test_189,i_mod_test_190,i_mod_test_191,i_mod_test_192,i_mod_test_193,i_mod_test_194,i_mod_test_195,i_mod_test_196,i_mod_test_197,i_mod_test_198,i_mod_test_199,i_mod_test_200,i_mod_test_201,i_mod_test_202,i_mod_test_203,i_mod_test_204,i_mod_test_205,i_mod_test_206,i_mod_test_207,i_mod_test_208,i_mod_test_209,i_mod_test_210,i_mod_test_211,i_mod_test_212,i_mod_test_213,i_mod_test_214,i_mod_test_215,i_mod_test_216,i_mod_test_217,i_mod_test_218,i_mod_test_219,i_mod_test_220,i_mod_test_221,i_mod_test_222,i_mod_test_223,i_mod_test_224,i_mod_test_225,i_mod_test_226,i_mod_test_227,i_mod_test_228,i_mod_test_229,i_mod_test_230,i_mod_test_231,i_mod_test_232,i_mod_test_233,i_mod_test_234,i_mod_test_235,i_mod_test_236,i_mod_test_237,i_mod_test_238,i_mod_test_239,i_mod_test_240,i_mod_test_241,i_mod_test_242,i_mod_test_243,i_mod_test_244,i_mod_test_245,i_mod_test_246,i_mod_test_247,i_mod_test_248,i_mod_test_249,i_mod_test_250,i_mod_test_251,i_mod_test_252,i_mod_test_253,i_mod_test_254,i_mod_test_255,i_mod_test_256,i_mod_test_257,i_mod_test_258,i_mod_test_259,i_mod_test_260,i_mod_test_261,i_mod_test_262,i_mod_test_263,i_mod_test_264,i_mod_test_265,i_mod_test_266,i_mod_test_267,i_mod_test_268,i_mod_test_269,i_mod_test_270,i_mod_test_271,i_mod_test_272,i_mod_test_273,i_mod_test_274,i_mod_test_275,i_mod_test_276,i_mod_test_277,i_mod_test_278,i_mod_test_279,i_mod_test_280,i_mod_test_281,i_mod_test_282,i_mod_test_283,i_mod_test_284,i_mod_test_285,i_mod_test_286,i_mod_test_287,i_mod_test_288,i_mod_test_289,i_mod_test_290,i_mod_test_291,i_mod_test_292,i_mod_test_293,i_mod_test_294,i_mod_test_295,i_mod_test_296,i_mod_test_297,i_mod_test_298,i_mod_test_299,i_mod_test_300,i_mod_test_301,i_mod_test_302,i_mod_test_303,i_mod_test_304,i_mod_test_305,i_mod_test_306,i_mod_test_307,i_mod_test_308,i_mod_test_309,i_mod_test_310,i_mod_test_311,i_mod_test_312,i_mod_test_313,i_mod_test_314,i_mod_test_315,i_mod_test_316,i_mod_test_317,i_mod_test_318,i_mod_test_319,i_mod_test_320,i_mod_test_321,i_mod_test_322,i_mod_test_323,i_mod_test_324,i_mod_test_325,i_mod_test_326,i_mod_test_327,i_mod_test_328,i_mod_test_329,i_mod_test_330,i_mod_test_331,i_mod_test_332,i_mod_test_333,i_mod_test_334,i_mod_test_335,i_mod_test_336,i_mod_test_337,i_mod_test_338,i_mod_test_339,i_mod_test_340,i_mod_test_341,i_mod_test_342,i_mod_test_343,i_mod_test_344,i_mod_test_345,i_mod_test_346,i_mod_test_347,i_mod_test_348,i_mod_test_349,i_mod_test_350,i_mod_test_351,i_mod_test_352,i_mod_test_353,i_mod_test_354,i_mod_test_355,i_mod_test_356,i_mod_test_357,i_mod_test_358,i_mod_test_359,i_mod_test_360,i_mod_test_361,i_mod_test_362,i_mod_test_363,i_mod_test_364,i_mod_test_365,i_mod_test_366,i_mod_test_367,i_mod_test_368,i_mod_test_369,i_mod_test_370,i_mod_test_371,i_mod_test_372,i_mod_test_373,i_mod_test_374,i_mod_test_375,i_mod_test_376,i_mod_