A1. Nonlinear Computational Unit (NCU)

Appendix

A1. Nonlinear computational unit (NCU)

The building block of the NCUC (nonlinear computational units cascaded) model is the nonlinear computational unit (NCU), which is a virtual unit. The concept of the NCU is simple and intuitive. The role of an NCU plays in the NCUC model is similar to that of a neuron in the ANN. For clarifying the characteristics of the NCUC model, the characteristics of the NCU are first explained.

An NCU possesses at least three basic components, namely, the entrance vent for the input, the elevation of the initial storage and several composite vents. An NCU that only possesses these three basic components is the simplest NCU which is called the type-A NCU (hereinafter referred to as the A-NCU). The A-NCU is depicted in Fig. A-1. The operations of the A-NCU are defined as follows:

(a1)

(a2)

where EIS is the elevation of the initial storage, is the sill of the NCU at time 0, is the output of the i-th composite vent at time 0, andm is the number of the composite vents. Eqs. (a1) and (a2) are initial conditions of the A-NCU. EIS is a constant, and the sill of the NCU is initialized as EIS. The output of the A-NCU is calculated by

(a3)

where t is time, is the output of the A-NCU at time t, q is the number of time intervals, and is the output of the i-th composite vent at time t. is calculated using

(a4)

where is the threshold function defined as

(a5)

CVCi is the composite vent coefficient of the i-th composite vent, SCVi is the sill of the i-th composite vent, and is the sill of the NCU at time t. A composite vent has two parameters: composite vent coefficient and sill. A composite vent generates its output only when the sill of the NCU is larger than that of the composite vent. The output of the NCU is the summation of the output of all composite vents. In Eq. (a4), is a state variable and is evaluated by

(a6)

where is the recovery function defined as:

(a7)

and is the input of the A-NCU at time t. In the initial stage (), the input of the NCU enters the entrance vent and then increases the sill of the NCU. From Eq. (a6), it can be found that is updated at every time step. is updated using the recovery rule. When the value of is larger than EIS, the sill is then updated as . Otherwise, the new sill is equal to . Eqs. (a1) to (a7) are referred to as the set of the basic operation rules of an NCU.

A slightly more complicated NCU is the type-B NCU (hereinafter referred to as the B-NCU). As shown in Fig. A-2, in addition to the three basic components of the A-NCU, the B-NCU possesses a single vent. The single vent can be regarded as a specialized composite vent whose sill is zero. The operations of the B-NCU are identical to the set of the basic operations of an NCU(Eqs. (a1) to (a7)) except Eq. (a3). Eq. (a3) should be replaced to account for the single vent. The output of the B-NCU at timetis calculated by

(a8)

whereis as defined in Eq. (a4), and

(a9)

where is the output of the single vent at time t, and CSV is the coefficient of the single vent t. In the B-NCU, the single vent generates an additional output of the NCU. The ingredients of the output between the A-NCU and the B-NCU are different.

Another slightly more complicated NCU is the type-C NCU (hereinafter referred to as the C-NCU). As illustrated in Fig. A-3, in addition to the three basic components of the A-NCU, the C-NCU possesses an intermediate single vent. Due to the intermediate single vent, in addition to the set of the basic operations of an NCU (Eqs. (a1) to (a7)), the following equation is added to the operations of the C-NCU:

(a10)

where is the output of the intermediate single vent at time t, and CISV is the coefficient of the intermediate single vent. The intermediate single vent is another specialized composite vent whose sill is zero. But in the C-NCU, the output of the intermediate single vent is not a part of the output of the C-NCU. Like the A-NCU, the output of the C-NCU is the summation of the output generated by the composite vents.

The most complicated NCU is the type-D NCU (hereinafter referred to as the D-NCU). As shown in Fig. A-4, in addition to the three basic components of an NCU, the D-NCU possesses a single vent and an intermediate single vent. The operations of the D-NCU are the combination of the operations the B-NCU and the C-NCU, i.e. Eqs. (a1) to (a2) and Eqs. (a4) to (a10). Like the C-NCU, the output of the intermediate single vent is not a part of the output of the D-NCU either. Moreover, like the B-NCU, the output of the D-NCU is also constituted by the output generated by the composite vents and the single vent.

