You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
title: MakeGridDH (Fortran)
keywords: spherical harmonics software package, spherical harmonic transform, legendre functions, multitaper spectral analysis, fortran, Python, gravity, magnetic field
sidebar: fortran_sidebar
permalink: makegriddh.html
summary:
tags: [fortran]
toc: false
editdoc: fdoc
---

Create a 2D map from a set of spherical harmonic coefficients that conforms with Driscoll and Healy's (1994) sampling theorem.

## Usage

call MakeGridDH (`griddh`, `n`, `cilm`, `lmax`, `norm`, `sampling`, `csphase`, `lmax_calc`, `extend`, `exitstatus`)

## Parameters

`griddh` : output, real(dp), dimension (nlat, nlong)
:   A 2D map of the input spherical harmonic coefficients `cilm` that conforms to the sampling theorem of Driscoll and Healy (1994). If `sampling` is 1, the grid is equally sampled and is dimensioned as (`n` by `n`), where `n` is `2lmax+2`. If sampling is 2, the grid is equally spaced and is dimensioned as (`n` by 2`n`). The first latitudinal band of the grid corresponds to 90 N, the latitudinal sampling interval is 180/`n` degrees, and the default behavior is to exclude the latitudinal band for 90 S. The first longitudinal band of the grid is 0 E, by default the longitudinal band for 360 E is not included, and the longitudinal sampling interval is 360/`n` for an equally sampled and 180/`n` for an equally spaced grid, respectively. If `extend` is 1, the longitudinal band for 360 E and the latitudinal band for 90 S will be included, which increases each of the dimensions of the grid by 1.

`n` : output, integer
:   The number of samples of the grid, which is equal to `2lmax+2`.

`cilm` : input, real(dp), dimension (2, `lmax`+1, `lmax`+1)
:   The real spherical harmonic coefficients of the function. The coefficients `c1lm` and `c2lm` refer to the cosine (`Clm`) and sine (`Slm`) coefficients, respectively, with `Clm=cilm(1,l+1,m+1)` and `Slm=cilm(2,l+1,m+1)`.

`lmax` : input, integer
:   The maximum spherical harmonic degree of the function. This determines the number of samples `n`.

`norm` : input, optional, integer, default = 1
:   1 (default) = 4-pi (geodesy) normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.

`sampling` : input, optional, integer, default = 1
:   If 1 (default) the input grid is equally sampled (`n` by `n`). If 2, the grid is equally spaced (`n` by 2`n`).

`csphase` : input, optional, integer, default = 1
:   1 (default) = do not apply the Condon-Shortley phase factor to the associated Legendre functions; -1 = append the Condon-Shortley phase factor of (-1)^m to the associated Legendre functions.

`lmax_calc` : input, optional, integer, default = `lmax`
:   The maximum spherical harmonic degree used in evaluating the function. This must be less than or equal to `lmax`.

`extend` : input, optional, integer, default = 0
:   If 1, compute the longitudinal band for 360 E and the latitudinal band for 90 S. This increases each of the dimensions of `griddh` by 1.

`exitstatus` : output, optional, integer
:   If present, instead of executing a STOP when an error is encountered, the variable exitstatus will be returned describing the error. 0 = No errors; 1 = Improper dimensions of input array; 2 = Improper bounds for input variable; 3 = Error allocating memory; 4 = File IO error.

## Description

`MakeGridDH` will create a 2-dimensional map equally sampled (`n` by `n`) or equally spaced (`n` by 2`n`) in latitude and longitude from a set of input spherical harmonic coefficients, where `n=2lmax+2`. This grid conforms with the sampling theorem of Driscoll and Healy (1994) and this routine is the inverse of `SHExpandDH`. The function is evaluated at each longitudinal band by inverse Fourier transforming the sin and cos terms for each degree `l`, and then summing over all degrees. When evaluating the function, the maximum spherical harmonic degree that is considered is the minimum of `lmax`, the size of `cilm-1`, or `lmax_calc` (if specified).

The default is to use an input grid that is equally sampled (`n` by `n`), but this can be changed to use an equally spaced grid (`n` by 2`n`) by the optional argument `sampling`. The redundant longitudinal band for 360 E and the latitudinal band for 90 S are excluded by default, but these can be computed by specifying the optional argument `extend`. The employed spherical harmonic normalization and Condon-Shortley phase convention can be set by the optional arguments `norm` and `csphase`; if not set, the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-Shortley phase of (-1)^m.

The normalized legendre functions are calculated using the scaling algorithm of Holmes and Featherstone (2002), which are accurate to about degree 2800. The unnormalized functions are accurate only to about degree 15.

## References

Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.

Holmes, S. A., and W. E. Featherstone, A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions, J. Geodesy, 76, 279-299, 2002.

## See also

[shexpanddh](shexpanddh.html), [makegriddhc](makegriddhc.html), [shexpanddhc](shexpanddhc.html), [makegridglq](makegridglq.html), [shexpandglq](shexpandglq.html), [makegridglqc](makegridglqc.html), [shexpandglqc](shexpandglqc.html), [shexpandlsq](shexpandlsq.html)