test_377,i_mod_test_378,i_mod_test_379,i_mod_test_380,i_mod_test_381,i_mod_test_382,i_mod_test_383,i_mod_test_384,i_mod_test_385,i_mod_test_386,i_mod_test_387,i_mod_test_388,i_mod_test_389,i_mod_test_390,i_mod_test_391,i_mod_test_392,i_mod_test_393,i_mod_test_394,i_mod_test_395,i_mod_test_396,i_mod_test_397,i_mod_test_398,i_mod_test_399,i_mod_test_400,i_mod_test_401,i_mod_test_402,i_mod_test_403,i_mod_test_404,i_mod_test_405,i_mod_test_406,i_mod_test_407,i_mod_test_408,i_mod_test_409,i_mod_test_410,i_mod_test_411,i_mod_test_412,i_mod_test_413,i_mod_test_414,i_mod_test_415,i_mod_test_416,i_mod_test_417,i_mod_test_418,i_mod_test_419,i_mod_test_420,i_mod_test_421,i_mod_test_422,i_mod_test_423,i_mod_test_424,i_mod_test_425,i_mod_test_426,i_mod_test_427,i_mod_test_428,i_mod_test_429,i_mod_test_430,i_mod_test_431,i_mod_test_432,i_mod_test_433,i_mod_test_434,i_mod_test_435,i_mod_test_436,i_mod_test_437,i_mod_test_438,i_mod_test_439,i_mod_test_440,i_mod_test_441,i_mod_test_442,i_mod_test_443,i_mod_test_444,i_mod_test_445,i_mod_test_446,i_mod_test_447,i_mod_test_448,i_mod_test_449,i_mod_test_450,i_mod_test_451,i_mod_test_452,i_mod_test_453,i_mod_test_454,i_mod_test_455,i_mod_test_456,i_mod_test_457,i_mod_test_458,i_mod_test_459,i_mod_test_460,i_mod_test_461,i_mod_test_462,i_mod_test_463,i_mod_test_464,i_mod_test_465,i_mod_test_466,i_mod_test_467,i_mod_test_468,i_mod_test_469,i_mod_test_470,i_mod_test_471,i_mod_test_472,i_mod_test_473,i_mod_test_474,i_mod_test_475,i_mod_test_476,i_mod_test_477,i_mod_test_478,i_mod_test_479,i_mod_test_480,i_mod_test_481,i_mod_test_482,i_mod_test_483,i_mod_test_484,i_mod_test_485,i_mod_test_486,i_mod_test_487,i_mod_test_488,i_mod_test_489,i_mod_test_490,i_mod_test_491,i_mod_test_492,i_mod_test_493,i_mod_test_494,i_mod_test_495,i_mod_test_496,i_mod_test_497,i_mod_test_498,i_mod_test_499,i_mod_test_500,i_mod_test_501,i_mod_test_502,i_mod_test_503,i_mod_test_504,i_mod_test_505,i_mod_test_506,i_mod_test_507,i_mod_test_508,i_mod_test_509,i_mod_test_510,i_mod_test_511,i_mod_test_512,i_mod_test_513,i_mod_test_514,i_mod_test_515,i_mod_test_516,i_mod_test_517,i_mod_test_518,i_mod_test_519,i_mod_test_520,i_mod_test_521,i_mod_test_522,i_mod_test_523,i_mod_test_524,i_mod_test_525,i_mod_test_526,i_mod_test_527,i_mod_test_528,i_mod_test_529,i_mod_test_530,i_mod_test_531,i_mod_test_532,i_mod_test_533,i_mod_test_534,i_mod_test_535,i_mod_test_536,i_mod_test_537,i_mod_test_538,i_mod_test_539,i_mod_test_540,i_mod_test_541,i_mod_test_542,i_mod_test_543,i_mod_test_544,i_mod_test_545,i_mod_test_546,i_mod_test_547,i_mod_test_548,i_mod_test_549,i_mod_test_550,i_mod_test_551,i_mod_test_552,i_mod_test_553,i_mod_test_554,i_mod_test_555,i_mod_test_556,i_mod_test_557,i_mod_test_558,i_mod_test_559,i_mod_test_560,i_mod_test_561,i_mod_test_562,i_mod_test_563,i_mod_test_564,i_mod_test