Restrictions on the parameters of the NCUs are

(a11)

(a12)

(a13)

(a14)

(a15)

where EIS, , , CSV and CISV have been defined earlier.

The absence and the presence of the single vent and the intermediated single vent within the four types of NCUs are fixed, but the number of the composite vents possessed by the NCUs is modifiable. The number of the composite vents possessed by the NCU is referred to as the configuration of an NCU. Modelers can choose the configuration of the NCU according to their requirements. For simplifying the notations of the four NCUs, a token that consists of a letter followed by an integer equal or larger than zero is used to symbolize an NCU. The letter represents the type of an NCU, and the integer indicates the configuration of an NCU. For examples, an A-NCU with 3 composite vents is denoted as {A3}; a B-NCU with 2 composite vents,{B2}; a C-NCU with 4 composite vents,{C4}; and a D-NCU with 1 composite vent,{D1}. The illustrations of the four types of NCUs are also simplified. As given in Fig. A-5, the tokens in the center of the circles (i.e., NCUs) are used to represent the types and the configurations of the NCUs. In the simplified illustrations (Fig. A-5), the composite vents and the single vents are omitted, but the other components of the NCUs are still drawn.

A2. Nonlinear computational units cascaded (NCUC) model

The NCUC model is constructed by a series of various NCUs. The way for connecting the NCUs is to cascade several NCUs in a specific order. In the NCUC model, the number of the cascaded NCUs can also be selected freely. Modelers can choose the number of the cascaded NCUs according to their requirements. An NCUC model that has r NCUs is shown in Fig. A-6 where the illustrations of the NCUs are all simplified. The operations of the NCUC model are defined by the following equations:

(a16)

(a17)

(a18)

(a19)

(a20)

where is the sill of the i-th NCUat time 0, is the elevation of the initial storage of the i-th NCU,is the output of the i-th NCU at time 0, is the output of the i-th NCU at time t, is the input of the i-th NCU at time t, is the output of the intermediate single vent of the i-th NCU at time t, is the output of the NCUC model at time t, is a function that indicates the type of the i-th NCU, and t, , and are as defined earlier.

The input of the first NCU is given by modelers. The outputs of the intermediate single vents are the inputs of the corresponding subsequent NCUs. The NCU without the intermediate single vent can also cascade another NCU whose input are always zero. At the beginning of modeling, the sills of the NCUs are initialized as the corresponding EISs and then updated continuously using Eq. (a19). The output of the NCUC model is the summation of all the output of all NCUs. The NCUC model is quite flexible to meet the modelers’ requirements. Modelers can freely choose the number of the cascaded NCUs and the configurations of the NCUs to build the NCUC model.

The configurations and the number of the NCUs cascaded in the NCUC model are referred to as the pattern of the NCUC model. For convenience, a more simplified notation to symbolize the pattern of an NCUC model is introduced herein. The tokens of the NCUs contained in an NCUC model are separated by commas and are concatenated sequentially to form a string. The string is used to denote the pattern of the NCUC model. For example, the string {D3, C2} indicates that the first NCU is {D3} and the second NCU is {C2} in an NCUC model. And, the string {C2, C2, A1} indicates that the first NCU is {C2}, the second NCU is {C2}, and the third NCU is {A1} in an NCUC model.

Figure A-1The architecture of the type-A NCU.

Figure A-2The architecture of the type-B NCU.

Figure A-3The architecture of the type-C NCU.

Figure A-4The architecture of the type-D NCU.

Figure A-5Simplified illustrations of the four types of NCUs: (a) A-NCU, (b) B-NCU, (c) C-NCU, and (d) D-NCU. The tokens in the centers of the circles, AN, BN, CN, and DN, are used to symbolize the corresponding NCUs. N is the configuration of the corresponding NCU.

Figure A-6An NCUC model with r NCUs.

A-1