_565,i_mod_test_566,i_mod_test_567,i_mod_test_568,i_mod_test_569,i_mod_test_570,i_mod_test_571,i_mod_test_572,i_mod_test_573,i_mod_test_574,i_mod_test_575,i_mod_test_576,i_mod_test_577,i_mod_test_578,i_mod_test_579,i_mod_test_580,i_mod_test_581,i_mod_test_582,i_mod_test_583,i_mod_test_584,i_mod_test_585,i_mod_test_586,i_mod_test_587,i_mod_test_588,i_mod_test_589,i_mod_test_590,i_mod_test_591,i_mod_test_592,i_mod_test_593,i_mod_test_594,i_mod_test_595,i_mod_test_596,i_mod_test_597,i_mod_test_598,i_mod_test_599,i_mod_test_600,i_mod_test_601,i_mod_test_602,i_mod_test_603,i_mod_test_604,i_mod_test_605,i_mod_test_606,i_mod_test_607,i_mod_test_608,i_mod_test_609,i_mod_test_610,i_mod_test_611,i_mod_test_612,i_mod_test_613,i_mod_test_614,i_mod_test_615,i_mod_test_616,i_mod_test_617,i_mod_test_618,i_mod_test_619,i_mod_test_620,i_mod_test_621,i_mod_test_622,i_mod_test_623,i_mod_test_624,i_mod_test_625,i_mod_test_626,i_mod_test_627,i_mod_test_628,i_mod_test_629,i_mod_test_630,i_mod_test_631,i_mod_test_632,i_mod_test_633,i_mod_test_634,i_mod_test_635,i_mod_test_636,i_mod_test_637,i_mod_test_638,i_mod_test_639,i_mod_test_640,i_mod_test_641,i_mod_test_642,i_mod_test_643,i_mod_test_644,i_mod_test_645,i_mod_test_646,i_mod_test_647,i_mod_test_648,i_mod_test_649,i_mod_test_650,i_mod_test_651,i_mod_test_652,i_mod_test_653,i_mod_test_654,i_mod_test_655,i_mod_test_656,i_mod_test_657,i_mod_test_658,i_mod_test_659,i_mod_test_660,i_mod_test_661,i_mod_test_662,i_mod_test_663,i_mod_test_664,i_mod_test_665,i_mod_test_666,i_mod_test_667,i_mod_test_668,i_mod_test_669,i_mod_test_670,i_mod_test_671,i_mod_test_672,i_mod_test_673,i_mod_test_674,i_mod_test_675,i_mod_test_676,i_mod_test_677,i_mod_test_678,i_mod_test_679,i_mod_test_680,i_mod_test_681,i_mod_test_682,i_mod_test_683,i_mod_test_684,i_mod_test_685,i_mod_test_686,i_mod_test_687,i_mod_test_688,i_mod_test_689,i_mod_test_690,i_mod_test_691,i_mod_test_692,i_mod_test_693,i_mod_test_694,i_mod_test_695,i_mod_test_696,i_mod_test_697,i_mod_test_698,i_mod_test_699,i_mod_test_700,i_mod_test_701,i_mod_test_702,i_mod_test_703,i_mod_test_704,i_mod_test_705,i_mod_test_706,i_mod_test_707,i_mod_test_708,i_mod_test_709,i_mod_test_710,i_mod_test_711,i_mod_test_712,i_mod_test_713,i_mod_test_714,i_mod_test_715,i_mod_test_716,i_mod_test_717,i_mod_test_718,i_mod_test_719,i_mod_test_720,i_mod_test_721,i_mod_test_722,i_mod_test_723,i_mod_test_724,i_mod_test_725,i_mod_test_726,i_mod_test_727,i_mod_test_728,i_mod_test_729,i_mod_test_730,i_mod_test_731) >> xaj.pst
pestpp xaj.pst
```

#### 2.2.2.2 Parameter uncertainty

PEST++ is used to analyze the uncertainty of the parameters of the Xin'anjiang model, and the linear analysis method is used. The results are stored in xaj.par.usum.csv, and can also be seen at the end of the running record file xaj.rec. As shown below.

▲The prior mean is the initial value of the parameter, the prior variance is 1/41/41/4 of the parameter domain, and the upper and lower bounds of the prior are the initial value of the parameter +/−2+/-2+/−2 times the prior variance; The posterior upper and lower bounds are calculated in the same way.

#### 2.2.2.3 Uncertainty of prediction results

PEST++ can perform uncertainty analysis on the forecast results of the flow process at the outlet section of the watershed from 1995 to 1996. The analysis results are stored in xaj.pred.usum.csv and can also be seen at the end of the operation record file xaj.rec. As shown below.

▲The prior mean is the calculation result of the initial value of the parameter, the prior variance is 1/41/41/4 of the calculation result of the upper and lower boundaries of the parameter domain, and the upper and lower boundaries are the initial value of the parameter +/−2+/-2+/− 2 times the prior variance; the posterior mean is the calculation result of the parameter optimization value, and the upper and lower bounds of the posterior are obtained by the posterior mean +/−2+/-2+/−2 times the posterior variance.

▲Uncertainty analysis results of the export flow process in Chengcun watershed in 1995

▲Uncertainty analysis results of export flow process in Chengcun watershed in 1996

▲Uncertainty analysis results of export flow process in Chengcun watershed in June 1995

▲Uncertainty analysis results of export flow process in Chengcun watershed in June 1996

# 3. Sensitivity analysis method

## 3.1 Local sensitivity analysis

### 3.1.1 Morse classification screening method

Morse classification screening method 4 (Morris,1991) is a widely used local sensitivity analysis method. The Morse classification screening method is simple and effective in determining the ordering of the sensitivity of each parameter of the model.

## 3.2 Global Sensitivity Analysis

### 3.2.1 RSA

RSA(Regional Sensitivity Analysis)3 The basic idea of ​​the algorithm is to change the objective function optimization into a credible parameter set search, that is, to set acceptable conditions for the objective function value, and to search for those parameters that make the objective function value meet the acceptable conditions through random sampling in the parameter space. , record these parameters to form a credible parameter set, and on the basis of obtaining the credible parameter set, study the uncertainty of input data and the transfer of parameter uncertainty to the simulation results. Through the lISA algorithm, the distribution of parameters that meet certain requirements can be obtained, rather than a single optimal parameter. The main steps are as follows:

(1) Determine the sampling space of the possible values ​​of the parameters, that is, determine the upper and lower limits of the parameter values ​​and the spatial statistical distribution characteristics;

(2) Design the objective function, and set acceptable conditions for the value of the objective function according to the existing monitoring data, which will be used to divide the simulation results and their corresponding parameter values ​​into two types: acceptable and unacceptable. type;

(3) The parameters are randomly sampled in the sampling space, and the system simulation is carried out with the sampled parameters;

(4) classify the parameters according to the results of the parameter simulation, respectively corresponding to the 2 division types in the (2) step;

(5) Repeat steps (3) and (4) until the required number of acceptable parameters are found.

Compare the cumulative frequency of acceptable parameters with the original uniform distribution. If the parameters have a greater impact on the objective function, the farther the distribution of the acceptable parameters is from the original distribution, the more significant the impact of the parameter on the objective function and the greater its sensitivity. The higher the value, the more important it is.

## 3.3 Application

### 3.3.1 Composite dimensionless sensitivity

Using PEST++ for sensitivity analysis, see the xaj.sen file, and the results are shown in the following figure.

▲See HILL and PEST columns for sensitivity. HILL_CSS_w_reg represents the Hill and Tiedeman method, and PEST_CSS_w_reg represents the method proposed by Doherty

Dimensionless scaled sensitivities indicate the importance of an observation (here, the daily mean recharge) to the estimation of a parameter or, conversely, the sensitivity of the simulated equivalent of the observation to the parameter.Poeter and Hill (1998) explained that one sensitivity equals the derivative of a simulated value with respect to one parameter.
Composite scaled sensitivities indicate the information content of all the observations for the estimation of a parameter.CSS summarize all the sensitivities for one parameter. CSS are calculated for each parameter using the dimensionless scaled sensitivities for all observations. Because they are dimensionless, CSS can be used to compare the amount of information provided by different types of parameters. Model simulation results will be more sensitive to parameters with large CSS relative to those for other parameters.9

▲DSS stands for dimensionless scale sensitivity, CSS stands for composite scale sensitivity.

PESTPP-GLM records composite parameter sensitivities in a file named case.sen where case is the filename base of the PEST control file. These are recorded during each iteration of the inversion process. Two composite parameter sensitivities are recorded. The first is the csp statistic of Doherty (2015). PESTPP-GLM also records the composite scaled sensitivity of Hill and Tiedeman (2007) in this same file; see that text for details of its computation. Where regularization is employed in the inversion process, two sets of these two composite sensitivities are calculated. Regularization observations and prior information equations are included in one of them, while these are excluded from the other. Where they are included, the weights applied to regularization are multiplied by the current regularization weight factor.

▲ Both methods are recorded in the xaj.sen file

### 3.3.2 morris global sensitivity analysis

morris global sensitivity analysis was performed on the model parameters using pestpp-sen.exe.
The command line is as follows:

```pestpp-sen.exe xaj.pst
```

See the results in xaj.msn:

▲For the global sensitivity of morris, see the sen_mean column.

# references

1. Liu Na. Application of GLUE method in Xin'anjiang model [D]. Jiangsu: Hohai University, 2008. DOI:10.7666/d.y1268526. ↩︎ ↩︎

2. Zhang Honggang, Guo Shenglian, Wang Caijun, et al. Research on parameter optimization technology of conceptual watershed hydrological model [J]. Journal of Wuhan University (Engineering Edition), 2004,37(3):18-22,26. DOI:10.3969/j.issn. 1671-8844.2004.03.005. ↩︎

3. Meng Die.Sensitivity Analysis of Hydrological Model Parameters[J].Water Conservancy and Hydropower Technology,2012,43(2):5-8. DOI:10.3969/j.issn.1000-0860.2012.02.002. ↩︎ ↩︎

4. Bo Huijuan, Dong Xiaohua, Deng Xia. Local Sensitivity Analysis of Xin'anjiang Model Parameters [J]. People's Yangtze River, 2010,41(1):25-28. DOI:10.3969/j.issn.1001-4179.2010.01.008. ↩︎ ↩︎

5. Wen Yahui, Li Zhijia, Huo Wenbo, et al. Research on parameter uncertainty of GLUE method based on different objective functions [J]. Hydropower, 2018,44(11):10-16. DOI:10.3969/j.issn.0559- 9342.2018.11.003. ↩︎

6. Wang Lili, Bao Hongjun, Li Zhijia. Research on parameter uncertainty of basin hydrological model based on GLUE [J]. Hydropower, 2018,44(9):12-15. DOI:10.3969/j.issn.0559-9342.2018.09.004. ↩︎

7. http://www.pesthomepage.org/Uncertainty_Analysis.php ↩︎

8. White, J.T., Welter, D.E. and Doherty, J., 2018. Manual for Version 4 of PEST++. Published by CAELUM. ↩︎ ↩︎ ↩︎

9. https://pubs.usgs.gov/sir/2006/5041/section4.html ↩︎

Posted by noobcody on Wed, 01 Jun 2022 02:16:30 +0530